introduction to modern physics phyx 2710 · “probability cheat sheet” ... introduction to...
TRANSCRIPT
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 1
Lecture 3 Slide 1
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Lecture 3
Distribution Functions
References Taylor Ch 5 (and Chs 10 and 11 for Reference)
Taylor Ch 6 and 7
Also refer to ldquoGlossary of Important Terms in Error Analysisrdquo
ldquoProbability Cheat Sheetrdquo
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 2
Lecture 3 Slide 2
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Distribution Functions
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 3
Lecture 3 Slide 3
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Practical Methods to Calculate Mean and St Deviation
We need to develop a good way to tally display and think about
a collection of repeated measurements of the same quantity
Here is where we are headed bull Develop the notion of a probability distribution function a distribution to
describe the probable outcomes of a measurement
bull Define what a distribution function is and its properties
bull Look at the properties of the most common distribution function the Gaussian
distribution for purely random events
bull Introduce other probability distribution functions
We will develop the mathematical basis for bull Mean
bull Standard deviation
bull Standard deviation of the mean (SDOM)
bull Moments and expectation values
bull Error propagation formulas
bull Addition of errors in quadrature (for independent and random measurements)
bull Schwartz inequality (ie the uncertainty principle) (next lecture)
bull Numerical values for confidence limits (t-test)
bull Principle of maximal likelihood
bull Central limit theorem
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 4
Lecture 3 Slide 4
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Two Practical Exercises in Probabilities
Roll a pair of dice 50 times and record the results
Flip Penny 50 times and record the results Flip a penny 50 times and record the results
Grab a partner and a
set of instructions
and complete the
exercise
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 5
Lecture 3 Slide 5
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Flip a penny 50 times and record the results
Two Practical Exercises in Probabilities
What is the asymmetry of the results
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 6
Lecture 3 Slide 6
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Two Practical Exercises in Probabilities
Flip a penny 50 times and record the results
What is the asymmetry of the results
4 asymmetry asymmetry
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 7
Lecture 3 Slide 7
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Two Practical Exercises in Probabilities
Roll a pair of dice 50 times and record the results
What is the mean value
The standard deviation
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 8
Lecture 3 Slide 8
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Two Practical Exercises in Probabilities
Roll a pair of dice 50 times and record the results
What is the mean value
The standard deviation
What is the asymmetry
(kurtosis)
What is the probability of
rolling a 4
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 9
Lecture 3 Slide 9
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Discrete Distribution Functions
A data set to play with Written in terms of ldquooccurrencerdquo F
The mean value In terms of fractional expectations
Fractional expectations
Normalization condition
Mean value
(This is just a weighted sum)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 10
Lecture 3 Slide 10
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Limit of Discrete Distribution Functions
ldquoNormalizingrdquo data sets
Binned data sets
fk equiv fractional occurrence
∆k equiv bin width
Mean value
Normalization
Expected value 119875119903119900119887 4 =1198654
119873= 1198914
119883 = 119865119896
119896
119909119896 = (119891119896119896
Δ119896)119909119896
1 = (119891119896119896 Δ119896)119909119896
Δ119896119896
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 11
Lecture 3 Slide 11
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Limits of Distribution Functions
Consider the limiting distribution function as N ȸ and ∆k0
Larger data sets
Mathcad Games
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 12
Lecture 3 Slide 12
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Continuous Distribution Functions
Central (mean) Value Width of Distribution
Meaning of Distribution Interval
Normalization of Distribution
Thus
and by extension
= fraction of measurements
that fall within altxltb
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 13
Lecture 3 Slide 13
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Moments of Distribution Functions
The first moment for a probability distribution function is
119909 equiv 119909 = 119891119894119903119904119905 119898119900119898119890119899119905 = 119909 119891(119909)+infin
minusinfin119889119909
For a general distribution function
119909 equiv 119909 = 119891119894119903119904119905 119898119900119898119890119899119905 = 119909 119892(119909)
+infinminusinfin
119889119909
119892(119909)+infinminusinfin
119889119909
Generalizing the n
th moment is
119909119899 equiv 119909119899 = 119899119905ℎ 119898119900119898119890119899119905 = 119909119899 119892(119909)
+infinminusinfin
119889119909
119892(119909)+infinminusinfin
119889119909= 119909119899 119891(119909)
+infin
minusinfin119889119909
Oth moment equiv N 2
nd moment equiv 119909 minus 119909 2 rarr 1199092
1st moment equiv 119909 3
rd moment equiv kurtosis
0 (for a centered distribution)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 14
Lecture 3 Slide 14
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Moments of Distribution Functions
Generalizing the nth
moment is
119909119899 equiv 119909119899 = 119899119905ℎ 119898119900119898119890119899119905 = 119909119899 119892(119909)
+infinminusinfin
119889119909
119892(119909)+infinminusinfin
119889119909= 119909119899 119891(119909)
+infin
minusinfin119889119909
Oth
moment equiv N 2nd
moment equiv 119909 minus 119909 2 rarr 1199092
1st moment equiv 119909 3
rd moment equiv kurtosis
The nth
moment about the mean is
120583119899 equiv 119909 minus 119909 119899 = 119899119905ℎ 119898119900119898119890119899119905 119886119887119900119906119905 119905ℎ119890 119898119890119886119899
= 119909minus119909 119899 119892(119909)
+infinminusinfin
119889119909
119892(119909)+infinminusinfin
119889119909= 119909 minus 119909 119899 119891(119909)
+infin
minusinfin119889119909
The standard deviation (or second moment about the mean) is
1205901199092 equiv 1205832 equiv 119909 minus 119909 2 = 2119899119889 119898119900119898119890119899119905 119886119887119900119906119905 119905ℎ119890 119898119890119886119899
= 119909minus119909 2 119892(119909)
+infinminusinfin
119889119909
119892(119909)+infinminusinfin
119889119909= 119909 minus 119909 2 119891(119909)
+infin
minusinfin119889119909
0
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 15
Lecture 3 Slide 15
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Example of Continuous Distribution Functions and Expectation Values Harmonic Oscillator Example from Mechanics
Expected Values
The expectation value of a function Ɍ(x) is
Ɍ x equiv Ɍ x 119892 119909
+infin
minusinfin119889119909
119892 119909 +infin
minusinfin119889119909
= Ɍ(x) 119891(119909)+infin
minusinfin119889119909
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 16
Lecture 3 Slide 16
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Example of Continuous Distribution Functions and Expectation Values Boltzmann Distribution Example from Kinetic Theory
Expected Values
The expectation value of a function
Ɍ(x) is
Ɍ x equiv Ɍ x 119892 119909
+infin
minusinfin119889119909
119892 119909 +infin
minusinfin119889119909
= Ɍ(x) 119891(119909)+infin
minusinfin119889119909
The Boltzmann distribution function for velocities of particles as
a function of temperature T is
119875 119907 119879 = 4120587 119872
2120587 119896119861119879
3 2
1199072119890119909119901
121198721199072
12119896119861119879
Then
v = 119907 119875(119907)+infin
minusinfin119889119907 =
8 119896119861119879120587 119872
1 2
v2 = v2 119875(119907)+infin
minusinfin119889119907 =
3 119896119861119879 119872
1 2
implies KE = 1
2119872 v2 =
3
2119896119861119879
vpeak=
2 119896119861119879 119872
1 2
= 2 3
1 2
v2
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 17
Lecture 3 Slide 17
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Example of Continuous Distribution Functions and Expectation Values Fermi-Dirac Distribution Example from Kinetic Theory
For a system of identical fermions the average number of fermions in a single-particle state is given
by the FermindashDirac (FndashD) distribution
where kB is Boltzmanns constant T is the absolute temperature is the energy of the single-particle
state and μ is the total chemical potential
Since the FndashD distribution was derived using the Pauli exclusion principle which allows at most one
electron to occupy each possible state a result is that
When a quasi-continuum of energies has an associated density of states (ie the number of
states per unit energy range per unit volume) the average number of fermions per unit energy range
per unit volume is
where is called the Fermi function
so that
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 18
Lecture 3 Slide 18
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Example of Continuous Distribution Functions and Expectation Values Finite Square Well Example from Quantum Mechanics
Expectation Values The expectation value of a QM operator O(x) is O x equiv Ψlowast x 119874 119909
+infin
minusinfinΨ x 119889119909
Ψlowast x Ψ x +infin
minusinfin119889119909
For a finite square well of width L Ψn x = 2L sin
n π x
L
Ψnlowast x |Ψn x equiv
Ψnlowast x 119874 119909
+infin
minusinfinΨn x 119889119909
Ψnlowast x Ψn x
+infin
minusinfin119889119909
= 1
119909 = Ψnlowast x |x|Ψn x equiv
Ψnlowast x 119909
+infin
minusinfinΨn x 119889119909
Ψnlowast x Ψn x
+infin
minusinfin119889119909
= 1198712
119901 = Ψnlowast x |
ℏ
i
part
partx|Ψn x equiv
Ψnlowast x
ℏi
partpartx
+infin
minusinfinΨn x 119889119909
Ψnlowast x Ψn x
+infin
minusinfin119889119909
= 0
119864119899 = Ψnlowast x |iℏ
part
partt|Ψn x equiv
Ψnlowast x iℏ
partpartt
+infin
minusinfinΨn x 119889119909
Ψnlowast x Ψn x
+infin
minusinfin119889119909
=11989921205872ℏ2
2 119898 1198712
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 19
Lecture 3 Slide 19
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Distribution Functions
Available
on web site
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 20
Lecture 3 Slide 20
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
The Gaussian Distribution
Function
References Taylor Ch 5
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 21
Lecture 3 Slide 21
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Gaussian Integrals
Factorial Approximations
119899 asymp 2120587119899 1 2 119899119899 119890119909119901 minus119899 + 1
12 119899+ 119874
1
1198992
119897119900119892 119899 asymp 1
2 119897119900119892 2120587 + 119899 +
1
2 log 119899 minus 119899 +
1
12 119899+ 119874
1
1198992
119897119900119892 119899 asymp 119899 log 119899 minus 119899 (for all terms decreasing faster than linearly with n)
Gaussian Integrals
119868119898 = 2 119909119898 exp(minus1199092)infin
0 119889119909 mgt-1
119868119898 = 2 119910119899 exp(minus119910)infin
0 119889119910 equiv Γ(119899 + 1) 1199092 equiv 119910 2 119889119909 = 1199101 2 119889119910 119899 equiv 1
2 (119898 minus 1)
1198680 = Γ 119899 =1
2 = 120587 m=0 119899 = minus1
2
1198682 119896 = Γ 119896 +1
2 = 119896 minus 1
2 119896 minus 32 3
2 12 120587 even m m=2 kgt0 119899 = 119896 minus 1
2
1198682 119896+1 = Γ 119896 + 1 = 119896 odd m m=2 k+1gt0 119899 = 119896 ge 0
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 22
Lecture 3 Slide 22
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Gaussian Distribution Function
Width of Distribution
(standard deviation)
Center of Distribution
(mean) Distribution
Function
Independent
Variable
Gaussian Distribution Function
Normalization
Constant
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 23
Lecture 3 Slide 23
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Effects of Increasing N on Gaussian Distribution Functions
Consider the limiting distribution function as N infin and dx0
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 24
Lecture 3 Slide 24
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Defining the Gaussian
distribution function
in Mathcad
I suggest you ldquoinvestigaterdquo
these with the Mathcad sheet
on the web site
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 25
Lecture 3 Slide 25
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Using Mathcad to
define other common
distribution functions
Consider the limiting
distribution function
as N ȸ and ∆k0
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 26
Lecture 3 Slide 26
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider the Gaussian distribution function
Use the normalization condition to
evaluate the normalization constant
(see Taylor p 132)
The mean Ẋ is the first moment
of the Gaussian distribution
function (see Taylor p 134)
The standard deviation σx is the standard
deviation of the mean of the Gaussian
distribution function (see Taylor p 143)
Gaussian Distribution Moments
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 27
Lecture 3 Slide 27
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
When is mean x not Xbest
Answer When the
distribution is not
symmetric about X
Example Cauchy
Distribution
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 28
Lecture 3 Slide 28
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
When is mean x not Xbest
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 29
Lecture 3 Slide 29
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
When is mean x not Xbest
Answer When the
distribution is has
more than one peak
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 30
Lecture 3 Slide 30
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
The Gaussian Distribution
Function and Its Relation to
Errors
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 31
Lecture 3 Slide 31
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
A Review of Probabilities in Combination
Probability of a data set of N like measurements (x1x2hellipxN)
P (x1x2hellipxN) = P(x)1P(x2)hellipP(xN)
1 head AND 1 Four
P(H4) = P(H) P(4)
1 head OR 1 Four
P(H4) = P(H) + P(4) - P(H and 4) (true for a ldquonon-mutually exclusiverdquo events)
NOT 1 Six
P(NOT 6) = 1 - P(6)
1 Six OR 1 Four
P(64) = P(6) + P(4) (true for a ldquomutually exclusiverdquo single role)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 32
Lecture 3 Slide 32
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
The Gaussian Distribution Function and Its Relation to Errors
We will use the Gaussian distribution as applied to random
variables to develop the mathematical basis for bull Mean
bull Standard deviation
bull Standard deviation of the mean (SDOM)
bull Moments and expectation values
bull Error propagation formulas
bull Addition of errors in quadrature (for independent and random
measurements)
bull Numerical values for confidence limits (t-test)
bull Principle of maximal likelihood
bull Central limit theorem
bull Weighted distributions and Chi squared
bull Schwartz inequality (ie the uncertainty principle) (next lecture)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 33
Lecture 3 Slide 33
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider the Gaussian distribution function
Use the normalization condition to
evaluate the normalization constant
(see Taylor p 132)
The mean Ẋ is the first moment
of the Gaussian distribution
function (see Taylor p 134)
The standard deviation σx is the standard
deviation of the mean of the Gaussian
distribution function (see Taylor p 143)
Gaussian Distribution Moments
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 34
Lecture 3 Slide 34
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Standard Deviation of Gaussian Distribution
Area under curve
(probability that
ndashσltxlt+σ) is 68
5 or ~2σ ldquoSignificantrdquo
1 or ~3σ ldquoHighly Significantrdquo
1σ ldquoWithin errorsrdquo
5 ppm or ~5σ ldquoValid for HEPrdquo
See Sec 106 Testing of
Hypotheses
More complete Table in App A and B
Ah thatrsquos
highly
significant
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 35
Lecture 3 Slide 35
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Function of Gaussian Distribution
Area under curve
(probability that
ndashtσltxlt+tσ) is given
in Table at right
Probable Error (probability that
ndash067σltxlt+067σ) is 50
Error Function (probability that ndashtσltxlt+tσ )
Error Function Tabulated values-see App A
Complementary Error Function
(probability that ndashxlt-tσ AND xgt+tσ )
Prob(x outside tσ) = 1 - Prob(x within tσ)
More complete Table in App A and B
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 36
Lecture 3 Slide 36
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Useful Points on Gaussian Distribution
Full Width at Half Maximum FWHM
(See Prob 512)
Points of Inflection Occur at plusmnσ
(See Prob 513)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 37
Lecture 3 Slide 37
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Analysis and Gaussian Distribution
Adding a Constant Multiplying by a Constant
XX+A XBX and σB σ
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 38
Lecture 3 Slide 38
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Sum of Two Variables
Consider the derived quantity
Z=X + Y
(with X=0 and Y=0)
Error Propagation Addition
Error in Z
Multiple two probabilities 119875119903119900119887 119909 119910 = 119875119903119900119887 119909 ∙ 119875119903119900119887 119909 prop 119890119909119901 minus
1
2
1199092
1205901199092+
1199102
1205901199102
prop 119890119909119901 minus1
2
(119909+119910)2
1205901199092+120590119910
2 minus1199112
prop 119890119909119901 minus1
2
(119909+119910)2
1205901199092+120590119910
2 119890119909119901 minus
1199112
2
ICBST (Eq 553)
X+YZ and σx2 + σy
2 σz
2
(addition in quadrature for random independent variables)
Integrates to radic2π
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 39
Lecture 3 Slide 39
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Error Propagation
General formula for error propagation see [Taylor Secs 35 and 39]
Uncertainty as a function of one variable [Taylor Sec 35]
1 Consider a graphical method of estimating error
a) Consider an arbitaray function q(x)
b) Plot q(x) vs x
c) On the graph label
(1) qbest = q(xbest)
(2) qhi = q(xbest + δx)
(3) qlow = q(xbest- δx)
d) Making a linear approximation
x
qqxslopeqq
x
qqxslopeqq
bestbestlow
bestbesthi
e) Therefore
xx
Note the absolute value
How do we determine the error of a derived quantity Z(XYhellip) from errors in XYhellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 40
Lecture 3 Slide 40
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Error Propagation
General formula for uncertainty of a function of one variable
xx
[Taylor Eq 323]
Can you now derive for specific rules of error propagation
1 Addition and Subtraction [Taylor p 49]
2 Multiplication and Division [Taylor p 51]
3 Multiplication by a constant (exact number) [Taylor p 54]
4 Exponentiation (powers) [Taylor p 56]
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 41
Lecture 3 Slide 41
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Multiple Variables
Uncertainty of a function of multiple variables [Taylor Sec 311]
1 It can easily (no really) be shown that (see Taylor Sec 311) for a
function of several variables
)(
z
z
qy
y
qx
x
qzyxq
[Taylor Eq 347]
2 More correctly it can be shown that (see Taylor Sec 311) for a
function of several variables
)(
z
z
qy
y
qx
x
qzyxq
[Taylor Eq 347]
where the equals sign represents an upper bound as discussed above
3 For a function of several independent and random variables
)(
222
z
z
qy
y
qx
x
qzyxq
[Taylor Eq 348]
Again the proof is left for Ch 5
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 42
Lecture 3 Slide 42
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Product of Two Variables
Consider the arbitrary derived quantity
q(xy) of two independent random variables x and y
Expand q(xy) in a Taylor series about the expected values of x and y
(ie at points near X and Y)
Error Propagation General Case
119902 119909 119910 = 119902 119883 119884 + 120597119902
120597119909
119883 119909 minus 119883 +
120597119902
120597119910
119884
(119910 minus 119884)
Fixed shifts peak of distribution
Fixed Distribution centered at X with width σX
120575119902 119909 119910 = 120590119902 = 119902 119883 119884 + 120597119902
120597119909
119883120590119909
2
+ 120597119902
120597119910
119884
120590119910
2
0
How do we determine the error of a derived quantity Z(XYhellip) from errors in XYhellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 43
Lecture 3 Slide 43
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
SDOM of Gaussian Distribution
Standard Deviation of the Mean
The SDOM decreases as the square root of the number of measurements
That is the relative width σẊ of the distribution gets narrower as more
measurements are made
Each measurement
has similar σXi= σẊ
and similar partial
derivatives
Thushellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 44
Lecture 3 Slide 44
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Two Key Theorems from Probability
Central Limit Theorem
For random independent measurements (each with a well-define
expectation value and well-defined variance) the arithmetic mean
(average) will be approximately normally distributed
Principle of Maximum Likelihood
Given the N observed measurements x1 x2hellipxN the best estimates
for Ẋ and σ are those values for which the observed x1 x2hellipxN are
most likely
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 45
Lecture 3 Slide 45
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Mean of Gaussian Distribution as ldquoBest Estimaterdquo
Principle of Maximum Likelihood
Best Estimate of X is from maximum probability or minimum summation
Consider a data set
Each randomly distributed with
x1 x2 x3 hellipxN
To find the most likely value of the mean (the best estimate of ẋ)
find X that yields the highest probability for the data set
The combined probability for the full data set is the product
Minimize
Sum
Solve for
derivative wrst
X set to 0
Best
estimate
of X
prop1
120590119890minus(1199091minus119883)2 2120590 times
1
120590119890minus(1199092minus119883)2 2120590 times hellip times
1
120590119890minus(119909119873minus119883)2 2120590 =
1
120590119873119890 minus(119909119894minus119883)2 2120590
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 46
Lecture 3 Slide 46
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Uncertainty of ldquoBest Estimatesrdquo of Gaussian Distribution Principle of Maximum Likelihood
Best Estimate of X is from maximum probability or minimum summation
Consider a data set x1 x2 x3 hellipxN
To find the most likely value of the standard deviation (the best estimate of
the width of the x distribution) find σx that yields the highest probability for
the data set
The combined probability for the full data set is the product
Minimize
Sum
Solve for
derivative
wrst X set to 0
Best
estimate
of X
Best Estimate of σ is from maximum probability or minimum summation
Minimize
Sum
Best
estimate
of σ
Solve for
derivative
wrst σ set to 0
See
Prob 526
prop1
120590119890minus(1199091minus119883)2 2120590 times
1
120590119890minus(1199092minus119883)2 2120590 times hellip times
1
120590119890minus(119909119873minus119883)2 2120590 =
1
120590119873119890 minus(119909119894minus119883)2 2120590
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 47
Lecture 3 Slide 47
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Combining Data Sets
Weighted Averages
References Taylor Ch 7
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 48
Lecture 3 Slide 48
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Weighted Averages
Question How can we properly combine two or more separate
independent measurements of the same randomly distributed
quantity to determine a best combined value with uncertainty
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 49
Lecture 3 Slide 49
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider two measurements of the same quantity described by a random Gaussian distribution
ltx1gt x1 and ltx2gt x2
The probability of measuring two such measurements is
119875119903119900119887119909 1199091 1199092 = 119875119903119900119887119909 1199091 119875119903119900119887119909 1199092
= 1
12059011205902119890minus12059422 119908ℎ119890119903119890 1205942 equiv
1199091 minus 119883
1205901 2 +
1199092 minus 119883
1205902 2
To find the best value for χ find the maximum Prob or minimum χ2
Weighted Averages
Assume negligible systematic errors
Note χ2 or Chi squared is the sum of the squares of the deviations from the mean divided by
the corresponding uncertainty
Such methods are called ldquoMethods of Least Squaresrdquo They follow directly from the Principle of
Maximum Likelihood
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 50
Lecture 3 Slide 50
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
The probability of measuring two such measurements is
119875119903119900119887119909 1199091 1199092 = 119875119903119900119887119909 1199091 119875119903119900119887119909 1199092
= 1
12059011205902119890minus12059422 119908ℎ119890119903119890 1205942 equiv
1199091 minus 119883
1205901 2 +
1199092 minus 119883
1205902 2
To find the best value for χ find the maximum Prob or minimum χ2
Weighted Averages
This leads to 119909119882_119886119907119892 = 11990811199091+11990821199092
1199081+1199082=
119908119894 119909119894
119908119894 119908ℎ119890119903119890 119908119894 = 1
120590119894 2
Best Estimate of χ is from maximum probibility or minimum summation
Minimize Sum Solve for best estimate of χ Solve for derivative wrst χ set to 0
Note If w1=w2 we recover the standard result Xwavg= (12) (x1+x2)
Finally the width of a weighted average distribution is
120590119908119894119890119892 ℎ119905119890119889 119886119907119892 = 1
119908119894119894
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 51
Lecture 3 Slide 51
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Weighted Averages on Steroids
A very powerful method for combining data from different sources
with different methods and uncertainties (or indeed data of
related measured and calculated quantities) is Kalman filtering
The Kalman filter also known as
linear quadratic estimation (LQE)
is an algorithm that uses a series
of measurements observed over
time containing noise (random
variations) and other
inaccuracies and produces
estimates of unknown variables
that tend to be more precise than
those based on a single
measurement alone
The Kalman filter keeps track of the estimated state of the system and the variance or uncertainty of the
estimate The estimate is updated using a state transition model and measurements xk|k-1 denotes the
estimate of the systems state at time step k before the kth measurement yk has been taken into account Pk|k-1
is the corresponding uncertainty --Wikipedia 2013
Ludger Scherliess of USU Physics is a world expert at using Kalman filtering for the assimilation
of satellite and ground-based data and the USU GAMES model to predict space weather
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 52
Lecture 3 Slide 52
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Rejecting Data
Chauvenetrsquos Criterion
References Taylor Ch 6
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 53
Lecture 3 Slide 53
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly
ldquounreasonablerdquo data point from a set of randomly distributed
measurements
bull Never
bull Whenever it makes things look better
bull Chauvenetrsquos criterion provides a (quantitative) compromise
Rejecting Data
What is a good criteria for
rejecting data
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 54
Lecture 3 Slide 54
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly
ldquounreasonablerdquo data point from a set of randomly distributed
measurements
bull Never
Rejecting Data
Zallenrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 55
Lecture 3 Slide 55
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly
ldquounreasonablerdquo data point from a set of randomly distributed
measurements
bull Whenever it makes things look better
Rejecting Data
Disneyrsquos Criterion
Disneyrsquos First Law
Wishing will make it so
Disneyrsquos Second Law
Dreams are more colorful than reality
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 56
Lecture 3 Slide 56
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Data may be rejected if the expected number of measurements at
least as deviant as the suspect measurement is less than 50
Consider a set of N measurements of a single quantity
x1x2 helliphellipxN
Calculate ltxgt and σx and then determine the fractional deviations from the mean of all
the points
For the suspect point(s) xsuspect find the probability of such a point occurring in N
measurements
n = (expected number as deviant as xsuspect)
= N Prob(outside xsuspectσx)
Rejecting Data
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 57
Lecture 3 Slide 57
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Function of Gaussian Distribution
Area under curve
(probability that
ndashtσltxlt+tσ) is given
in Table at right
Probable Error (probability that
ndash067σltxlt+067σ) is 50
Error Function (probability that ndashtσltxlt+tσ )
Error Function Tabulated values-see App A
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 58
Lecture 3 Slide 58
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 59
Lecture 3 Slide 59
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashExample 1
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 60
Lecture 3 Slide 60
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Details (1)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 61
Lecture 3 Slide 61
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashDetails (2)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 62
Lecture 3 Slide 62
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos
Criterionmdash
Details (3)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 63
Lecture 3 Slide 63
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos
Criterionmdash
Example 2
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 64
Lecture 3 Slide 64
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Summary of Probability Theory
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 65
Lecture 3 Slide 65
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of
Probability
Theory-I
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 66
Lecture 3 Slide 66
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of
Probability
Theory-II
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 67
Lecture 3 Slide 67
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-III
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 68
Lecture 3 Slide 68
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-IV
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 2
Lecture 3 Slide 2
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Distribution Functions
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 3
Lecture 3 Slide 3
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Practical Methods to Calculate Mean and St Deviation
We need to develop a good way to tally display and think about
a collection of repeated measurements of the same quantity
Here is where we are headed bull Develop the notion of a probability distribution function a distribution to
describe the probable outcomes of a measurement
bull Define what a distribution function is and its properties
bull Look at the properties of the most common distribution function the Gaussian
distribution for purely random events
bull Introduce other probability distribution functions
We will develop the mathematical basis for bull Mean
bull Standard deviation
bull Standard deviation of the mean (SDOM)
bull Moments and expectation values
bull Error propagation formulas
bull Addition of errors in quadrature (for independent and random measurements)
bull Schwartz inequality (ie the uncertainty principle) (next lecture)
bull Numerical values for confidence limits (t-test)
bull Principle of maximal likelihood
bull Central limit theorem
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 4
Lecture 3 Slide 4
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Two Practical Exercises in Probabilities
Roll a pair of dice 50 times and record the results
Flip Penny 50 times and record the results Flip a penny 50 times and record the results
Grab a partner and a
set of instructions
and complete the
exercise
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 5
Lecture 3 Slide 5
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Flip a penny 50 times and record the results
Two Practical Exercises in Probabilities
What is the asymmetry of the results
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 6
Lecture 3 Slide 6
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Two Practical Exercises in Probabilities
Flip a penny 50 times and record the results
What is the asymmetry of the results
4 asymmetry asymmetry
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 7
Lecture 3 Slide 7
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Two Practical Exercises in Probabilities
Roll a pair of dice 50 times and record the results
What is the mean value
The standard deviation
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 8
Lecture 3 Slide 8
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Two Practical Exercises in Probabilities
Roll a pair of dice 50 times and record the results
What is the mean value
The standard deviation
What is the asymmetry
(kurtosis)
What is the probability of
rolling a 4
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 9
Lecture 3 Slide 9
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Discrete Distribution Functions
A data set to play with Written in terms of ldquooccurrencerdquo F
The mean value In terms of fractional expectations
Fractional expectations
Normalization condition
Mean value
(This is just a weighted sum)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 10
Lecture 3 Slide 10
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Limit of Discrete Distribution Functions
ldquoNormalizingrdquo data sets
Binned data sets
fk equiv fractional occurrence
∆k equiv bin width
Mean value
Normalization
Expected value 119875119903119900119887 4 =1198654
119873= 1198914
119883 = 119865119896
119896
119909119896 = (119891119896119896
Δ119896)119909119896
1 = (119891119896119896 Δ119896)119909119896
Δ119896119896
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 11
Lecture 3 Slide 11
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Limits of Distribution Functions
Consider the limiting distribution function as N ȸ and ∆k0
Larger data sets
Mathcad Games
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 12
Lecture 3 Slide 12
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Continuous Distribution Functions
Central (mean) Value Width of Distribution
Meaning of Distribution Interval
Normalization of Distribution
Thus
and by extension
= fraction of measurements
that fall within altxltb
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 13
Lecture 3 Slide 13
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Moments of Distribution Functions
The first moment for a probability distribution function is
119909 equiv 119909 = 119891119894119903119904119905 119898119900119898119890119899119905 = 119909 119891(119909)+infin
minusinfin119889119909
For a general distribution function
119909 equiv 119909 = 119891119894119903119904119905 119898119900119898119890119899119905 = 119909 119892(119909)
+infinminusinfin
119889119909
119892(119909)+infinminusinfin
119889119909
Generalizing the n
th moment is
119909119899 equiv 119909119899 = 119899119905ℎ 119898119900119898119890119899119905 = 119909119899 119892(119909)
+infinminusinfin
119889119909
119892(119909)+infinminusinfin
119889119909= 119909119899 119891(119909)
+infin
minusinfin119889119909
Oth moment equiv N 2
nd moment equiv 119909 minus 119909 2 rarr 1199092
1st moment equiv 119909 3
rd moment equiv kurtosis
0 (for a centered distribution)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 14
Lecture 3 Slide 14
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Moments of Distribution Functions
Generalizing the nth
moment is
119909119899 equiv 119909119899 = 119899119905ℎ 119898119900119898119890119899119905 = 119909119899 119892(119909)
+infinminusinfin
119889119909
119892(119909)+infinminusinfin
119889119909= 119909119899 119891(119909)
+infin
minusinfin119889119909
Oth
moment equiv N 2nd
moment equiv 119909 minus 119909 2 rarr 1199092
1st moment equiv 119909 3
rd moment equiv kurtosis
The nth
moment about the mean is
120583119899 equiv 119909 minus 119909 119899 = 119899119905ℎ 119898119900119898119890119899119905 119886119887119900119906119905 119905ℎ119890 119898119890119886119899
= 119909minus119909 119899 119892(119909)
+infinminusinfin
119889119909
119892(119909)+infinminusinfin
119889119909= 119909 minus 119909 119899 119891(119909)
+infin
minusinfin119889119909
The standard deviation (or second moment about the mean) is
1205901199092 equiv 1205832 equiv 119909 minus 119909 2 = 2119899119889 119898119900119898119890119899119905 119886119887119900119906119905 119905ℎ119890 119898119890119886119899
= 119909minus119909 2 119892(119909)
+infinminusinfin
119889119909
119892(119909)+infinminusinfin
119889119909= 119909 minus 119909 2 119891(119909)
+infin
minusinfin119889119909
0
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 15
Lecture 3 Slide 15
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Example of Continuous Distribution Functions and Expectation Values Harmonic Oscillator Example from Mechanics
Expected Values
The expectation value of a function Ɍ(x) is
Ɍ x equiv Ɍ x 119892 119909
+infin
minusinfin119889119909
119892 119909 +infin
minusinfin119889119909
= Ɍ(x) 119891(119909)+infin
minusinfin119889119909
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 16
Lecture 3 Slide 16
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Example of Continuous Distribution Functions and Expectation Values Boltzmann Distribution Example from Kinetic Theory
Expected Values
The expectation value of a function
Ɍ(x) is
Ɍ x equiv Ɍ x 119892 119909
+infin
minusinfin119889119909
119892 119909 +infin
minusinfin119889119909
= Ɍ(x) 119891(119909)+infin
minusinfin119889119909
The Boltzmann distribution function for velocities of particles as
a function of temperature T is
119875 119907 119879 = 4120587 119872
2120587 119896119861119879
3 2
1199072119890119909119901
121198721199072
12119896119861119879
Then
v = 119907 119875(119907)+infin
minusinfin119889119907 =
8 119896119861119879120587 119872
1 2
v2 = v2 119875(119907)+infin
minusinfin119889119907 =
3 119896119861119879 119872
1 2
implies KE = 1
2119872 v2 =
3
2119896119861119879
vpeak=
2 119896119861119879 119872
1 2
= 2 3
1 2
v2
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 17
Lecture 3 Slide 17
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Example of Continuous Distribution Functions and Expectation Values Fermi-Dirac Distribution Example from Kinetic Theory
For a system of identical fermions the average number of fermions in a single-particle state is given
by the FermindashDirac (FndashD) distribution
where kB is Boltzmanns constant T is the absolute temperature is the energy of the single-particle
state and μ is the total chemical potential
Since the FndashD distribution was derived using the Pauli exclusion principle which allows at most one
electron to occupy each possible state a result is that
When a quasi-continuum of energies has an associated density of states (ie the number of
states per unit energy range per unit volume) the average number of fermions per unit energy range
per unit volume is
where is called the Fermi function
so that
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 18
Lecture 3 Slide 18
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Example of Continuous Distribution Functions and Expectation Values Finite Square Well Example from Quantum Mechanics
Expectation Values The expectation value of a QM operator O(x) is O x equiv Ψlowast x 119874 119909
+infin
minusinfinΨ x 119889119909
Ψlowast x Ψ x +infin
minusinfin119889119909
For a finite square well of width L Ψn x = 2L sin
n π x
L
Ψnlowast x |Ψn x equiv
Ψnlowast x 119874 119909
+infin
minusinfinΨn x 119889119909
Ψnlowast x Ψn x
+infin
minusinfin119889119909
= 1
119909 = Ψnlowast x |x|Ψn x equiv
Ψnlowast x 119909
+infin
minusinfinΨn x 119889119909
Ψnlowast x Ψn x
+infin
minusinfin119889119909
= 1198712
119901 = Ψnlowast x |
ℏ
i
part
partx|Ψn x equiv
Ψnlowast x
ℏi
partpartx
+infin
minusinfinΨn x 119889119909
Ψnlowast x Ψn x
+infin
minusinfin119889119909
= 0
119864119899 = Ψnlowast x |iℏ
part
partt|Ψn x equiv
Ψnlowast x iℏ
partpartt
+infin
minusinfinΨn x 119889119909
Ψnlowast x Ψn x
+infin
minusinfin119889119909
=11989921205872ℏ2
2 119898 1198712
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 19
Lecture 3 Slide 19
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Distribution Functions
Available
on web site
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 20
Lecture 3 Slide 20
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
The Gaussian Distribution
Function
References Taylor Ch 5
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 21
Lecture 3 Slide 21
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Gaussian Integrals
Factorial Approximations
119899 asymp 2120587119899 1 2 119899119899 119890119909119901 minus119899 + 1
12 119899+ 119874
1
1198992
119897119900119892 119899 asymp 1
2 119897119900119892 2120587 + 119899 +
1
2 log 119899 minus 119899 +
1
12 119899+ 119874
1
1198992
119897119900119892 119899 asymp 119899 log 119899 minus 119899 (for all terms decreasing faster than linearly with n)
Gaussian Integrals
119868119898 = 2 119909119898 exp(minus1199092)infin
0 119889119909 mgt-1
119868119898 = 2 119910119899 exp(minus119910)infin
0 119889119910 equiv Γ(119899 + 1) 1199092 equiv 119910 2 119889119909 = 1199101 2 119889119910 119899 equiv 1
2 (119898 minus 1)
1198680 = Γ 119899 =1
2 = 120587 m=0 119899 = minus1
2
1198682 119896 = Γ 119896 +1
2 = 119896 minus 1
2 119896 minus 32 3
2 12 120587 even m m=2 kgt0 119899 = 119896 minus 1
2
1198682 119896+1 = Γ 119896 + 1 = 119896 odd m m=2 k+1gt0 119899 = 119896 ge 0
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 22
Lecture 3 Slide 22
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Gaussian Distribution Function
Width of Distribution
(standard deviation)
Center of Distribution
(mean) Distribution
Function
Independent
Variable
Gaussian Distribution Function
Normalization
Constant
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 23
Lecture 3 Slide 23
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Effects of Increasing N on Gaussian Distribution Functions
Consider the limiting distribution function as N infin and dx0
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 24
Lecture 3 Slide 24
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Defining the Gaussian
distribution function
in Mathcad
I suggest you ldquoinvestigaterdquo
these with the Mathcad sheet
on the web site
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 25
Lecture 3 Slide 25
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Using Mathcad to
define other common
distribution functions
Consider the limiting
distribution function
as N ȸ and ∆k0
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 26
Lecture 3 Slide 26
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider the Gaussian distribution function
Use the normalization condition to
evaluate the normalization constant
(see Taylor p 132)
The mean Ẋ is the first moment
of the Gaussian distribution
function (see Taylor p 134)
The standard deviation σx is the standard
deviation of the mean of the Gaussian
distribution function (see Taylor p 143)
Gaussian Distribution Moments
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 27
Lecture 3 Slide 27
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
When is mean x not Xbest
Answer When the
distribution is not
symmetric about X
Example Cauchy
Distribution
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 28
Lecture 3 Slide 28
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
When is mean x not Xbest
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 29
Lecture 3 Slide 29
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
When is mean x not Xbest
Answer When the
distribution is has
more than one peak
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 30
Lecture 3 Slide 30
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
The Gaussian Distribution
Function and Its Relation to
Errors
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 31
Lecture 3 Slide 31
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
A Review of Probabilities in Combination
Probability of a data set of N like measurements (x1x2hellipxN)
P (x1x2hellipxN) = P(x)1P(x2)hellipP(xN)
1 head AND 1 Four
P(H4) = P(H) P(4)
1 head OR 1 Four
P(H4) = P(H) + P(4) - P(H and 4) (true for a ldquonon-mutually exclusiverdquo events)
NOT 1 Six
P(NOT 6) = 1 - P(6)
1 Six OR 1 Four
P(64) = P(6) + P(4) (true for a ldquomutually exclusiverdquo single role)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 32
Lecture 3 Slide 32
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
The Gaussian Distribution Function and Its Relation to Errors
We will use the Gaussian distribution as applied to random
variables to develop the mathematical basis for bull Mean
bull Standard deviation
bull Standard deviation of the mean (SDOM)
bull Moments and expectation values
bull Error propagation formulas
bull Addition of errors in quadrature (for independent and random
measurements)
bull Numerical values for confidence limits (t-test)
bull Principle of maximal likelihood
bull Central limit theorem
bull Weighted distributions and Chi squared
bull Schwartz inequality (ie the uncertainty principle) (next lecture)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 33
Lecture 3 Slide 33
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider the Gaussian distribution function
Use the normalization condition to
evaluate the normalization constant
(see Taylor p 132)
The mean Ẋ is the first moment
of the Gaussian distribution
function (see Taylor p 134)
The standard deviation σx is the standard
deviation of the mean of the Gaussian
distribution function (see Taylor p 143)
Gaussian Distribution Moments
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 34
Lecture 3 Slide 34
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Standard Deviation of Gaussian Distribution
Area under curve
(probability that
ndashσltxlt+σ) is 68
5 or ~2σ ldquoSignificantrdquo
1 or ~3σ ldquoHighly Significantrdquo
1σ ldquoWithin errorsrdquo
5 ppm or ~5σ ldquoValid for HEPrdquo
See Sec 106 Testing of
Hypotheses
More complete Table in App A and B
Ah thatrsquos
highly
significant
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 35
Lecture 3 Slide 35
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Function of Gaussian Distribution
Area under curve
(probability that
ndashtσltxlt+tσ) is given
in Table at right
Probable Error (probability that
ndash067σltxlt+067σ) is 50
Error Function (probability that ndashtσltxlt+tσ )
Error Function Tabulated values-see App A
Complementary Error Function
(probability that ndashxlt-tσ AND xgt+tσ )
Prob(x outside tσ) = 1 - Prob(x within tσ)
More complete Table in App A and B
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 36
Lecture 3 Slide 36
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Useful Points on Gaussian Distribution
Full Width at Half Maximum FWHM
(See Prob 512)
Points of Inflection Occur at plusmnσ
(See Prob 513)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 37
Lecture 3 Slide 37
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Analysis and Gaussian Distribution
Adding a Constant Multiplying by a Constant
XX+A XBX and σB σ
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 38
Lecture 3 Slide 38
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Sum of Two Variables
Consider the derived quantity
Z=X + Y
(with X=0 and Y=0)
Error Propagation Addition
Error in Z
Multiple two probabilities 119875119903119900119887 119909 119910 = 119875119903119900119887 119909 ∙ 119875119903119900119887 119909 prop 119890119909119901 minus
1
2
1199092
1205901199092+
1199102
1205901199102
prop 119890119909119901 minus1
2
(119909+119910)2
1205901199092+120590119910
2 minus1199112
prop 119890119909119901 minus1
2
(119909+119910)2
1205901199092+120590119910
2 119890119909119901 minus
1199112
2
ICBST (Eq 553)
X+YZ and σx2 + σy
2 σz
2
(addition in quadrature for random independent variables)
Integrates to radic2π
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 39
Lecture 3 Slide 39
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Error Propagation
General formula for error propagation see [Taylor Secs 35 and 39]
Uncertainty as a function of one variable [Taylor Sec 35]
1 Consider a graphical method of estimating error
a) Consider an arbitaray function q(x)
b) Plot q(x) vs x
c) On the graph label
(1) qbest = q(xbest)
(2) qhi = q(xbest + δx)
(3) qlow = q(xbest- δx)
d) Making a linear approximation
x
qqxslopeqq
x
qqxslopeqq
bestbestlow
bestbesthi
e) Therefore
xx
Note the absolute value
How do we determine the error of a derived quantity Z(XYhellip) from errors in XYhellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 40
Lecture 3 Slide 40
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Error Propagation
General formula for uncertainty of a function of one variable
xx
[Taylor Eq 323]
Can you now derive for specific rules of error propagation
1 Addition and Subtraction [Taylor p 49]
2 Multiplication and Division [Taylor p 51]
3 Multiplication by a constant (exact number) [Taylor p 54]
4 Exponentiation (powers) [Taylor p 56]
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 41
Lecture 3 Slide 41
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Multiple Variables
Uncertainty of a function of multiple variables [Taylor Sec 311]
1 It can easily (no really) be shown that (see Taylor Sec 311) for a
function of several variables
)(
z
z
qy
y
qx
x
qzyxq
[Taylor Eq 347]
2 More correctly it can be shown that (see Taylor Sec 311) for a
function of several variables
)(
z
z
qy
y
qx
x
qzyxq
[Taylor Eq 347]
where the equals sign represents an upper bound as discussed above
3 For a function of several independent and random variables
)(
222
z
z
qy
y
qx
x
qzyxq
[Taylor Eq 348]
Again the proof is left for Ch 5
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 42
Lecture 3 Slide 42
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Product of Two Variables
Consider the arbitrary derived quantity
q(xy) of two independent random variables x and y
Expand q(xy) in a Taylor series about the expected values of x and y
(ie at points near X and Y)
Error Propagation General Case
119902 119909 119910 = 119902 119883 119884 + 120597119902
120597119909
119883 119909 minus 119883 +
120597119902
120597119910
119884
(119910 minus 119884)
Fixed shifts peak of distribution
Fixed Distribution centered at X with width σX
120575119902 119909 119910 = 120590119902 = 119902 119883 119884 + 120597119902
120597119909
119883120590119909
2
+ 120597119902
120597119910
119884
120590119910
2
0
How do we determine the error of a derived quantity Z(XYhellip) from errors in XYhellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 43
Lecture 3 Slide 43
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
SDOM of Gaussian Distribution
Standard Deviation of the Mean
The SDOM decreases as the square root of the number of measurements
That is the relative width σẊ of the distribution gets narrower as more
measurements are made
Each measurement
has similar σXi= σẊ
and similar partial
derivatives
Thushellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 44
Lecture 3 Slide 44
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Two Key Theorems from Probability
Central Limit Theorem
For random independent measurements (each with a well-define
expectation value and well-defined variance) the arithmetic mean
(average) will be approximately normally distributed
Principle of Maximum Likelihood
Given the N observed measurements x1 x2hellipxN the best estimates
for Ẋ and σ are those values for which the observed x1 x2hellipxN are
most likely
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 45
Lecture 3 Slide 45
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Mean of Gaussian Distribution as ldquoBest Estimaterdquo
Principle of Maximum Likelihood
Best Estimate of X is from maximum probability or minimum summation
Consider a data set
Each randomly distributed with
x1 x2 x3 hellipxN
To find the most likely value of the mean (the best estimate of ẋ)
find X that yields the highest probability for the data set
The combined probability for the full data set is the product
Minimize
Sum
Solve for
derivative wrst
X set to 0
Best
estimate
of X
prop1
120590119890minus(1199091minus119883)2 2120590 times
1
120590119890minus(1199092minus119883)2 2120590 times hellip times
1
120590119890minus(119909119873minus119883)2 2120590 =
1
120590119873119890 minus(119909119894minus119883)2 2120590
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 46
Lecture 3 Slide 46
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Uncertainty of ldquoBest Estimatesrdquo of Gaussian Distribution Principle of Maximum Likelihood
Best Estimate of X is from maximum probability or minimum summation
Consider a data set x1 x2 x3 hellipxN
To find the most likely value of the standard deviation (the best estimate of
the width of the x distribution) find σx that yields the highest probability for
the data set
The combined probability for the full data set is the product
Minimize
Sum
Solve for
derivative
wrst X set to 0
Best
estimate
of X
Best Estimate of σ is from maximum probability or minimum summation
Minimize
Sum
Best
estimate
of σ
Solve for
derivative
wrst σ set to 0
See
Prob 526
prop1
120590119890minus(1199091minus119883)2 2120590 times
1
120590119890minus(1199092minus119883)2 2120590 times hellip times
1
120590119890minus(119909119873minus119883)2 2120590 =
1
120590119873119890 minus(119909119894minus119883)2 2120590
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 47
Lecture 3 Slide 47
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Combining Data Sets
Weighted Averages
References Taylor Ch 7
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 48
Lecture 3 Slide 48
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Weighted Averages
Question How can we properly combine two or more separate
independent measurements of the same randomly distributed
quantity to determine a best combined value with uncertainty
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 49
Lecture 3 Slide 49
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider two measurements of the same quantity described by a random Gaussian distribution
ltx1gt x1 and ltx2gt x2
The probability of measuring two such measurements is
119875119903119900119887119909 1199091 1199092 = 119875119903119900119887119909 1199091 119875119903119900119887119909 1199092
= 1
12059011205902119890minus12059422 119908ℎ119890119903119890 1205942 equiv
1199091 minus 119883
1205901 2 +
1199092 minus 119883
1205902 2
To find the best value for χ find the maximum Prob or minimum χ2
Weighted Averages
Assume negligible systematic errors
Note χ2 or Chi squared is the sum of the squares of the deviations from the mean divided by
the corresponding uncertainty
Such methods are called ldquoMethods of Least Squaresrdquo They follow directly from the Principle of
Maximum Likelihood
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 50
Lecture 3 Slide 50
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
The probability of measuring two such measurements is
119875119903119900119887119909 1199091 1199092 = 119875119903119900119887119909 1199091 119875119903119900119887119909 1199092
= 1
12059011205902119890minus12059422 119908ℎ119890119903119890 1205942 equiv
1199091 minus 119883
1205901 2 +
1199092 minus 119883
1205902 2
To find the best value for χ find the maximum Prob or minimum χ2
Weighted Averages
This leads to 119909119882_119886119907119892 = 11990811199091+11990821199092
1199081+1199082=
119908119894 119909119894
119908119894 119908ℎ119890119903119890 119908119894 = 1
120590119894 2
Best Estimate of χ is from maximum probibility or minimum summation
Minimize Sum Solve for best estimate of χ Solve for derivative wrst χ set to 0
Note If w1=w2 we recover the standard result Xwavg= (12) (x1+x2)
Finally the width of a weighted average distribution is
120590119908119894119890119892 ℎ119905119890119889 119886119907119892 = 1
119908119894119894
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 51
Lecture 3 Slide 51
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Weighted Averages on Steroids
A very powerful method for combining data from different sources
with different methods and uncertainties (or indeed data of
related measured and calculated quantities) is Kalman filtering
The Kalman filter also known as
linear quadratic estimation (LQE)
is an algorithm that uses a series
of measurements observed over
time containing noise (random
variations) and other
inaccuracies and produces
estimates of unknown variables
that tend to be more precise than
those based on a single
measurement alone
The Kalman filter keeps track of the estimated state of the system and the variance or uncertainty of the
estimate The estimate is updated using a state transition model and measurements xk|k-1 denotes the
estimate of the systems state at time step k before the kth measurement yk has been taken into account Pk|k-1
is the corresponding uncertainty --Wikipedia 2013
Ludger Scherliess of USU Physics is a world expert at using Kalman filtering for the assimilation
of satellite and ground-based data and the USU GAMES model to predict space weather
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 52
Lecture 3 Slide 52
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Rejecting Data
Chauvenetrsquos Criterion
References Taylor Ch 6
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 53
Lecture 3 Slide 53
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly
ldquounreasonablerdquo data point from a set of randomly distributed
measurements
bull Never
bull Whenever it makes things look better
bull Chauvenetrsquos criterion provides a (quantitative) compromise
Rejecting Data
What is a good criteria for
rejecting data
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 54
Lecture 3 Slide 54
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly
ldquounreasonablerdquo data point from a set of randomly distributed
measurements
bull Never
Rejecting Data
Zallenrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 55
Lecture 3 Slide 55
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly
ldquounreasonablerdquo data point from a set of randomly distributed
measurements
bull Whenever it makes things look better
Rejecting Data
Disneyrsquos Criterion
Disneyrsquos First Law
Wishing will make it so
Disneyrsquos Second Law
Dreams are more colorful than reality
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 56
Lecture 3 Slide 56
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Data may be rejected if the expected number of measurements at
least as deviant as the suspect measurement is less than 50
Consider a set of N measurements of a single quantity
x1x2 helliphellipxN
Calculate ltxgt and σx and then determine the fractional deviations from the mean of all
the points
For the suspect point(s) xsuspect find the probability of such a point occurring in N
measurements
n = (expected number as deviant as xsuspect)
= N Prob(outside xsuspectσx)
Rejecting Data
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 57
Lecture 3 Slide 57
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Function of Gaussian Distribution
Area under curve
(probability that
ndashtσltxlt+tσ) is given
in Table at right
Probable Error (probability that
ndash067σltxlt+067σ) is 50
Error Function (probability that ndashtσltxlt+tσ )
Error Function Tabulated values-see App A
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 58
Lecture 3 Slide 58
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 59
Lecture 3 Slide 59
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashExample 1
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 60
Lecture 3 Slide 60
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Details (1)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 61
Lecture 3 Slide 61
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashDetails (2)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 62
Lecture 3 Slide 62
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos
Criterionmdash
Details (3)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 63
Lecture 3 Slide 63
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos
Criterionmdash
Example 2
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 64
Lecture 3 Slide 64
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Summary of Probability Theory
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 65
Lecture 3 Slide 65
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of
Probability
Theory-I
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 66
Lecture 3 Slide 66
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of
Probability
Theory-II
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 67
Lecture 3 Slide 67
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-III
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 68
Lecture 3 Slide 68
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-IV
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 3
Lecture 3 Slide 3
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Practical Methods to Calculate Mean and St Deviation
We need to develop a good way to tally display and think about
a collection of repeated measurements of the same quantity
Here is where we are headed bull Develop the notion of a probability distribution function a distribution to
describe the probable outcomes of a measurement
bull Define what a distribution function is and its properties
bull Look at the properties of the most common distribution function the Gaussian
distribution for purely random events
bull Introduce other probability distribution functions
We will develop the mathematical basis for bull Mean
bull Standard deviation
bull Standard deviation of the mean (SDOM)
bull Moments and expectation values
bull Error propagation formulas
bull Addition of errors in quadrature (for independent and random measurements)
bull Schwartz inequality (ie the uncertainty principle) (next lecture)
bull Numerical values for confidence limits (t-test)
bull Principle of maximal likelihood
bull Central limit theorem
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 4
Lecture 3 Slide 4
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Two Practical Exercises in Probabilities
Roll a pair of dice 50 times and record the results
Flip Penny 50 times and record the results Flip a penny 50 times and record the results
Grab a partner and a
set of instructions
and complete the
exercise
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 5
Lecture 3 Slide 5
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Flip a penny 50 times and record the results
Two Practical Exercises in Probabilities
What is the asymmetry of the results
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 6
Lecture 3 Slide 6
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Two Practical Exercises in Probabilities
Flip a penny 50 times and record the results
What is the asymmetry of the results
4 asymmetry asymmetry
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 7
Lecture 3 Slide 7
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Two Practical Exercises in Probabilities
Roll a pair of dice 50 times and record the results
What is the mean value
The standard deviation
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 8
Lecture 3 Slide 8
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Two Practical Exercises in Probabilities
Roll a pair of dice 50 times and record the results
What is the mean value
The standard deviation
What is the asymmetry
(kurtosis)
What is the probability of
rolling a 4
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 9
Lecture 3 Slide 9
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Discrete Distribution Functions
A data set to play with Written in terms of ldquooccurrencerdquo F
The mean value In terms of fractional expectations
Fractional expectations
Normalization condition
Mean value
(This is just a weighted sum)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 10
Lecture 3 Slide 10
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Limit of Discrete Distribution Functions
ldquoNormalizingrdquo data sets
Binned data sets
fk equiv fractional occurrence
∆k equiv bin width
Mean value
Normalization
Expected value 119875119903119900119887 4 =1198654
119873= 1198914
119883 = 119865119896
119896
119909119896 = (119891119896119896
Δ119896)119909119896
1 = (119891119896119896 Δ119896)119909119896
Δ119896119896
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 11
Lecture 3 Slide 11
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Limits of Distribution Functions
Consider the limiting distribution function as N ȸ and ∆k0
Larger data sets
Mathcad Games
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 12
Lecture 3 Slide 12
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Continuous Distribution Functions
Central (mean) Value Width of Distribution
Meaning of Distribution Interval
Normalization of Distribution
Thus
and by extension
= fraction of measurements
that fall within altxltb
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 13
Lecture 3 Slide 13
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Moments of Distribution Functions
The first moment for a probability distribution function is
119909 equiv 119909 = 119891119894119903119904119905 119898119900119898119890119899119905 = 119909 119891(119909)+infin
minusinfin119889119909
For a general distribution function
119909 equiv 119909 = 119891119894119903119904119905 119898119900119898119890119899119905 = 119909 119892(119909)
+infinminusinfin
119889119909
119892(119909)+infinminusinfin
119889119909
Generalizing the n
th moment is
119909119899 equiv 119909119899 = 119899119905ℎ 119898119900119898119890119899119905 = 119909119899 119892(119909)
+infinminusinfin
119889119909
119892(119909)+infinminusinfin
119889119909= 119909119899 119891(119909)
+infin
minusinfin119889119909
Oth moment equiv N 2
nd moment equiv 119909 minus 119909 2 rarr 1199092
1st moment equiv 119909 3
rd moment equiv kurtosis
0 (for a centered distribution)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 14
Lecture 3 Slide 14
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Moments of Distribution Functions
Generalizing the nth
moment is
119909119899 equiv 119909119899 = 119899119905ℎ 119898119900119898119890119899119905 = 119909119899 119892(119909)
+infinminusinfin
119889119909
119892(119909)+infinminusinfin
119889119909= 119909119899 119891(119909)
+infin
minusinfin119889119909
Oth
moment equiv N 2nd
moment equiv 119909 minus 119909 2 rarr 1199092
1st moment equiv 119909 3
rd moment equiv kurtosis
The nth
moment about the mean is
120583119899 equiv 119909 minus 119909 119899 = 119899119905ℎ 119898119900119898119890119899119905 119886119887119900119906119905 119905ℎ119890 119898119890119886119899
= 119909minus119909 119899 119892(119909)
+infinminusinfin
119889119909
119892(119909)+infinminusinfin
119889119909= 119909 minus 119909 119899 119891(119909)
+infin
minusinfin119889119909
The standard deviation (or second moment about the mean) is
1205901199092 equiv 1205832 equiv 119909 minus 119909 2 = 2119899119889 119898119900119898119890119899119905 119886119887119900119906119905 119905ℎ119890 119898119890119886119899
= 119909minus119909 2 119892(119909)
+infinminusinfin
119889119909
119892(119909)+infinminusinfin
119889119909= 119909 minus 119909 2 119891(119909)
+infin
minusinfin119889119909
0
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 15
Lecture 3 Slide 15
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Example of Continuous Distribution Functions and Expectation Values Harmonic Oscillator Example from Mechanics
Expected Values
The expectation value of a function Ɍ(x) is
Ɍ x equiv Ɍ x 119892 119909
+infin
minusinfin119889119909
119892 119909 +infin
minusinfin119889119909
= Ɍ(x) 119891(119909)+infin
minusinfin119889119909
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 16
Lecture 3 Slide 16
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Example of Continuous Distribution Functions and Expectation Values Boltzmann Distribution Example from Kinetic Theory
Expected Values
The expectation value of a function
Ɍ(x) is
Ɍ x equiv Ɍ x 119892 119909
+infin
minusinfin119889119909
119892 119909 +infin
minusinfin119889119909
= Ɍ(x) 119891(119909)+infin
minusinfin119889119909
The Boltzmann distribution function for velocities of particles as
a function of temperature T is
119875 119907 119879 = 4120587 119872
2120587 119896119861119879
3 2
1199072119890119909119901
121198721199072
12119896119861119879
Then
v = 119907 119875(119907)+infin
minusinfin119889119907 =
8 119896119861119879120587 119872
1 2
v2 = v2 119875(119907)+infin
minusinfin119889119907 =
3 119896119861119879 119872
1 2
implies KE = 1
2119872 v2 =
3
2119896119861119879
vpeak=
2 119896119861119879 119872
1 2
= 2 3
1 2
v2
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 17
Lecture 3 Slide 17
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Example of Continuous Distribution Functions and Expectation Values Fermi-Dirac Distribution Example from Kinetic Theory
For a system of identical fermions the average number of fermions in a single-particle state is given
by the FermindashDirac (FndashD) distribution
where kB is Boltzmanns constant T is the absolute temperature is the energy of the single-particle
state and μ is the total chemical potential
Since the FndashD distribution was derived using the Pauli exclusion principle which allows at most one
electron to occupy each possible state a result is that
When a quasi-continuum of energies has an associated density of states (ie the number of
states per unit energy range per unit volume) the average number of fermions per unit energy range
per unit volume is
where is called the Fermi function
so that
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 18
Lecture 3 Slide 18
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Example of Continuous Distribution Functions and Expectation Values Finite Square Well Example from Quantum Mechanics
Expectation Values The expectation value of a QM operator O(x) is O x equiv Ψlowast x 119874 119909
+infin
minusinfinΨ x 119889119909
Ψlowast x Ψ x +infin
minusinfin119889119909
For a finite square well of width L Ψn x = 2L sin
n π x
L
Ψnlowast x |Ψn x equiv
Ψnlowast x 119874 119909
+infin
minusinfinΨn x 119889119909
Ψnlowast x Ψn x
+infin
minusinfin119889119909
= 1
119909 = Ψnlowast x |x|Ψn x equiv
Ψnlowast x 119909
+infin
minusinfinΨn x 119889119909
Ψnlowast x Ψn x
+infin
minusinfin119889119909
= 1198712
119901 = Ψnlowast x |
ℏ
i
part
partx|Ψn x equiv
Ψnlowast x
ℏi
partpartx
+infin
minusinfinΨn x 119889119909
Ψnlowast x Ψn x
+infin
minusinfin119889119909
= 0
119864119899 = Ψnlowast x |iℏ
part
partt|Ψn x equiv
Ψnlowast x iℏ
partpartt
+infin
minusinfinΨn x 119889119909
Ψnlowast x Ψn x
+infin
minusinfin119889119909
=11989921205872ℏ2
2 119898 1198712
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 19
Lecture 3 Slide 19
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Distribution Functions
Available
on web site
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 20
Lecture 3 Slide 20
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
The Gaussian Distribution
Function
References Taylor Ch 5
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 21
Lecture 3 Slide 21
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Gaussian Integrals
Factorial Approximations
119899 asymp 2120587119899 1 2 119899119899 119890119909119901 minus119899 + 1
12 119899+ 119874
1
1198992
119897119900119892 119899 asymp 1
2 119897119900119892 2120587 + 119899 +
1
2 log 119899 minus 119899 +
1
12 119899+ 119874
1
1198992
119897119900119892 119899 asymp 119899 log 119899 minus 119899 (for all terms decreasing faster than linearly with n)
Gaussian Integrals
119868119898 = 2 119909119898 exp(minus1199092)infin
0 119889119909 mgt-1
119868119898 = 2 119910119899 exp(minus119910)infin
0 119889119910 equiv Γ(119899 + 1) 1199092 equiv 119910 2 119889119909 = 1199101 2 119889119910 119899 equiv 1
2 (119898 minus 1)
1198680 = Γ 119899 =1
2 = 120587 m=0 119899 = minus1
2
1198682 119896 = Γ 119896 +1
2 = 119896 minus 1
2 119896 minus 32 3
2 12 120587 even m m=2 kgt0 119899 = 119896 minus 1
2
1198682 119896+1 = Γ 119896 + 1 = 119896 odd m m=2 k+1gt0 119899 = 119896 ge 0
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 22
Lecture 3 Slide 22
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Gaussian Distribution Function
Width of Distribution
(standard deviation)
Center of Distribution
(mean) Distribution
Function
Independent
Variable
Gaussian Distribution Function
Normalization
Constant
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 23
Lecture 3 Slide 23
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Effects of Increasing N on Gaussian Distribution Functions
Consider the limiting distribution function as N infin and dx0
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 24
Lecture 3 Slide 24
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Defining the Gaussian
distribution function
in Mathcad
I suggest you ldquoinvestigaterdquo
these with the Mathcad sheet
on the web site
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 25
Lecture 3 Slide 25
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Using Mathcad to
define other common
distribution functions
Consider the limiting
distribution function
as N ȸ and ∆k0
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 26
Lecture 3 Slide 26
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider the Gaussian distribution function
Use the normalization condition to
evaluate the normalization constant
(see Taylor p 132)
The mean Ẋ is the first moment
of the Gaussian distribution
function (see Taylor p 134)
The standard deviation σx is the standard
deviation of the mean of the Gaussian
distribution function (see Taylor p 143)
Gaussian Distribution Moments
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 27
Lecture 3 Slide 27
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
When is mean x not Xbest
Answer When the
distribution is not
symmetric about X
Example Cauchy
Distribution
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 28
Lecture 3 Slide 28
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
When is mean x not Xbest
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 29
Lecture 3 Slide 29
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
When is mean x not Xbest
Answer When the
distribution is has
more than one peak
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 30
Lecture 3 Slide 30
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
The Gaussian Distribution
Function and Its Relation to
Errors
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 31
Lecture 3 Slide 31
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
A Review of Probabilities in Combination
Probability of a data set of N like measurements (x1x2hellipxN)
P (x1x2hellipxN) = P(x)1P(x2)hellipP(xN)
1 head AND 1 Four
P(H4) = P(H) P(4)
1 head OR 1 Four
P(H4) = P(H) + P(4) - P(H and 4) (true for a ldquonon-mutually exclusiverdquo events)
NOT 1 Six
P(NOT 6) = 1 - P(6)
1 Six OR 1 Four
P(64) = P(6) + P(4) (true for a ldquomutually exclusiverdquo single role)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 32
Lecture 3 Slide 32
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
The Gaussian Distribution Function and Its Relation to Errors
We will use the Gaussian distribution as applied to random
variables to develop the mathematical basis for bull Mean
bull Standard deviation
bull Standard deviation of the mean (SDOM)
bull Moments and expectation values
bull Error propagation formulas
bull Addition of errors in quadrature (for independent and random
measurements)
bull Numerical values for confidence limits (t-test)
bull Principle of maximal likelihood
bull Central limit theorem
bull Weighted distributions and Chi squared
bull Schwartz inequality (ie the uncertainty principle) (next lecture)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 33
Lecture 3 Slide 33
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider the Gaussian distribution function
Use the normalization condition to
evaluate the normalization constant
(see Taylor p 132)
The mean Ẋ is the first moment
of the Gaussian distribution
function (see Taylor p 134)
The standard deviation σx is the standard
deviation of the mean of the Gaussian
distribution function (see Taylor p 143)
Gaussian Distribution Moments
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 34
Lecture 3 Slide 34
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Standard Deviation of Gaussian Distribution
Area under curve
(probability that
ndashσltxlt+σ) is 68
5 or ~2σ ldquoSignificantrdquo
1 or ~3σ ldquoHighly Significantrdquo
1σ ldquoWithin errorsrdquo
5 ppm or ~5σ ldquoValid for HEPrdquo
See Sec 106 Testing of
Hypotheses
More complete Table in App A and B
Ah thatrsquos
highly
significant
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 35
Lecture 3 Slide 35
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Function of Gaussian Distribution
Area under curve
(probability that
ndashtσltxlt+tσ) is given
in Table at right
Probable Error (probability that
ndash067σltxlt+067σ) is 50
Error Function (probability that ndashtσltxlt+tσ )
Error Function Tabulated values-see App A
Complementary Error Function
(probability that ndashxlt-tσ AND xgt+tσ )
Prob(x outside tσ) = 1 - Prob(x within tσ)
More complete Table in App A and B
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 36
Lecture 3 Slide 36
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Useful Points on Gaussian Distribution
Full Width at Half Maximum FWHM
(See Prob 512)
Points of Inflection Occur at plusmnσ
(See Prob 513)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 37
Lecture 3 Slide 37
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Analysis and Gaussian Distribution
Adding a Constant Multiplying by a Constant
XX+A XBX and σB σ
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 38
Lecture 3 Slide 38
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Sum of Two Variables
Consider the derived quantity
Z=X + Y
(with X=0 and Y=0)
Error Propagation Addition
Error in Z
Multiple two probabilities 119875119903119900119887 119909 119910 = 119875119903119900119887 119909 ∙ 119875119903119900119887 119909 prop 119890119909119901 minus
1
2
1199092
1205901199092+
1199102
1205901199102
prop 119890119909119901 minus1
2
(119909+119910)2
1205901199092+120590119910
2 minus1199112
prop 119890119909119901 minus1
2
(119909+119910)2
1205901199092+120590119910
2 119890119909119901 minus
1199112
2
ICBST (Eq 553)
X+YZ and σx2 + σy
2 σz
2
(addition in quadrature for random independent variables)
Integrates to radic2π
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 39
Lecture 3 Slide 39
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Error Propagation
General formula for error propagation see [Taylor Secs 35 and 39]
Uncertainty as a function of one variable [Taylor Sec 35]
1 Consider a graphical method of estimating error
a) Consider an arbitaray function q(x)
b) Plot q(x) vs x
c) On the graph label
(1) qbest = q(xbest)
(2) qhi = q(xbest + δx)
(3) qlow = q(xbest- δx)
d) Making a linear approximation
x
qqxslopeqq
x
qqxslopeqq
bestbestlow
bestbesthi
e) Therefore
xx
Note the absolute value
How do we determine the error of a derived quantity Z(XYhellip) from errors in XYhellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 40
Lecture 3 Slide 40
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Error Propagation
General formula for uncertainty of a function of one variable
xx
[Taylor Eq 323]
Can you now derive for specific rules of error propagation
1 Addition and Subtraction [Taylor p 49]
2 Multiplication and Division [Taylor p 51]
3 Multiplication by a constant (exact number) [Taylor p 54]
4 Exponentiation (powers) [Taylor p 56]
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 41
Lecture 3 Slide 41
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Multiple Variables
Uncertainty of a function of multiple variables [Taylor Sec 311]
1 It can easily (no really) be shown that (see Taylor Sec 311) for a
function of several variables
)(
z
z
qy
y
qx
x
qzyxq
[Taylor Eq 347]
2 More correctly it can be shown that (see Taylor Sec 311) for a
function of several variables
)(
z
z
qy
y
qx
x
qzyxq
[Taylor Eq 347]
where the equals sign represents an upper bound as discussed above
3 For a function of several independent and random variables
)(
222
z
z
qy
y
qx
x
qzyxq
[Taylor Eq 348]
Again the proof is left for Ch 5
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 42
Lecture 3 Slide 42
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Product of Two Variables
Consider the arbitrary derived quantity
q(xy) of two independent random variables x and y
Expand q(xy) in a Taylor series about the expected values of x and y
(ie at points near X and Y)
Error Propagation General Case
119902 119909 119910 = 119902 119883 119884 + 120597119902
120597119909
119883 119909 minus 119883 +
120597119902
120597119910
119884
(119910 minus 119884)
Fixed shifts peak of distribution
Fixed Distribution centered at X with width σX
120575119902 119909 119910 = 120590119902 = 119902 119883 119884 + 120597119902
120597119909
119883120590119909
2
+ 120597119902
120597119910
119884
120590119910
2
0
How do we determine the error of a derived quantity Z(XYhellip) from errors in XYhellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 43
Lecture 3 Slide 43
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
SDOM of Gaussian Distribution
Standard Deviation of the Mean
The SDOM decreases as the square root of the number of measurements
That is the relative width σẊ of the distribution gets narrower as more
measurements are made
Each measurement
has similar σXi= σẊ
and similar partial
derivatives
Thushellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 44
Lecture 3 Slide 44
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Two Key Theorems from Probability
Central Limit Theorem
For random independent measurements (each with a well-define
expectation value and well-defined variance) the arithmetic mean
(average) will be approximately normally distributed
Principle of Maximum Likelihood
Given the N observed measurements x1 x2hellipxN the best estimates
for Ẋ and σ are those values for which the observed x1 x2hellipxN are
most likely
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 45
Lecture 3 Slide 45
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Mean of Gaussian Distribution as ldquoBest Estimaterdquo
Principle of Maximum Likelihood
Best Estimate of X is from maximum probability or minimum summation
Consider a data set
Each randomly distributed with
x1 x2 x3 hellipxN
To find the most likely value of the mean (the best estimate of ẋ)
find X that yields the highest probability for the data set
The combined probability for the full data set is the product
Minimize
Sum
Solve for
derivative wrst
X set to 0
Best
estimate
of X
prop1
120590119890minus(1199091minus119883)2 2120590 times
1
120590119890minus(1199092minus119883)2 2120590 times hellip times
1
120590119890minus(119909119873minus119883)2 2120590 =
1
120590119873119890 minus(119909119894minus119883)2 2120590
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 46
Lecture 3 Slide 46
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Uncertainty of ldquoBest Estimatesrdquo of Gaussian Distribution Principle of Maximum Likelihood
Best Estimate of X is from maximum probability or minimum summation
Consider a data set x1 x2 x3 hellipxN
To find the most likely value of the standard deviation (the best estimate of
the width of the x distribution) find σx that yields the highest probability for
the data set
The combined probability for the full data set is the product
Minimize
Sum
Solve for
derivative
wrst X set to 0
Best
estimate
of X
Best Estimate of σ is from maximum probability or minimum summation
Minimize
Sum
Best
estimate
of σ
Solve for
derivative
wrst σ set to 0
See
Prob 526
prop1
120590119890minus(1199091minus119883)2 2120590 times
1
120590119890minus(1199092minus119883)2 2120590 times hellip times
1
120590119890minus(119909119873minus119883)2 2120590 =
1
120590119873119890 minus(119909119894minus119883)2 2120590
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 47
Lecture 3 Slide 47
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Combining Data Sets
Weighted Averages
References Taylor Ch 7
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 48
Lecture 3 Slide 48
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Weighted Averages
Question How can we properly combine two or more separate
independent measurements of the same randomly distributed
quantity to determine a best combined value with uncertainty
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 49
Lecture 3 Slide 49
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider two measurements of the same quantity described by a random Gaussian distribution
ltx1gt x1 and ltx2gt x2
The probability of measuring two such measurements is
119875119903119900119887119909 1199091 1199092 = 119875119903119900119887119909 1199091 119875119903119900119887119909 1199092
= 1
12059011205902119890minus12059422 119908ℎ119890119903119890 1205942 equiv
1199091 minus 119883
1205901 2 +
1199092 minus 119883
1205902 2
To find the best value for χ find the maximum Prob or minimum χ2
Weighted Averages
Assume negligible systematic errors
Note χ2 or Chi squared is the sum of the squares of the deviations from the mean divided by
the corresponding uncertainty
Such methods are called ldquoMethods of Least Squaresrdquo They follow directly from the Principle of
Maximum Likelihood
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 50
Lecture 3 Slide 50
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
The probability of measuring two such measurements is
119875119903119900119887119909 1199091 1199092 = 119875119903119900119887119909 1199091 119875119903119900119887119909 1199092
= 1
12059011205902119890minus12059422 119908ℎ119890119903119890 1205942 equiv
1199091 minus 119883
1205901 2 +
1199092 minus 119883
1205902 2
To find the best value for χ find the maximum Prob or minimum χ2
Weighted Averages
This leads to 119909119882_119886119907119892 = 11990811199091+11990821199092
1199081+1199082=
119908119894 119909119894
119908119894 119908ℎ119890119903119890 119908119894 = 1
120590119894 2
Best Estimate of χ is from maximum probibility or minimum summation
Minimize Sum Solve for best estimate of χ Solve for derivative wrst χ set to 0
Note If w1=w2 we recover the standard result Xwavg= (12) (x1+x2)
Finally the width of a weighted average distribution is
120590119908119894119890119892 ℎ119905119890119889 119886119907119892 = 1
119908119894119894
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 51
Lecture 3 Slide 51
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Weighted Averages on Steroids
A very powerful method for combining data from different sources
with different methods and uncertainties (or indeed data of
related measured and calculated quantities) is Kalman filtering
The Kalman filter also known as
linear quadratic estimation (LQE)
is an algorithm that uses a series
of measurements observed over
time containing noise (random
variations) and other
inaccuracies and produces
estimates of unknown variables
that tend to be more precise than
those based on a single
measurement alone
The Kalman filter keeps track of the estimated state of the system and the variance or uncertainty of the
estimate The estimate is updated using a state transition model and measurements xk|k-1 denotes the
estimate of the systems state at time step k before the kth measurement yk has been taken into account Pk|k-1
is the corresponding uncertainty --Wikipedia 2013
Ludger Scherliess of USU Physics is a world expert at using Kalman filtering for the assimilation
of satellite and ground-based data and the USU GAMES model to predict space weather
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 52
Lecture 3 Slide 52
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Rejecting Data
Chauvenetrsquos Criterion
References Taylor Ch 6
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 53
Lecture 3 Slide 53
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly
ldquounreasonablerdquo data point from a set of randomly distributed
measurements
bull Never
bull Whenever it makes things look better
bull Chauvenetrsquos criterion provides a (quantitative) compromise
Rejecting Data
What is a good criteria for
rejecting data
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 54
Lecture 3 Slide 54
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly
ldquounreasonablerdquo data point from a set of randomly distributed
measurements
bull Never
Rejecting Data
Zallenrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 55
Lecture 3 Slide 55
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly
ldquounreasonablerdquo data point from a set of randomly distributed
measurements
bull Whenever it makes things look better
Rejecting Data
Disneyrsquos Criterion
Disneyrsquos First Law
Wishing will make it so
Disneyrsquos Second Law
Dreams are more colorful than reality
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 56
Lecture 3 Slide 56
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Data may be rejected if the expected number of measurements at
least as deviant as the suspect measurement is less than 50
Consider a set of N measurements of a single quantity
x1x2 helliphellipxN
Calculate ltxgt and σx and then determine the fractional deviations from the mean of all
the points
For the suspect point(s) xsuspect find the probability of such a point occurring in N
measurements
n = (expected number as deviant as xsuspect)
= N Prob(outside xsuspectσx)
Rejecting Data
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 57
Lecture 3 Slide 57
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Function of Gaussian Distribution
Area under curve
(probability that
ndashtσltxlt+tσ) is given
in Table at right
Probable Error (probability that
ndash067σltxlt+067σ) is 50
Error Function (probability that ndashtσltxlt+tσ )
Error Function Tabulated values-see App A
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 58
Lecture 3 Slide 58
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 59
Lecture 3 Slide 59
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashExample 1
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 60
Lecture 3 Slide 60
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Details (1)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 61
Lecture 3 Slide 61
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashDetails (2)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 62
Lecture 3 Slide 62
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos
Criterionmdash
Details (3)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 63
Lecture 3 Slide 63
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos
Criterionmdash
Example 2
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 64
Lecture 3 Slide 64
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Summary of Probability Theory
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 65
Lecture 3 Slide 65
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of
Probability
Theory-I
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 66
Lecture 3 Slide 66
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of
Probability
Theory-II
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 67
Lecture 3 Slide 67
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-III
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 68
Lecture 3 Slide 68
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-IV
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 4
Lecture 3 Slide 4
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Two Practical Exercises in Probabilities
Roll a pair of dice 50 times and record the results
Flip Penny 50 times and record the results Flip a penny 50 times and record the results
Grab a partner and a
set of instructions
and complete the
exercise
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 5
Lecture 3 Slide 5
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Flip a penny 50 times and record the results
Two Practical Exercises in Probabilities
What is the asymmetry of the results
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 6
Lecture 3 Slide 6
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Two Practical Exercises in Probabilities
Flip a penny 50 times and record the results
What is the asymmetry of the results
4 asymmetry asymmetry
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 7
Lecture 3 Slide 7
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Two Practical Exercises in Probabilities
Roll a pair of dice 50 times and record the results
What is the mean value
The standard deviation
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 8
Lecture 3 Slide 8
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Two Practical Exercises in Probabilities
Roll a pair of dice 50 times and record the results
What is the mean value
The standard deviation
What is the asymmetry
(kurtosis)
What is the probability of
rolling a 4
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 9
Lecture 3 Slide 9
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Discrete Distribution Functions
A data set to play with Written in terms of ldquooccurrencerdquo F
The mean value In terms of fractional expectations
Fractional expectations
Normalization condition
Mean value
(This is just a weighted sum)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 10
Lecture 3 Slide 10
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Limit of Discrete Distribution Functions
ldquoNormalizingrdquo data sets
Binned data sets
fk equiv fractional occurrence
∆k equiv bin width
Mean value
Normalization
Expected value 119875119903119900119887 4 =1198654
119873= 1198914
119883 = 119865119896
119896
119909119896 = (119891119896119896
Δ119896)119909119896
1 = (119891119896119896 Δ119896)119909119896
Δ119896119896
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 11
Lecture 3 Slide 11
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Limits of Distribution Functions
Consider the limiting distribution function as N ȸ and ∆k0
Larger data sets
Mathcad Games
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 12
Lecture 3 Slide 12
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Continuous Distribution Functions
Central (mean) Value Width of Distribution
Meaning of Distribution Interval
Normalization of Distribution
Thus
and by extension
= fraction of measurements
that fall within altxltb
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 13
Lecture 3 Slide 13
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Moments of Distribution Functions
The first moment for a probability distribution function is
119909 equiv 119909 = 119891119894119903119904119905 119898119900119898119890119899119905 = 119909 119891(119909)+infin
minusinfin119889119909
For a general distribution function
119909 equiv 119909 = 119891119894119903119904119905 119898119900119898119890119899119905 = 119909 119892(119909)
+infinminusinfin
119889119909
119892(119909)+infinminusinfin
119889119909
Generalizing the n
th moment is
119909119899 equiv 119909119899 = 119899119905ℎ 119898119900119898119890119899119905 = 119909119899 119892(119909)
+infinminusinfin
119889119909
119892(119909)+infinminusinfin
119889119909= 119909119899 119891(119909)
+infin
minusinfin119889119909
Oth moment equiv N 2
nd moment equiv 119909 minus 119909 2 rarr 1199092
1st moment equiv 119909 3
rd moment equiv kurtosis
0 (for a centered distribution)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 14
Lecture 3 Slide 14
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Moments of Distribution Functions
Generalizing the nth
moment is
119909119899 equiv 119909119899 = 119899119905ℎ 119898119900119898119890119899119905 = 119909119899 119892(119909)
+infinminusinfin
119889119909
119892(119909)+infinminusinfin
119889119909= 119909119899 119891(119909)
+infin
minusinfin119889119909
Oth
moment equiv N 2nd
moment equiv 119909 minus 119909 2 rarr 1199092
1st moment equiv 119909 3
rd moment equiv kurtosis
The nth
moment about the mean is
120583119899 equiv 119909 minus 119909 119899 = 119899119905ℎ 119898119900119898119890119899119905 119886119887119900119906119905 119905ℎ119890 119898119890119886119899
= 119909minus119909 119899 119892(119909)
+infinminusinfin
119889119909
119892(119909)+infinminusinfin
119889119909= 119909 minus 119909 119899 119891(119909)
+infin
minusinfin119889119909
The standard deviation (or second moment about the mean) is
1205901199092 equiv 1205832 equiv 119909 minus 119909 2 = 2119899119889 119898119900119898119890119899119905 119886119887119900119906119905 119905ℎ119890 119898119890119886119899
= 119909minus119909 2 119892(119909)
+infinminusinfin
119889119909
119892(119909)+infinminusinfin
119889119909= 119909 minus 119909 2 119891(119909)
+infin
minusinfin119889119909
0
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 15
Lecture 3 Slide 15
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Example of Continuous Distribution Functions and Expectation Values Harmonic Oscillator Example from Mechanics
Expected Values
The expectation value of a function Ɍ(x) is
Ɍ x equiv Ɍ x 119892 119909
+infin
minusinfin119889119909
119892 119909 +infin
minusinfin119889119909
= Ɍ(x) 119891(119909)+infin
minusinfin119889119909
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 16
Lecture 3 Slide 16
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Example of Continuous Distribution Functions and Expectation Values Boltzmann Distribution Example from Kinetic Theory
Expected Values
The expectation value of a function
Ɍ(x) is
Ɍ x equiv Ɍ x 119892 119909
+infin
minusinfin119889119909
119892 119909 +infin
minusinfin119889119909
= Ɍ(x) 119891(119909)+infin
minusinfin119889119909
The Boltzmann distribution function for velocities of particles as
a function of temperature T is
119875 119907 119879 = 4120587 119872
2120587 119896119861119879
3 2
1199072119890119909119901
121198721199072
12119896119861119879
Then
v = 119907 119875(119907)+infin
minusinfin119889119907 =
8 119896119861119879120587 119872
1 2
v2 = v2 119875(119907)+infin
minusinfin119889119907 =
3 119896119861119879 119872
1 2
implies KE = 1
2119872 v2 =
3
2119896119861119879
vpeak=
2 119896119861119879 119872
1 2
= 2 3
1 2
v2
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 17
Lecture 3 Slide 17
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Example of Continuous Distribution Functions and Expectation Values Fermi-Dirac Distribution Example from Kinetic Theory
For a system of identical fermions the average number of fermions in a single-particle state is given
by the FermindashDirac (FndashD) distribution
where kB is Boltzmanns constant T is the absolute temperature is the energy of the single-particle
state and μ is the total chemical potential
Since the FndashD distribution was derived using the Pauli exclusion principle which allows at most one
electron to occupy each possible state a result is that
When a quasi-continuum of energies has an associated density of states (ie the number of
states per unit energy range per unit volume) the average number of fermions per unit energy range
per unit volume is
where is called the Fermi function
so that
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 18
Lecture 3 Slide 18
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Example of Continuous Distribution Functions and Expectation Values Finite Square Well Example from Quantum Mechanics
Expectation Values The expectation value of a QM operator O(x) is O x equiv Ψlowast x 119874 119909
+infin
minusinfinΨ x 119889119909
Ψlowast x Ψ x +infin
minusinfin119889119909
For a finite square well of width L Ψn x = 2L sin
n π x
L
Ψnlowast x |Ψn x equiv
Ψnlowast x 119874 119909
+infin
minusinfinΨn x 119889119909
Ψnlowast x Ψn x
+infin
minusinfin119889119909
= 1
119909 = Ψnlowast x |x|Ψn x equiv
Ψnlowast x 119909
+infin
minusinfinΨn x 119889119909
Ψnlowast x Ψn x
+infin
minusinfin119889119909
= 1198712
119901 = Ψnlowast x |
ℏ
i
part
partx|Ψn x equiv
Ψnlowast x
ℏi
partpartx
+infin
minusinfinΨn x 119889119909
Ψnlowast x Ψn x
+infin
minusinfin119889119909
= 0
119864119899 = Ψnlowast x |iℏ
part
partt|Ψn x equiv
Ψnlowast x iℏ
partpartt
+infin
minusinfinΨn x 119889119909
Ψnlowast x Ψn x
+infin
minusinfin119889119909
=11989921205872ℏ2
2 119898 1198712
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 19
Lecture 3 Slide 19
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Distribution Functions
Available
on web site
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 20
Lecture 3 Slide 20
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
The Gaussian Distribution
Function
References Taylor Ch 5
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 21
Lecture 3 Slide 21
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Gaussian Integrals
Factorial Approximations
119899 asymp 2120587119899 1 2 119899119899 119890119909119901 minus119899 + 1
12 119899+ 119874
1
1198992
119897119900119892 119899 asymp 1
2 119897119900119892 2120587 + 119899 +
1
2 log 119899 minus 119899 +
1
12 119899+ 119874
1
1198992
119897119900119892 119899 asymp 119899 log 119899 minus 119899 (for all terms decreasing faster than linearly with n)
Gaussian Integrals
119868119898 = 2 119909119898 exp(minus1199092)infin
0 119889119909 mgt-1
119868119898 = 2 119910119899 exp(minus119910)infin
0 119889119910 equiv Γ(119899 + 1) 1199092 equiv 119910 2 119889119909 = 1199101 2 119889119910 119899 equiv 1
2 (119898 minus 1)
1198680 = Γ 119899 =1
2 = 120587 m=0 119899 = minus1
2
1198682 119896 = Γ 119896 +1
2 = 119896 minus 1
2 119896 minus 32 3
2 12 120587 even m m=2 kgt0 119899 = 119896 minus 1
2
1198682 119896+1 = Γ 119896 + 1 = 119896 odd m m=2 k+1gt0 119899 = 119896 ge 0
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 22
Lecture 3 Slide 22
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Gaussian Distribution Function
Width of Distribution
(standard deviation)
Center of Distribution
(mean) Distribution
Function
Independent
Variable
Gaussian Distribution Function
Normalization
Constant
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 23
Lecture 3 Slide 23
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Effects of Increasing N on Gaussian Distribution Functions
Consider the limiting distribution function as N infin and dx0
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 24
Lecture 3 Slide 24
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Defining the Gaussian
distribution function
in Mathcad
I suggest you ldquoinvestigaterdquo
these with the Mathcad sheet
on the web site
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 25
Lecture 3 Slide 25
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Using Mathcad to
define other common
distribution functions
Consider the limiting
distribution function
as N ȸ and ∆k0
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 26
Lecture 3 Slide 26
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider the Gaussian distribution function
Use the normalization condition to
evaluate the normalization constant
(see Taylor p 132)
The mean Ẋ is the first moment
of the Gaussian distribution
function (see Taylor p 134)
The standard deviation σx is the standard
deviation of the mean of the Gaussian
distribution function (see Taylor p 143)
Gaussian Distribution Moments
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 27
Lecture 3 Slide 27
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
When is mean x not Xbest
Answer When the
distribution is not
symmetric about X
Example Cauchy
Distribution
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 28
Lecture 3 Slide 28
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
When is mean x not Xbest
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 29
Lecture 3 Slide 29
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
When is mean x not Xbest
Answer When the
distribution is has
more than one peak
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 30
Lecture 3 Slide 30
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
The Gaussian Distribution
Function and Its Relation to
Errors
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 31
Lecture 3 Slide 31
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
A Review of Probabilities in Combination
Probability of a data set of N like measurements (x1x2hellipxN)
P (x1x2hellipxN) = P(x)1P(x2)hellipP(xN)
1 head AND 1 Four
P(H4) = P(H) P(4)
1 head OR 1 Four
P(H4) = P(H) + P(4) - P(H and 4) (true for a ldquonon-mutually exclusiverdquo events)
NOT 1 Six
P(NOT 6) = 1 - P(6)
1 Six OR 1 Four
P(64) = P(6) + P(4) (true for a ldquomutually exclusiverdquo single role)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 32
Lecture 3 Slide 32
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
The Gaussian Distribution Function and Its Relation to Errors
We will use the Gaussian distribution as applied to random
variables to develop the mathematical basis for bull Mean
bull Standard deviation
bull Standard deviation of the mean (SDOM)
bull Moments and expectation values
bull Error propagation formulas
bull Addition of errors in quadrature (for independent and random
measurements)
bull Numerical values for confidence limits (t-test)
bull Principle of maximal likelihood
bull Central limit theorem
bull Weighted distributions and Chi squared
bull Schwartz inequality (ie the uncertainty principle) (next lecture)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 33
Lecture 3 Slide 33
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider the Gaussian distribution function
Use the normalization condition to
evaluate the normalization constant
(see Taylor p 132)
The mean Ẋ is the first moment
of the Gaussian distribution
function (see Taylor p 134)
The standard deviation σx is the standard
deviation of the mean of the Gaussian
distribution function (see Taylor p 143)
Gaussian Distribution Moments
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 34
Lecture 3 Slide 34
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Standard Deviation of Gaussian Distribution
Area under curve
(probability that
ndashσltxlt+σ) is 68
5 or ~2σ ldquoSignificantrdquo
1 or ~3σ ldquoHighly Significantrdquo
1σ ldquoWithin errorsrdquo
5 ppm or ~5σ ldquoValid for HEPrdquo
See Sec 106 Testing of
Hypotheses
More complete Table in App A and B
Ah thatrsquos
highly
significant
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 35
Lecture 3 Slide 35
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Function of Gaussian Distribution
Area under curve
(probability that
ndashtσltxlt+tσ) is given
in Table at right
Probable Error (probability that
ndash067σltxlt+067σ) is 50
Error Function (probability that ndashtσltxlt+tσ )
Error Function Tabulated values-see App A
Complementary Error Function
(probability that ndashxlt-tσ AND xgt+tσ )
Prob(x outside tσ) = 1 - Prob(x within tσ)
More complete Table in App A and B
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 36
Lecture 3 Slide 36
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Useful Points on Gaussian Distribution
Full Width at Half Maximum FWHM
(See Prob 512)
Points of Inflection Occur at plusmnσ
(See Prob 513)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 37
Lecture 3 Slide 37
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Analysis and Gaussian Distribution
Adding a Constant Multiplying by a Constant
XX+A XBX and σB σ
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 38
Lecture 3 Slide 38
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Sum of Two Variables
Consider the derived quantity
Z=X + Y
(with X=0 and Y=0)
Error Propagation Addition
Error in Z
Multiple two probabilities 119875119903119900119887 119909 119910 = 119875119903119900119887 119909 ∙ 119875119903119900119887 119909 prop 119890119909119901 minus
1
2
1199092
1205901199092+
1199102
1205901199102
prop 119890119909119901 minus1
2
(119909+119910)2
1205901199092+120590119910
2 minus1199112
prop 119890119909119901 minus1
2
(119909+119910)2
1205901199092+120590119910
2 119890119909119901 minus
1199112
2
ICBST (Eq 553)
X+YZ and σx2 + σy
2 σz
2
(addition in quadrature for random independent variables)
Integrates to radic2π
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 39
Lecture 3 Slide 39
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Error Propagation
General formula for error propagation see [Taylor Secs 35 and 39]
Uncertainty as a function of one variable [Taylor Sec 35]
1 Consider a graphical method of estimating error
a) Consider an arbitaray function q(x)
b) Plot q(x) vs x
c) On the graph label
(1) qbest = q(xbest)
(2) qhi = q(xbest + δx)
(3) qlow = q(xbest- δx)
d) Making a linear approximation
x
qqxslopeqq
x
qqxslopeqq
bestbestlow
bestbesthi
e) Therefore
xx
Note the absolute value
How do we determine the error of a derived quantity Z(XYhellip) from errors in XYhellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 40
Lecture 3 Slide 40
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Error Propagation
General formula for uncertainty of a function of one variable
xx
[Taylor Eq 323]
Can you now derive for specific rules of error propagation
1 Addition and Subtraction [Taylor p 49]
2 Multiplication and Division [Taylor p 51]
3 Multiplication by a constant (exact number) [Taylor p 54]
4 Exponentiation (powers) [Taylor p 56]
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 41
Lecture 3 Slide 41
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Multiple Variables
Uncertainty of a function of multiple variables [Taylor Sec 311]
1 It can easily (no really) be shown that (see Taylor Sec 311) for a
function of several variables
)(
z
z
qy
y
qx
x
qzyxq
[Taylor Eq 347]
2 More correctly it can be shown that (see Taylor Sec 311) for a
function of several variables
)(
z
z
qy
y
qx
x
qzyxq
[Taylor Eq 347]
where the equals sign represents an upper bound as discussed above
3 For a function of several independent and random variables
)(
222
z
z
qy
y
qx
x
qzyxq
[Taylor Eq 348]
Again the proof is left for Ch 5
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 42
Lecture 3 Slide 42
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Product of Two Variables
Consider the arbitrary derived quantity
q(xy) of two independent random variables x and y
Expand q(xy) in a Taylor series about the expected values of x and y
(ie at points near X and Y)
Error Propagation General Case
119902 119909 119910 = 119902 119883 119884 + 120597119902
120597119909
119883 119909 minus 119883 +
120597119902
120597119910
119884
(119910 minus 119884)
Fixed shifts peak of distribution
Fixed Distribution centered at X with width σX
120575119902 119909 119910 = 120590119902 = 119902 119883 119884 + 120597119902
120597119909
119883120590119909
2
+ 120597119902
120597119910
119884
120590119910
2
0
How do we determine the error of a derived quantity Z(XYhellip) from errors in XYhellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 43
Lecture 3 Slide 43
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
SDOM of Gaussian Distribution
Standard Deviation of the Mean
The SDOM decreases as the square root of the number of measurements
That is the relative width σẊ of the distribution gets narrower as more
measurements are made
Each measurement
has similar σXi= σẊ
and similar partial
derivatives
Thushellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 44
Lecture 3 Slide 44
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Two Key Theorems from Probability
Central Limit Theorem
For random independent measurements (each with a well-define
expectation value and well-defined variance) the arithmetic mean
(average) will be approximately normally distributed
Principle of Maximum Likelihood
Given the N observed measurements x1 x2hellipxN the best estimates
for Ẋ and σ are those values for which the observed x1 x2hellipxN are
most likely
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 45
Lecture 3 Slide 45
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Mean of Gaussian Distribution as ldquoBest Estimaterdquo
Principle of Maximum Likelihood
Best Estimate of X is from maximum probability or minimum summation
Consider a data set
Each randomly distributed with
x1 x2 x3 hellipxN
To find the most likely value of the mean (the best estimate of ẋ)
find X that yields the highest probability for the data set
The combined probability for the full data set is the product
Minimize
Sum
Solve for
derivative wrst
X set to 0
Best
estimate
of X
prop1
120590119890minus(1199091minus119883)2 2120590 times
1
120590119890minus(1199092minus119883)2 2120590 times hellip times
1
120590119890minus(119909119873minus119883)2 2120590 =
1
120590119873119890 minus(119909119894minus119883)2 2120590
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 46
Lecture 3 Slide 46
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Uncertainty of ldquoBest Estimatesrdquo of Gaussian Distribution Principle of Maximum Likelihood
Best Estimate of X is from maximum probability or minimum summation
Consider a data set x1 x2 x3 hellipxN
To find the most likely value of the standard deviation (the best estimate of
the width of the x distribution) find σx that yields the highest probability for
the data set
The combined probability for the full data set is the product
Minimize
Sum
Solve for
derivative
wrst X set to 0
Best
estimate
of X
Best Estimate of σ is from maximum probability or minimum summation
Minimize
Sum
Best
estimate
of σ
Solve for
derivative
wrst σ set to 0
See
Prob 526
prop1
120590119890minus(1199091minus119883)2 2120590 times
1
120590119890minus(1199092minus119883)2 2120590 times hellip times
1
120590119890minus(119909119873minus119883)2 2120590 =
1
120590119873119890 minus(119909119894minus119883)2 2120590
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 47
Lecture 3 Slide 47
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Combining Data Sets
Weighted Averages
References Taylor Ch 7
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 48
Lecture 3 Slide 48
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Weighted Averages
Question How can we properly combine two or more separate
independent measurements of the same randomly distributed
quantity to determine a best combined value with uncertainty
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 49
Lecture 3 Slide 49
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider two measurements of the same quantity described by a random Gaussian distribution
ltx1gt x1 and ltx2gt x2
The probability of measuring two such measurements is
119875119903119900119887119909 1199091 1199092 = 119875119903119900119887119909 1199091 119875119903119900119887119909 1199092
= 1
12059011205902119890minus12059422 119908ℎ119890119903119890 1205942 equiv
1199091 minus 119883
1205901 2 +
1199092 minus 119883
1205902 2
To find the best value for χ find the maximum Prob or minimum χ2
Weighted Averages
Assume negligible systematic errors
Note χ2 or Chi squared is the sum of the squares of the deviations from the mean divided by
the corresponding uncertainty
Such methods are called ldquoMethods of Least Squaresrdquo They follow directly from the Principle of
Maximum Likelihood
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 50
Lecture 3 Slide 50
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
The probability of measuring two such measurements is
119875119903119900119887119909 1199091 1199092 = 119875119903119900119887119909 1199091 119875119903119900119887119909 1199092
= 1
12059011205902119890minus12059422 119908ℎ119890119903119890 1205942 equiv
1199091 minus 119883
1205901 2 +
1199092 minus 119883
1205902 2
To find the best value for χ find the maximum Prob or minimum χ2
Weighted Averages
This leads to 119909119882_119886119907119892 = 11990811199091+11990821199092
1199081+1199082=
119908119894 119909119894
119908119894 119908ℎ119890119903119890 119908119894 = 1
120590119894 2
Best Estimate of χ is from maximum probibility or minimum summation
Minimize Sum Solve for best estimate of χ Solve for derivative wrst χ set to 0
Note If w1=w2 we recover the standard result Xwavg= (12) (x1+x2)
Finally the width of a weighted average distribution is
120590119908119894119890119892 ℎ119905119890119889 119886119907119892 = 1
119908119894119894
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 51
Lecture 3 Slide 51
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Weighted Averages on Steroids
A very powerful method for combining data from different sources
with different methods and uncertainties (or indeed data of
related measured and calculated quantities) is Kalman filtering
The Kalman filter also known as
linear quadratic estimation (LQE)
is an algorithm that uses a series
of measurements observed over
time containing noise (random
variations) and other
inaccuracies and produces
estimates of unknown variables
that tend to be more precise than
those based on a single
measurement alone
The Kalman filter keeps track of the estimated state of the system and the variance or uncertainty of the
estimate The estimate is updated using a state transition model and measurements xk|k-1 denotes the
estimate of the systems state at time step k before the kth measurement yk has been taken into account Pk|k-1
is the corresponding uncertainty --Wikipedia 2013
Ludger Scherliess of USU Physics is a world expert at using Kalman filtering for the assimilation
of satellite and ground-based data and the USU GAMES model to predict space weather
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 52
Lecture 3 Slide 52
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Rejecting Data
Chauvenetrsquos Criterion
References Taylor Ch 6
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 53
Lecture 3 Slide 53
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly
ldquounreasonablerdquo data point from a set of randomly distributed
measurements
bull Never
bull Whenever it makes things look better
bull Chauvenetrsquos criterion provides a (quantitative) compromise
Rejecting Data
What is a good criteria for
rejecting data
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 54
Lecture 3 Slide 54
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly
ldquounreasonablerdquo data point from a set of randomly distributed
measurements
bull Never
Rejecting Data
Zallenrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 55
Lecture 3 Slide 55
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly
ldquounreasonablerdquo data point from a set of randomly distributed
measurements
bull Whenever it makes things look better
Rejecting Data
Disneyrsquos Criterion
Disneyrsquos First Law
Wishing will make it so
Disneyrsquos Second Law
Dreams are more colorful than reality
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 56
Lecture 3 Slide 56
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Data may be rejected if the expected number of measurements at
least as deviant as the suspect measurement is less than 50
Consider a set of N measurements of a single quantity
x1x2 helliphellipxN
Calculate ltxgt and σx and then determine the fractional deviations from the mean of all
the points
For the suspect point(s) xsuspect find the probability of such a point occurring in N
measurements
n = (expected number as deviant as xsuspect)
= N Prob(outside xsuspectσx)
Rejecting Data
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 57
Lecture 3 Slide 57
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Function of Gaussian Distribution
Area under curve
(probability that
ndashtσltxlt+tσ) is given
in Table at right
Probable Error (probability that
ndash067σltxlt+067σ) is 50
Error Function (probability that ndashtσltxlt+tσ )
Error Function Tabulated values-see App A
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 58
Lecture 3 Slide 58
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 59
Lecture 3 Slide 59
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashExample 1
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 60
Lecture 3 Slide 60
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Details (1)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 61
Lecture 3 Slide 61
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashDetails (2)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 62
Lecture 3 Slide 62
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos
Criterionmdash
Details (3)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 63
Lecture 3 Slide 63
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos
Criterionmdash
Example 2
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 64
Lecture 3 Slide 64
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Summary of Probability Theory
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 65
Lecture 3 Slide 65
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of
Probability
Theory-I
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 66
Lecture 3 Slide 66
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of
Probability
Theory-II
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 67
Lecture 3 Slide 67
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-III
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 68
Lecture 3 Slide 68
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-IV
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 5
Lecture 3 Slide 5
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Flip a penny 50 times and record the results
Two Practical Exercises in Probabilities
What is the asymmetry of the results
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 6
Lecture 3 Slide 6
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Two Practical Exercises in Probabilities
Flip a penny 50 times and record the results
What is the asymmetry of the results
4 asymmetry asymmetry
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 7
Lecture 3 Slide 7
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Two Practical Exercises in Probabilities
Roll a pair of dice 50 times and record the results
What is the mean value
The standard deviation
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 8
Lecture 3 Slide 8
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Two Practical Exercises in Probabilities
Roll a pair of dice 50 times and record the results
What is the mean value
The standard deviation
What is the asymmetry
(kurtosis)
What is the probability of
rolling a 4
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 9
Lecture 3 Slide 9
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Discrete Distribution Functions
A data set to play with Written in terms of ldquooccurrencerdquo F
The mean value In terms of fractional expectations
Fractional expectations
Normalization condition
Mean value
(This is just a weighted sum)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 10
Lecture 3 Slide 10
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Limit of Discrete Distribution Functions
ldquoNormalizingrdquo data sets
Binned data sets
fk equiv fractional occurrence
∆k equiv bin width
Mean value
Normalization
Expected value 119875119903119900119887 4 =1198654
119873= 1198914
119883 = 119865119896
119896
119909119896 = (119891119896119896
Δ119896)119909119896
1 = (119891119896119896 Δ119896)119909119896
Δ119896119896
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 11
Lecture 3 Slide 11
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Limits of Distribution Functions
Consider the limiting distribution function as N ȸ and ∆k0
Larger data sets
Mathcad Games
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 12
Lecture 3 Slide 12
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Continuous Distribution Functions
Central (mean) Value Width of Distribution
Meaning of Distribution Interval
Normalization of Distribution
Thus
and by extension
= fraction of measurements
that fall within altxltb
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 13
Lecture 3 Slide 13
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Moments of Distribution Functions
The first moment for a probability distribution function is
119909 equiv 119909 = 119891119894119903119904119905 119898119900119898119890119899119905 = 119909 119891(119909)+infin
minusinfin119889119909
For a general distribution function
119909 equiv 119909 = 119891119894119903119904119905 119898119900119898119890119899119905 = 119909 119892(119909)
+infinminusinfin
119889119909
119892(119909)+infinminusinfin
119889119909
Generalizing the n
th moment is
119909119899 equiv 119909119899 = 119899119905ℎ 119898119900119898119890119899119905 = 119909119899 119892(119909)
+infinminusinfin
119889119909
119892(119909)+infinminusinfin
119889119909= 119909119899 119891(119909)
+infin
minusinfin119889119909
Oth moment equiv N 2
nd moment equiv 119909 minus 119909 2 rarr 1199092
1st moment equiv 119909 3
rd moment equiv kurtosis
0 (for a centered distribution)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 14
Lecture 3 Slide 14
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Moments of Distribution Functions
Generalizing the nth
moment is
119909119899 equiv 119909119899 = 119899119905ℎ 119898119900119898119890119899119905 = 119909119899 119892(119909)
+infinminusinfin
119889119909
119892(119909)+infinminusinfin
119889119909= 119909119899 119891(119909)
+infin
minusinfin119889119909
Oth
moment equiv N 2nd
moment equiv 119909 minus 119909 2 rarr 1199092
1st moment equiv 119909 3
rd moment equiv kurtosis
The nth
moment about the mean is
120583119899 equiv 119909 minus 119909 119899 = 119899119905ℎ 119898119900119898119890119899119905 119886119887119900119906119905 119905ℎ119890 119898119890119886119899
= 119909minus119909 119899 119892(119909)
+infinminusinfin
119889119909
119892(119909)+infinminusinfin
119889119909= 119909 minus 119909 119899 119891(119909)
+infin
minusinfin119889119909
The standard deviation (or second moment about the mean) is
1205901199092 equiv 1205832 equiv 119909 minus 119909 2 = 2119899119889 119898119900119898119890119899119905 119886119887119900119906119905 119905ℎ119890 119898119890119886119899
= 119909minus119909 2 119892(119909)
+infinminusinfin
119889119909
119892(119909)+infinminusinfin
119889119909= 119909 minus 119909 2 119891(119909)
+infin
minusinfin119889119909
0
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 15
Lecture 3 Slide 15
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Example of Continuous Distribution Functions and Expectation Values Harmonic Oscillator Example from Mechanics
Expected Values
The expectation value of a function Ɍ(x) is
Ɍ x equiv Ɍ x 119892 119909
+infin
minusinfin119889119909
119892 119909 +infin
minusinfin119889119909
= Ɍ(x) 119891(119909)+infin
minusinfin119889119909
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 16
Lecture 3 Slide 16
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Example of Continuous Distribution Functions and Expectation Values Boltzmann Distribution Example from Kinetic Theory
Expected Values
The expectation value of a function
Ɍ(x) is
Ɍ x equiv Ɍ x 119892 119909
+infin
minusinfin119889119909
119892 119909 +infin
minusinfin119889119909
= Ɍ(x) 119891(119909)+infin
minusinfin119889119909
The Boltzmann distribution function for velocities of particles as
a function of temperature T is
119875 119907 119879 = 4120587 119872
2120587 119896119861119879
3 2
1199072119890119909119901
121198721199072
12119896119861119879
Then
v = 119907 119875(119907)+infin
minusinfin119889119907 =
8 119896119861119879120587 119872
1 2
v2 = v2 119875(119907)+infin
minusinfin119889119907 =
3 119896119861119879 119872
1 2
implies KE = 1
2119872 v2 =
3
2119896119861119879
vpeak=
2 119896119861119879 119872
1 2
= 2 3
1 2
v2
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 17
Lecture 3 Slide 17
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Example of Continuous Distribution Functions and Expectation Values Fermi-Dirac Distribution Example from Kinetic Theory
For a system of identical fermions the average number of fermions in a single-particle state is given
by the FermindashDirac (FndashD) distribution
where kB is Boltzmanns constant T is the absolute temperature is the energy of the single-particle
state and μ is the total chemical potential
Since the FndashD distribution was derived using the Pauli exclusion principle which allows at most one
electron to occupy each possible state a result is that
When a quasi-continuum of energies has an associated density of states (ie the number of
states per unit energy range per unit volume) the average number of fermions per unit energy range
per unit volume is
where is called the Fermi function
so that
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 18
Lecture 3 Slide 18
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Example of Continuous Distribution Functions and Expectation Values Finite Square Well Example from Quantum Mechanics
Expectation Values The expectation value of a QM operator O(x) is O x equiv Ψlowast x 119874 119909
+infin
minusinfinΨ x 119889119909
Ψlowast x Ψ x +infin
minusinfin119889119909
For a finite square well of width L Ψn x = 2L sin
n π x
L
Ψnlowast x |Ψn x equiv
Ψnlowast x 119874 119909
+infin
minusinfinΨn x 119889119909
Ψnlowast x Ψn x
+infin
minusinfin119889119909
= 1
119909 = Ψnlowast x |x|Ψn x equiv
Ψnlowast x 119909
+infin
minusinfinΨn x 119889119909
Ψnlowast x Ψn x
+infin
minusinfin119889119909
= 1198712
119901 = Ψnlowast x |
ℏ
i
part
partx|Ψn x equiv
Ψnlowast x
ℏi
partpartx
+infin
minusinfinΨn x 119889119909
Ψnlowast x Ψn x
+infin
minusinfin119889119909
= 0
119864119899 = Ψnlowast x |iℏ
part
partt|Ψn x equiv
Ψnlowast x iℏ
partpartt
+infin
minusinfinΨn x 119889119909
Ψnlowast x Ψn x
+infin
minusinfin119889119909
=11989921205872ℏ2
2 119898 1198712
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 19
Lecture 3 Slide 19
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Distribution Functions
Available
on web site
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 20
Lecture 3 Slide 20
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
The Gaussian Distribution
Function
References Taylor Ch 5
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 21
Lecture 3 Slide 21
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Gaussian Integrals
Factorial Approximations
119899 asymp 2120587119899 1 2 119899119899 119890119909119901 minus119899 + 1
12 119899+ 119874
1
1198992
119897119900119892 119899 asymp 1
2 119897119900119892 2120587 + 119899 +
1
2 log 119899 minus 119899 +
1
12 119899+ 119874
1
1198992
119897119900119892 119899 asymp 119899 log 119899 minus 119899 (for all terms decreasing faster than linearly with n)
Gaussian Integrals
119868119898 = 2 119909119898 exp(minus1199092)infin
0 119889119909 mgt-1
119868119898 = 2 119910119899 exp(minus119910)infin
0 119889119910 equiv Γ(119899 + 1) 1199092 equiv 119910 2 119889119909 = 1199101 2 119889119910 119899 equiv 1
2 (119898 minus 1)
1198680 = Γ 119899 =1
2 = 120587 m=0 119899 = minus1
2
1198682 119896 = Γ 119896 +1
2 = 119896 minus 1
2 119896 minus 32 3
2 12 120587 even m m=2 kgt0 119899 = 119896 minus 1
2
1198682 119896+1 = Γ 119896 + 1 = 119896 odd m m=2 k+1gt0 119899 = 119896 ge 0
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 22
Lecture 3 Slide 22
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Gaussian Distribution Function
Width of Distribution
(standard deviation)
Center of Distribution
(mean) Distribution
Function
Independent
Variable
Gaussian Distribution Function
Normalization
Constant
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 23
Lecture 3 Slide 23
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Effects of Increasing N on Gaussian Distribution Functions
Consider the limiting distribution function as N infin and dx0
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 24
Lecture 3 Slide 24
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Defining the Gaussian
distribution function
in Mathcad
I suggest you ldquoinvestigaterdquo
these with the Mathcad sheet
on the web site
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 25
Lecture 3 Slide 25
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Using Mathcad to
define other common
distribution functions
Consider the limiting
distribution function
as N ȸ and ∆k0
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 26
Lecture 3 Slide 26
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider the Gaussian distribution function
Use the normalization condition to
evaluate the normalization constant
(see Taylor p 132)
The mean Ẋ is the first moment
of the Gaussian distribution
function (see Taylor p 134)
The standard deviation σx is the standard
deviation of the mean of the Gaussian
distribution function (see Taylor p 143)
Gaussian Distribution Moments
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 27
Lecture 3 Slide 27
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
When is mean x not Xbest
Answer When the
distribution is not
symmetric about X
Example Cauchy
Distribution
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 28
Lecture 3 Slide 28
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
When is mean x not Xbest
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 29
Lecture 3 Slide 29
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
When is mean x not Xbest
Answer When the
distribution is has
more than one peak
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 30
Lecture 3 Slide 30
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
The Gaussian Distribution
Function and Its Relation to
Errors
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 31
Lecture 3 Slide 31
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
A Review of Probabilities in Combination
Probability of a data set of N like measurements (x1x2hellipxN)
P (x1x2hellipxN) = P(x)1P(x2)hellipP(xN)
1 head AND 1 Four
P(H4) = P(H) P(4)
1 head OR 1 Four
P(H4) = P(H) + P(4) - P(H and 4) (true for a ldquonon-mutually exclusiverdquo events)
NOT 1 Six
P(NOT 6) = 1 - P(6)
1 Six OR 1 Four
P(64) = P(6) + P(4) (true for a ldquomutually exclusiverdquo single role)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 32
Lecture 3 Slide 32
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
The Gaussian Distribution Function and Its Relation to Errors
We will use the Gaussian distribution as applied to random
variables to develop the mathematical basis for bull Mean
bull Standard deviation
bull Standard deviation of the mean (SDOM)
bull Moments and expectation values
bull Error propagation formulas
bull Addition of errors in quadrature (for independent and random
measurements)
bull Numerical values for confidence limits (t-test)
bull Principle of maximal likelihood
bull Central limit theorem
bull Weighted distributions and Chi squared
bull Schwartz inequality (ie the uncertainty principle) (next lecture)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 33
Lecture 3 Slide 33
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider the Gaussian distribution function
Use the normalization condition to
evaluate the normalization constant
(see Taylor p 132)
The mean Ẋ is the first moment
of the Gaussian distribution
function (see Taylor p 134)
The standard deviation σx is the standard
deviation of the mean of the Gaussian
distribution function (see Taylor p 143)
Gaussian Distribution Moments
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 34
Lecture 3 Slide 34
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Standard Deviation of Gaussian Distribution
Area under curve
(probability that
ndashσltxlt+σ) is 68
5 or ~2σ ldquoSignificantrdquo
1 or ~3σ ldquoHighly Significantrdquo
1σ ldquoWithin errorsrdquo
5 ppm or ~5σ ldquoValid for HEPrdquo
See Sec 106 Testing of
Hypotheses
More complete Table in App A and B
Ah thatrsquos
highly
significant
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 35
Lecture 3 Slide 35
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Function of Gaussian Distribution
Area under curve
(probability that
ndashtσltxlt+tσ) is given
in Table at right
Probable Error (probability that
ndash067σltxlt+067σ) is 50
Error Function (probability that ndashtσltxlt+tσ )
Error Function Tabulated values-see App A
Complementary Error Function
(probability that ndashxlt-tσ AND xgt+tσ )
Prob(x outside tσ) = 1 - Prob(x within tσ)
More complete Table in App A and B
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 36
Lecture 3 Slide 36
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Useful Points on Gaussian Distribution
Full Width at Half Maximum FWHM
(See Prob 512)
Points of Inflection Occur at plusmnσ
(See Prob 513)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 37
Lecture 3 Slide 37
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Analysis and Gaussian Distribution
Adding a Constant Multiplying by a Constant
XX+A XBX and σB σ
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 38
Lecture 3 Slide 38
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Sum of Two Variables
Consider the derived quantity
Z=X + Y
(with X=0 and Y=0)
Error Propagation Addition
Error in Z
Multiple two probabilities 119875119903119900119887 119909 119910 = 119875119903119900119887 119909 ∙ 119875119903119900119887 119909 prop 119890119909119901 minus
1
2
1199092
1205901199092+
1199102
1205901199102
prop 119890119909119901 minus1
2
(119909+119910)2
1205901199092+120590119910
2 minus1199112
prop 119890119909119901 minus1
2
(119909+119910)2
1205901199092+120590119910
2 119890119909119901 minus
1199112
2
ICBST (Eq 553)
X+YZ and σx2 + σy
2 σz
2
(addition in quadrature for random independent variables)
Integrates to radic2π
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 39
Lecture 3 Slide 39
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Error Propagation
General formula for error propagation see [Taylor Secs 35 and 39]
Uncertainty as a function of one variable [Taylor Sec 35]
1 Consider a graphical method of estimating error
a) Consider an arbitaray function q(x)
b) Plot q(x) vs x
c) On the graph label
(1) qbest = q(xbest)
(2) qhi = q(xbest + δx)
(3) qlow = q(xbest- δx)
d) Making a linear approximation
x
qqxslopeqq
x
qqxslopeqq
bestbestlow
bestbesthi
e) Therefore
xx
Note the absolute value
How do we determine the error of a derived quantity Z(XYhellip) from errors in XYhellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 40
Lecture 3 Slide 40
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Error Propagation
General formula for uncertainty of a function of one variable
xx
[Taylor Eq 323]
Can you now derive for specific rules of error propagation
1 Addition and Subtraction [Taylor p 49]
2 Multiplication and Division [Taylor p 51]
3 Multiplication by a constant (exact number) [Taylor p 54]
4 Exponentiation (powers) [Taylor p 56]
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 41
Lecture 3 Slide 41
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Multiple Variables
Uncertainty of a function of multiple variables [Taylor Sec 311]
1 It can easily (no really) be shown that (see Taylor Sec 311) for a
function of several variables
)(
z
z
qy
y
qx
x
qzyxq
[Taylor Eq 347]
2 More correctly it can be shown that (see Taylor Sec 311) for a
function of several variables
)(
z
z
qy
y
qx
x
qzyxq
[Taylor Eq 347]
where the equals sign represents an upper bound as discussed above
3 For a function of several independent and random variables
)(
222
z
z
qy
y
qx
x
qzyxq
[Taylor Eq 348]
Again the proof is left for Ch 5
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 42
Lecture 3 Slide 42
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Product of Two Variables
Consider the arbitrary derived quantity
q(xy) of two independent random variables x and y
Expand q(xy) in a Taylor series about the expected values of x and y
(ie at points near X and Y)
Error Propagation General Case
119902 119909 119910 = 119902 119883 119884 + 120597119902
120597119909
119883 119909 minus 119883 +
120597119902
120597119910
119884
(119910 minus 119884)
Fixed shifts peak of distribution
Fixed Distribution centered at X with width σX
120575119902 119909 119910 = 120590119902 = 119902 119883 119884 + 120597119902
120597119909
119883120590119909
2
+ 120597119902
120597119910
119884
120590119910
2
0
How do we determine the error of a derived quantity Z(XYhellip) from errors in XYhellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 43
Lecture 3 Slide 43
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
SDOM of Gaussian Distribution
Standard Deviation of the Mean
The SDOM decreases as the square root of the number of measurements
That is the relative width σẊ of the distribution gets narrower as more
measurements are made
Each measurement
has similar σXi= σẊ
and similar partial
derivatives
Thushellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 44
Lecture 3 Slide 44
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Two Key Theorems from Probability
Central Limit Theorem
For random independent measurements (each with a well-define
expectation value and well-defined variance) the arithmetic mean
(average) will be approximately normally distributed
Principle of Maximum Likelihood
Given the N observed measurements x1 x2hellipxN the best estimates
for Ẋ and σ are those values for which the observed x1 x2hellipxN are
most likely
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 45
Lecture 3 Slide 45
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Mean of Gaussian Distribution as ldquoBest Estimaterdquo
Principle of Maximum Likelihood
Best Estimate of X is from maximum probability or minimum summation
Consider a data set
Each randomly distributed with
x1 x2 x3 hellipxN
To find the most likely value of the mean (the best estimate of ẋ)
find X that yields the highest probability for the data set
The combined probability for the full data set is the product
Minimize
Sum
Solve for
derivative wrst
X set to 0
Best
estimate
of X
prop1
120590119890minus(1199091minus119883)2 2120590 times
1
120590119890minus(1199092minus119883)2 2120590 times hellip times
1
120590119890minus(119909119873minus119883)2 2120590 =
1
120590119873119890 minus(119909119894minus119883)2 2120590
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 46
Lecture 3 Slide 46
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Uncertainty of ldquoBest Estimatesrdquo of Gaussian Distribution Principle of Maximum Likelihood
Best Estimate of X is from maximum probability or minimum summation
Consider a data set x1 x2 x3 hellipxN
To find the most likely value of the standard deviation (the best estimate of
the width of the x distribution) find σx that yields the highest probability for
the data set
The combined probability for the full data set is the product
Minimize
Sum
Solve for
derivative
wrst X set to 0
Best
estimate
of X
Best Estimate of σ is from maximum probability or minimum summation
Minimize
Sum
Best
estimate
of σ
Solve for
derivative
wrst σ set to 0
See
Prob 526
prop1
120590119890minus(1199091minus119883)2 2120590 times
1
120590119890minus(1199092minus119883)2 2120590 times hellip times
1
120590119890minus(119909119873minus119883)2 2120590 =
1
120590119873119890 minus(119909119894minus119883)2 2120590
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 47
Lecture 3 Slide 47
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Combining Data Sets
Weighted Averages
References Taylor Ch 7
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 48
Lecture 3 Slide 48
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Weighted Averages
Question How can we properly combine two or more separate
independent measurements of the same randomly distributed
quantity to determine a best combined value with uncertainty
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 49
Lecture 3 Slide 49
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider two measurements of the same quantity described by a random Gaussian distribution
ltx1gt x1 and ltx2gt x2
The probability of measuring two such measurements is
119875119903119900119887119909 1199091 1199092 = 119875119903119900119887119909 1199091 119875119903119900119887119909 1199092
= 1
12059011205902119890minus12059422 119908ℎ119890119903119890 1205942 equiv
1199091 minus 119883
1205901 2 +
1199092 minus 119883
1205902 2
To find the best value for χ find the maximum Prob or minimum χ2
Weighted Averages
Assume negligible systematic errors
Note χ2 or Chi squared is the sum of the squares of the deviations from the mean divided by
the corresponding uncertainty
Such methods are called ldquoMethods of Least Squaresrdquo They follow directly from the Principle of
Maximum Likelihood
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 50
Lecture 3 Slide 50
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
The probability of measuring two such measurements is
119875119903119900119887119909 1199091 1199092 = 119875119903119900119887119909 1199091 119875119903119900119887119909 1199092
= 1
12059011205902119890minus12059422 119908ℎ119890119903119890 1205942 equiv
1199091 minus 119883
1205901 2 +
1199092 minus 119883
1205902 2
To find the best value for χ find the maximum Prob or minimum χ2
Weighted Averages
This leads to 119909119882_119886119907119892 = 11990811199091+11990821199092
1199081+1199082=
119908119894 119909119894
119908119894 119908ℎ119890119903119890 119908119894 = 1
120590119894 2
Best Estimate of χ is from maximum probibility or minimum summation
Minimize Sum Solve for best estimate of χ Solve for derivative wrst χ set to 0
Note If w1=w2 we recover the standard result Xwavg= (12) (x1+x2)
Finally the width of a weighted average distribution is
120590119908119894119890119892 ℎ119905119890119889 119886119907119892 = 1
119908119894119894
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 51
Lecture 3 Slide 51
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Weighted Averages on Steroids
A very powerful method for combining data from different sources
with different methods and uncertainties (or indeed data of
related measured and calculated quantities) is Kalman filtering
The Kalman filter also known as
linear quadratic estimation (LQE)
is an algorithm that uses a series
of measurements observed over
time containing noise (random
variations) and other
inaccuracies and produces
estimates of unknown variables
that tend to be more precise than
those based on a single
measurement alone
The Kalman filter keeps track of the estimated state of the system and the variance or uncertainty of the
estimate The estimate is updated using a state transition model and measurements xk|k-1 denotes the
estimate of the systems state at time step k before the kth measurement yk has been taken into account Pk|k-1
is the corresponding uncertainty --Wikipedia 2013
Ludger Scherliess of USU Physics is a world expert at using Kalman filtering for the assimilation
of satellite and ground-based data and the USU GAMES model to predict space weather
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 52
Lecture 3 Slide 52
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Rejecting Data
Chauvenetrsquos Criterion
References Taylor Ch 6
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 53
Lecture 3 Slide 53
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly
ldquounreasonablerdquo data point from a set of randomly distributed
measurements
bull Never
bull Whenever it makes things look better
bull Chauvenetrsquos criterion provides a (quantitative) compromise
Rejecting Data
What is a good criteria for
rejecting data
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 54
Lecture 3 Slide 54
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly
ldquounreasonablerdquo data point from a set of randomly distributed
measurements
bull Never
Rejecting Data
Zallenrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 55
Lecture 3 Slide 55
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly
ldquounreasonablerdquo data point from a set of randomly distributed
measurements
bull Whenever it makes things look better
Rejecting Data
Disneyrsquos Criterion
Disneyrsquos First Law
Wishing will make it so
Disneyrsquos Second Law
Dreams are more colorful than reality
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 56
Lecture 3 Slide 56
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Data may be rejected if the expected number of measurements at
least as deviant as the suspect measurement is less than 50
Consider a set of N measurements of a single quantity
x1x2 helliphellipxN
Calculate ltxgt and σx and then determine the fractional deviations from the mean of all
the points
For the suspect point(s) xsuspect find the probability of such a point occurring in N
measurements
n = (expected number as deviant as xsuspect)
= N Prob(outside xsuspectσx)
Rejecting Data
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 57
Lecture 3 Slide 57
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Function of Gaussian Distribution
Area under curve
(probability that
ndashtσltxlt+tσ) is given
in Table at right
Probable Error (probability that
ndash067σltxlt+067σ) is 50
Error Function (probability that ndashtσltxlt+tσ )
Error Function Tabulated values-see App A
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 58
Lecture 3 Slide 58
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 59
Lecture 3 Slide 59
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashExample 1
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 60
Lecture 3 Slide 60
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Details (1)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 61
Lecture 3 Slide 61
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashDetails (2)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 62
Lecture 3 Slide 62
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos
Criterionmdash
Details (3)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 63
Lecture 3 Slide 63
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos
Criterionmdash
Example 2
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 64
Lecture 3 Slide 64
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Summary of Probability Theory
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 65
Lecture 3 Slide 65
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of
Probability
Theory-I
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 66
Lecture 3 Slide 66
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of
Probability
Theory-II
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 67
Lecture 3 Slide 67
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-III
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 68
Lecture 3 Slide 68
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-IV
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 6
Lecture 3 Slide 6
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Two Practical Exercises in Probabilities
Flip a penny 50 times and record the results
What is the asymmetry of the results
4 asymmetry asymmetry
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 7
Lecture 3 Slide 7
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Two Practical Exercises in Probabilities
Roll a pair of dice 50 times and record the results
What is the mean value
The standard deviation
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 8
Lecture 3 Slide 8
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Two Practical Exercises in Probabilities
Roll a pair of dice 50 times and record the results
What is the mean value
The standard deviation
What is the asymmetry
(kurtosis)
What is the probability of
rolling a 4
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 9
Lecture 3 Slide 9
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Discrete Distribution Functions
A data set to play with Written in terms of ldquooccurrencerdquo F
The mean value In terms of fractional expectations
Fractional expectations
Normalization condition
Mean value
(This is just a weighted sum)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 10
Lecture 3 Slide 10
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Limit of Discrete Distribution Functions
ldquoNormalizingrdquo data sets
Binned data sets
fk equiv fractional occurrence
∆k equiv bin width
Mean value
Normalization
Expected value 119875119903119900119887 4 =1198654
119873= 1198914
119883 = 119865119896
119896
119909119896 = (119891119896119896
Δ119896)119909119896
1 = (119891119896119896 Δ119896)119909119896
Δ119896119896
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 11
Lecture 3 Slide 11
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Limits of Distribution Functions
Consider the limiting distribution function as N ȸ and ∆k0
Larger data sets
Mathcad Games
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 12
Lecture 3 Slide 12
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Continuous Distribution Functions
Central (mean) Value Width of Distribution
Meaning of Distribution Interval
Normalization of Distribution
Thus
and by extension
= fraction of measurements
that fall within altxltb
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 13
Lecture 3 Slide 13
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Moments of Distribution Functions
The first moment for a probability distribution function is
119909 equiv 119909 = 119891119894119903119904119905 119898119900119898119890119899119905 = 119909 119891(119909)+infin
minusinfin119889119909
For a general distribution function
119909 equiv 119909 = 119891119894119903119904119905 119898119900119898119890119899119905 = 119909 119892(119909)
+infinminusinfin
119889119909
119892(119909)+infinminusinfin
119889119909
Generalizing the n
th moment is
119909119899 equiv 119909119899 = 119899119905ℎ 119898119900119898119890119899119905 = 119909119899 119892(119909)
+infinminusinfin
119889119909
119892(119909)+infinminusinfin
119889119909= 119909119899 119891(119909)
+infin
minusinfin119889119909
Oth moment equiv N 2
nd moment equiv 119909 minus 119909 2 rarr 1199092
1st moment equiv 119909 3
rd moment equiv kurtosis
0 (for a centered distribution)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 14
Lecture 3 Slide 14
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Moments of Distribution Functions
Generalizing the nth
moment is
119909119899 equiv 119909119899 = 119899119905ℎ 119898119900119898119890119899119905 = 119909119899 119892(119909)
+infinminusinfin
119889119909
119892(119909)+infinminusinfin
119889119909= 119909119899 119891(119909)
+infin
minusinfin119889119909
Oth
moment equiv N 2nd
moment equiv 119909 minus 119909 2 rarr 1199092
1st moment equiv 119909 3
rd moment equiv kurtosis
The nth
moment about the mean is
120583119899 equiv 119909 minus 119909 119899 = 119899119905ℎ 119898119900119898119890119899119905 119886119887119900119906119905 119905ℎ119890 119898119890119886119899
= 119909minus119909 119899 119892(119909)
+infinminusinfin
119889119909
119892(119909)+infinminusinfin
119889119909= 119909 minus 119909 119899 119891(119909)
+infin
minusinfin119889119909
The standard deviation (or second moment about the mean) is
1205901199092 equiv 1205832 equiv 119909 minus 119909 2 = 2119899119889 119898119900119898119890119899119905 119886119887119900119906119905 119905ℎ119890 119898119890119886119899
= 119909minus119909 2 119892(119909)
+infinminusinfin
119889119909
119892(119909)+infinminusinfin
119889119909= 119909 minus 119909 2 119891(119909)
+infin
minusinfin119889119909
0
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 15
Lecture 3 Slide 15
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Example of Continuous Distribution Functions and Expectation Values Harmonic Oscillator Example from Mechanics
Expected Values
The expectation value of a function Ɍ(x) is
Ɍ x equiv Ɍ x 119892 119909
+infin
minusinfin119889119909
119892 119909 +infin
minusinfin119889119909
= Ɍ(x) 119891(119909)+infin
minusinfin119889119909
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 16
Lecture 3 Slide 16
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Example of Continuous Distribution Functions and Expectation Values Boltzmann Distribution Example from Kinetic Theory
Expected Values
The expectation value of a function
Ɍ(x) is
Ɍ x equiv Ɍ x 119892 119909
+infin
minusinfin119889119909
119892 119909 +infin
minusinfin119889119909
= Ɍ(x) 119891(119909)+infin
minusinfin119889119909
The Boltzmann distribution function for velocities of particles as
a function of temperature T is
119875 119907 119879 = 4120587 119872
2120587 119896119861119879
3 2
1199072119890119909119901
121198721199072
12119896119861119879
Then
v = 119907 119875(119907)+infin
minusinfin119889119907 =
8 119896119861119879120587 119872
1 2
v2 = v2 119875(119907)+infin
minusinfin119889119907 =
3 119896119861119879 119872
1 2
implies KE = 1
2119872 v2 =
3
2119896119861119879
vpeak=
2 119896119861119879 119872
1 2
= 2 3
1 2
v2
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 17
Lecture 3 Slide 17
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Example of Continuous Distribution Functions and Expectation Values Fermi-Dirac Distribution Example from Kinetic Theory
For a system of identical fermions the average number of fermions in a single-particle state is given
by the FermindashDirac (FndashD) distribution
where kB is Boltzmanns constant T is the absolute temperature is the energy of the single-particle
state and μ is the total chemical potential
Since the FndashD distribution was derived using the Pauli exclusion principle which allows at most one
electron to occupy each possible state a result is that
When a quasi-continuum of energies has an associated density of states (ie the number of
states per unit energy range per unit volume) the average number of fermions per unit energy range
per unit volume is
where is called the Fermi function
so that
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 18
Lecture 3 Slide 18
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Example of Continuous Distribution Functions and Expectation Values Finite Square Well Example from Quantum Mechanics
Expectation Values The expectation value of a QM operator O(x) is O x equiv Ψlowast x 119874 119909
+infin
minusinfinΨ x 119889119909
Ψlowast x Ψ x +infin
minusinfin119889119909
For a finite square well of width L Ψn x = 2L sin
n π x
L
Ψnlowast x |Ψn x equiv
Ψnlowast x 119874 119909
+infin
minusinfinΨn x 119889119909
Ψnlowast x Ψn x
+infin
minusinfin119889119909
= 1
119909 = Ψnlowast x |x|Ψn x equiv
Ψnlowast x 119909
+infin
minusinfinΨn x 119889119909
Ψnlowast x Ψn x
+infin
minusinfin119889119909
= 1198712
119901 = Ψnlowast x |
ℏ
i
part
partx|Ψn x equiv
Ψnlowast x
ℏi
partpartx
+infin
minusinfinΨn x 119889119909
Ψnlowast x Ψn x
+infin
minusinfin119889119909
= 0
119864119899 = Ψnlowast x |iℏ
part
partt|Ψn x equiv
Ψnlowast x iℏ
partpartt
+infin
minusinfinΨn x 119889119909
Ψnlowast x Ψn x
+infin
minusinfin119889119909
=11989921205872ℏ2
2 119898 1198712
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 19
Lecture 3 Slide 19
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Distribution Functions
Available
on web site
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 20
Lecture 3 Slide 20
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
The Gaussian Distribution
Function
References Taylor Ch 5
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 21
Lecture 3 Slide 21
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Gaussian Integrals
Factorial Approximations
119899 asymp 2120587119899 1 2 119899119899 119890119909119901 minus119899 + 1
12 119899+ 119874
1
1198992
119897119900119892 119899 asymp 1
2 119897119900119892 2120587 + 119899 +
1
2 log 119899 minus 119899 +
1
12 119899+ 119874
1
1198992
119897119900119892 119899 asymp 119899 log 119899 minus 119899 (for all terms decreasing faster than linearly with n)
Gaussian Integrals
119868119898 = 2 119909119898 exp(minus1199092)infin
0 119889119909 mgt-1
119868119898 = 2 119910119899 exp(minus119910)infin
0 119889119910 equiv Γ(119899 + 1) 1199092 equiv 119910 2 119889119909 = 1199101 2 119889119910 119899 equiv 1
2 (119898 minus 1)
1198680 = Γ 119899 =1
2 = 120587 m=0 119899 = minus1
2
1198682 119896 = Γ 119896 +1
2 = 119896 minus 1
2 119896 minus 32 3
2 12 120587 even m m=2 kgt0 119899 = 119896 minus 1
2
1198682 119896+1 = Γ 119896 + 1 = 119896 odd m m=2 k+1gt0 119899 = 119896 ge 0
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 22
Lecture 3 Slide 22
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Gaussian Distribution Function
Width of Distribution
(standard deviation)
Center of Distribution
(mean) Distribution
Function
Independent
Variable
Gaussian Distribution Function
Normalization
Constant
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 23
Lecture 3 Slide 23
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Effects of Increasing N on Gaussian Distribution Functions
Consider the limiting distribution function as N infin and dx0
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 24
Lecture 3 Slide 24
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Defining the Gaussian
distribution function
in Mathcad
I suggest you ldquoinvestigaterdquo
these with the Mathcad sheet
on the web site
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 25
Lecture 3 Slide 25
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Using Mathcad to
define other common
distribution functions
Consider the limiting
distribution function
as N ȸ and ∆k0
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 26
Lecture 3 Slide 26
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider the Gaussian distribution function
Use the normalization condition to
evaluate the normalization constant
(see Taylor p 132)
The mean Ẋ is the first moment
of the Gaussian distribution
function (see Taylor p 134)
The standard deviation σx is the standard
deviation of the mean of the Gaussian
distribution function (see Taylor p 143)
Gaussian Distribution Moments
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 27
Lecture 3 Slide 27
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
When is mean x not Xbest
Answer When the
distribution is not
symmetric about X
Example Cauchy
Distribution
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 28
Lecture 3 Slide 28
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
When is mean x not Xbest
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 29
Lecture 3 Slide 29
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
When is mean x not Xbest
Answer When the
distribution is has
more than one peak
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 30
Lecture 3 Slide 30
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
The Gaussian Distribution
Function and Its Relation to
Errors
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 31
Lecture 3 Slide 31
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
A Review of Probabilities in Combination
Probability of a data set of N like measurements (x1x2hellipxN)
P (x1x2hellipxN) = P(x)1P(x2)hellipP(xN)
1 head AND 1 Four
P(H4) = P(H) P(4)
1 head OR 1 Four
P(H4) = P(H) + P(4) - P(H and 4) (true for a ldquonon-mutually exclusiverdquo events)
NOT 1 Six
P(NOT 6) = 1 - P(6)
1 Six OR 1 Four
P(64) = P(6) + P(4) (true for a ldquomutually exclusiverdquo single role)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 32
Lecture 3 Slide 32
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
The Gaussian Distribution Function and Its Relation to Errors
We will use the Gaussian distribution as applied to random
variables to develop the mathematical basis for bull Mean
bull Standard deviation
bull Standard deviation of the mean (SDOM)
bull Moments and expectation values
bull Error propagation formulas
bull Addition of errors in quadrature (for independent and random
measurements)
bull Numerical values for confidence limits (t-test)
bull Principle of maximal likelihood
bull Central limit theorem
bull Weighted distributions and Chi squared
bull Schwartz inequality (ie the uncertainty principle) (next lecture)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 33
Lecture 3 Slide 33
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider the Gaussian distribution function
Use the normalization condition to
evaluate the normalization constant
(see Taylor p 132)
The mean Ẋ is the first moment
of the Gaussian distribution
function (see Taylor p 134)
The standard deviation σx is the standard
deviation of the mean of the Gaussian
distribution function (see Taylor p 143)
Gaussian Distribution Moments
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 34
Lecture 3 Slide 34
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Standard Deviation of Gaussian Distribution
Area under curve
(probability that
ndashσltxlt+σ) is 68
5 or ~2σ ldquoSignificantrdquo
1 or ~3σ ldquoHighly Significantrdquo
1σ ldquoWithin errorsrdquo
5 ppm or ~5σ ldquoValid for HEPrdquo
See Sec 106 Testing of
Hypotheses
More complete Table in App A and B
Ah thatrsquos
highly
significant
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 35
Lecture 3 Slide 35
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Function of Gaussian Distribution
Area under curve
(probability that
ndashtσltxlt+tσ) is given
in Table at right
Probable Error (probability that
ndash067σltxlt+067σ) is 50
Error Function (probability that ndashtσltxlt+tσ )
Error Function Tabulated values-see App A
Complementary Error Function
(probability that ndashxlt-tσ AND xgt+tσ )
Prob(x outside tσ) = 1 - Prob(x within tσ)
More complete Table in App A and B
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 36
Lecture 3 Slide 36
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Useful Points on Gaussian Distribution
Full Width at Half Maximum FWHM
(See Prob 512)
Points of Inflection Occur at plusmnσ
(See Prob 513)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 37
Lecture 3 Slide 37
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Analysis and Gaussian Distribution
Adding a Constant Multiplying by a Constant
XX+A XBX and σB σ
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 38
Lecture 3 Slide 38
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Sum of Two Variables
Consider the derived quantity
Z=X + Y
(with X=0 and Y=0)
Error Propagation Addition
Error in Z
Multiple two probabilities 119875119903119900119887 119909 119910 = 119875119903119900119887 119909 ∙ 119875119903119900119887 119909 prop 119890119909119901 minus
1
2
1199092
1205901199092+
1199102
1205901199102
prop 119890119909119901 minus1
2
(119909+119910)2
1205901199092+120590119910
2 minus1199112
prop 119890119909119901 minus1
2
(119909+119910)2
1205901199092+120590119910
2 119890119909119901 minus
1199112
2
ICBST (Eq 553)
X+YZ and σx2 + σy
2 σz
2
(addition in quadrature for random independent variables)
Integrates to radic2π
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 39
Lecture 3 Slide 39
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Error Propagation
General formula for error propagation see [Taylor Secs 35 and 39]
Uncertainty as a function of one variable [Taylor Sec 35]
1 Consider a graphical method of estimating error
a) Consider an arbitaray function q(x)
b) Plot q(x) vs x
c) On the graph label
(1) qbest = q(xbest)
(2) qhi = q(xbest + δx)
(3) qlow = q(xbest- δx)
d) Making a linear approximation
x
qqxslopeqq
x
qqxslopeqq
bestbestlow
bestbesthi
e) Therefore
xx
Note the absolute value
How do we determine the error of a derived quantity Z(XYhellip) from errors in XYhellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 40
Lecture 3 Slide 40
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Error Propagation
General formula for uncertainty of a function of one variable
xx
[Taylor Eq 323]
Can you now derive for specific rules of error propagation
1 Addition and Subtraction [Taylor p 49]
2 Multiplication and Division [Taylor p 51]
3 Multiplication by a constant (exact number) [Taylor p 54]
4 Exponentiation (powers) [Taylor p 56]
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 41
Lecture 3 Slide 41
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Multiple Variables
Uncertainty of a function of multiple variables [Taylor Sec 311]
1 It can easily (no really) be shown that (see Taylor Sec 311) for a
function of several variables
)(
z
z
qy
y
qx
x
qzyxq
[Taylor Eq 347]
2 More correctly it can be shown that (see Taylor Sec 311) for a
function of several variables
)(
z
z
qy
y
qx
x
qzyxq
[Taylor Eq 347]
where the equals sign represents an upper bound as discussed above
3 For a function of several independent and random variables
)(
222
z
z
qy
y
qx
x
qzyxq
[Taylor Eq 348]
Again the proof is left for Ch 5
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 42
Lecture 3 Slide 42
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Product of Two Variables
Consider the arbitrary derived quantity
q(xy) of two independent random variables x and y
Expand q(xy) in a Taylor series about the expected values of x and y
(ie at points near X and Y)
Error Propagation General Case
119902 119909 119910 = 119902 119883 119884 + 120597119902
120597119909
119883 119909 minus 119883 +
120597119902
120597119910
119884
(119910 minus 119884)
Fixed shifts peak of distribution
Fixed Distribution centered at X with width σX
120575119902 119909 119910 = 120590119902 = 119902 119883 119884 + 120597119902
120597119909
119883120590119909
2
+ 120597119902
120597119910
119884
120590119910
2
0
How do we determine the error of a derived quantity Z(XYhellip) from errors in XYhellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 43
Lecture 3 Slide 43
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
SDOM of Gaussian Distribution
Standard Deviation of the Mean
The SDOM decreases as the square root of the number of measurements
That is the relative width σẊ of the distribution gets narrower as more
measurements are made
Each measurement
has similar σXi= σẊ
and similar partial
derivatives
Thushellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 44
Lecture 3 Slide 44
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Two Key Theorems from Probability
Central Limit Theorem
For random independent measurements (each with a well-define
expectation value and well-defined variance) the arithmetic mean
(average) will be approximately normally distributed
Principle of Maximum Likelihood
Given the N observed measurements x1 x2hellipxN the best estimates
for Ẋ and σ are those values for which the observed x1 x2hellipxN are
most likely
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 45
Lecture 3 Slide 45
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Mean of Gaussian Distribution as ldquoBest Estimaterdquo
Principle of Maximum Likelihood
Best Estimate of X is from maximum probability or minimum summation
Consider a data set
Each randomly distributed with
x1 x2 x3 hellipxN
To find the most likely value of the mean (the best estimate of ẋ)
find X that yields the highest probability for the data set
The combined probability for the full data set is the product
Minimize
Sum
Solve for
derivative wrst
X set to 0
Best
estimate
of X
prop1
120590119890minus(1199091minus119883)2 2120590 times
1
120590119890minus(1199092minus119883)2 2120590 times hellip times
1
120590119890minus(119909119873minus119883)2 2120590 =
1
120590119873119890 minus(119909119894minus119883)2 2120590
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 46
Lecture 3 Slide 46
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Uncertainty of ldquoBest Estimatesrdquo of Gaussian Distribution Principle of Maximum Likelihood
Best Estimate of X is from maximum probability or minimum summation
Consider a data set x1 x2 x3 hellipxN
To find the most likely value of the standard deviation (the best estimate of
the width of the x distribution) find σx that yields the highest probability for
the data set
The combined probability for the full data set is the product
Minimize
Sum
Solve for
derivative
wrst X set to 0
Best
estimate
of X
Best Estimate of σ is from maximum probability or minimum summation
Minimize
Sum
Best
estimate
of σ
Solve for
derivative
wrst σ set to 0
See
Prob 526
prop1
120590119890minus(1199091minus119883)2 2120590 times
1
120590119890minus(1199092minus119883)2 2120590 times hellip times
1
120590119890minus(119909119873minus119883)2 2120590 =
1
120590119873119890 minus(119909119894minus119883)2 2120590
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 47
Lecture 3 Slide 47
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Combining Data Sets
Weighted Averages
References Taylor Ch 7
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 48
Lecture 3 Slide 48
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Weighted Averages
Question How can we properly combine two or more separate
independent measurements of the same randomly distributed
quantity to determine a best combined value with uncertainty
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 49
Lecture 3 Slide 49
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider two measurements of the same quantity described by a random Gaussian distribution
ltx1gt x1 and ltx2gt x2
The probability of measuring two such measurements is
119875119903119900119887119909 1199091 1199092 = 119875119903119900119887119909 1199091 119875119903119900119887119909 1199092
= 1
12059011205902119890minus12059422 119908ℎ119890119903119890 1205942 equiv
1199091 minus 119883
1205901 2 +
1199092 minus 119883
1205902 2
To find the best value for χ find the maximum Prob or minimum χ2
Weighted Averages
Assume negligible systematic errors
Note χ2 or Chi squared is the sum of the squares of the deviations from the mean divided by
the corresponding uncertainty
Such methods are called ldquoMethods of Least Squaresrdquo They follow directly from the Principle of
Maximum Likelihood
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 50
Lecture 3 Slide 50
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
The probability of measuring two such measurements is
119875119903119900119887119909 1199091 1199092 = 119875119903119900119887119909 1199091 119875119903119900119887119909 1199092
= 1
12059011205902119890minus12059422 119908ℎ119890119903119890 1205942 equiv
1199091 minus 119883
1205901 2 +
1199092 minus 119883
1205902 2
To find the best value for χ find the maximum Prob or minimum χ2
Weighted Averages
This leads to 119909119882_119886119907119892 = 11990811199091+11990821199092
1199081+1199082=
119908119894 119909119894
119908119894 119908ℎ119890119903119890 119908119894 = 1
120590119894 2
Best Estimate of χ is from maximum probibility or minimum summation
Minimize Sum Solve for best estimate of χ Solve for derivative wrst χ set to 0
Note If w1=w2 we recover the standard result Xwavg= (12) (x1+x2)
Finally the width of a weighted average distribution is
120590119908119894119890119892 ℎ119905119890119889 119886119907119892 = 1
119908119894119894
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 51
Lecture 3 Slide 51
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Weighted Averages on Steroids
A very powerful method for combining data from different sources
with different methods and uncertainties (or indeed data of
related measured and calculated quantities) is Kalman filtering
The Kalman filter also known as
linear quadratic estimation (LQE)
is an algorithm that uses a series
of measurements observed over
time containing noise (random
variations) and other
inaccuracies and produces
estimates of unknown variables
that tend to be more precise than
those based on a single
measurement alone
The Kalman filter keeps track of the estimated state of the system and the variance or uncertainty of the
estimate The estimate is updated using a state transition model and measurements xk|k-1 denotes the
estimate of the systems state at time step k before the kth measurement yk has been taken into account Pk|k-1
is the corresponding uncertainty --Wikipedia 2013
Ludger Scherliess of USU Physics is a world expert at using Kalman filtering for the assimilation
of satellite and ground-based data and the USU GAMES model to predict space weather
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 52
Lecture 3 Slide 52
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Rejecting Data
Chauvenetrsquos Criterion
References Taylor Ch 6
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 53
Lecture 3 Slide 53
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly
ldquounreasonablerdquo data point from a set of randomly distributed
measurements
bull Never
bull Whenever it makes things look better
bull Chauvenetrsquos criterion provides a (quantitative) compromise
Rejecting Data
What is a good criteria for
rejecting data
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 54
Lecture 3 Slide 54
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly
ldquounreasonablerdquo data point from a set of randomly distributed
measurements
bull Never
Rejecting Data
Zallenrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 55
Lecture 3 Slide 55
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly
ldquounreasonablerdquo data point from a set of randomly distributed
measurements
bull Whenever it makes things look better
Rejecting Data
Disneyrsquos Criterion
Disneyrsquos First Law
Wishing will make it so
Disneyrsquos Second Law
Dreams are more colorful than reality
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 56
Lecture 3 Slide 56
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Data may be rejected if the expected number of measurements at
least as deviant as the suspect measurement is less than 50
Consider a set of N measurements of a single quantity
x1x2 helliphellipxN
Calculate ltxgt and σx and then determine the fractional deviations from the mean of all
the points
For the suspect point(s) xsuspect find the probability of such a point occurring in N
measurements
n = (expected number as deviant as xsuspect)
= N Prob(outside xsuspectσx)
Rejecting Data
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 57
Lecture 3 Slide 57
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Function of Gaussian Distribution
Area under curve
(probability that
ndashtσltxlt+tσ) is given
in Table at right
Probable Error (probability that
ndash067σltxlt+067σ) is 50
Error Function (probability that ndashtσltxlt+tσ )
Error Function Tabulated values-see App A
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 58
Lecture 3 Slide 58
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 59
Lecture 3 Slide 59
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashExample 1
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 60
Lecture 3 Slide 60
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Details (1)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 61
Lecture 3 Slide 61
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashDetails (2)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 62
Lecture 3 Slide 62
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos
Criterionmdash
Details (3)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 63
Lecture 3 Slide 63
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos
Criterionmdash
Example 2
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 64
Lecture 3 Slide 64
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Summary of Probability Theory
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 65
Lecture 3 Slide 65
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of
Probability
Theory-I
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 66
Lecture 3 Slide 66
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of
Probability
Theory-II
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 67
Lecture 3 Slide 67
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-III
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 68
Lecture 3 Slide 68
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-IV
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 7
Lecture 3 Slide 7
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Two Practical Exercises in Probabilities
Roll a pair of dice 50 times and record the results
What is the mean value
The standard deviation
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 8
Lecture 3 Slide 8
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Two Practical Exercises in Probabilities
Roll a pair of dice 50 times and record the results
What is the mean value
The standard deviation
What is the asymmetry
(kurtosis)
What is the probability of
rolling a 4
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 9
Lecture 3 Slide 9
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Discrete Distribution Functions
A data set to play with Written in terms of ldquooccurrencerdquo F
The mean value In terms of fractional expectations
Fractional expectations
Normalization condition
Mean value
(This is just a weighted sum)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 10
Lecture 3 Slide 10
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Limit of Discrete Distribution Functions
ldquoNormalizingrdquo data sets
Binned data sets
fk equiv fractional occurrence
∆k equiv bin width
Mean value
Normalization
Expected value 119875119903119900119887 4 =1198654
119873= 1198914
119883 = 119865119896
119896
119909119896 = (119891119896119896
Δ119896)119909119896
1 = (119891119896119896 Δ119896)119909119896
Δ119896119896
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 11
Lecture 3 Slide 11
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Limits of Distribution Functions
Consider the limiting distribution function as N ȸ and ∆k0
Larger data sets
Mathcad Games
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 12
Lecture 3 Slide 12
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Continuous Distribution Functions
Central (mean) Value Width of Distribution
Meaning of Distribution Interval
Normalization of Distribution
Thus
and by extension
= fraction of measurements
that fall within altxltb
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 13
Lecture 3 Slide 13
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Moments of Distribution Functions
The first moment for a probability distribution function is
119909 equiv 119909 = 119891119894119903119904119905 119898119900119898119890119899119905 = 119909 119891(119909)+infin
minusinfin119889119909
For a general distribution function
119909 equiv 119909 = 119891119894119903119904119905 119898119900119898119890119899119905 = 119909 119892(119909)
+infinminusinfin
119889119909
119892(119909)+infinminusinfin
119889119909
Generalizing the n
th moment is
119909119899 equiv 119909119899 = 119899119905ℎ 119898119900119898119890119899119905 = 119909119899 119892(119909)
+infinminusinfin
119889119909
119892(119909)+infinminusinfin
119889119909= 119909119899 119891(119909)
+infin
minusinfin119889119909
Oth moment equiv N 2
nd moment equiv 119909 minus 119909 2 rarr 1199092
1st moment equiv 119909 3
rd moment equiv kurtosis
0 (for a centered distribution)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 14
Lecture 3 Slide 14
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Moments of Distribution Functions
Generalizing the nth
moment is
119909119899 equiv 119909119899 = 119899119905ℎ 119898119900119898119890119899119905 = 119909119899 119892(119909)
+infinminusinfin
119889119909
119892(119909)+infinminusinfin
119889119909= 119909119899 119891(119909)
+infin
minusinfin119889119909
Oth
moment equiv N 2nd
moment equiv 119909 minus 119909 2 rarr 1199092
1st moment equiv 119909 3
rd moment equiv kurtosis
The nth
moment about the mean is
120583119899 equiv 119909 minus 119909 119899 = 119899119905ℎ 119898119900119898119890119899119905 119886119887119900119906119905 119905ℎ119890 119898119890119886119899
= 119909minus119909 119899 119892(119909)
+infinminusinfin
119889119909
119892(119909)+infinminusinfin
119889119909= 119909 minus 119909 119899 119891(119909)
+infin
minusinfin119889119909
The standard deviation (or second moment about the mean) is
1205901199092 equiv 1205832 equiv 119909 minus 119909 2 = 2119899119889 119898119900119898119890119899119905 119886119887119900119906119905 119905ℎ119890 119898119890119886119899
= 119909minus119909 2 119892(119909)
+infinminusinfin
119889119909
119892(119909)+infinminusinfin
119889119909= 119909 minus 119909 2 119891(119909)
+infin
minusinfin119889119909
0
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 15
Lecture 3 Slide 15
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Example of Continuous Distribution Functions and Expectation Values Harmonic Oscillator Example from Mechanics
Expected Values
The expectation value of a function Ɍ(x) is
Ɍ x equiv Ɍ x 119892 119909
+infin
minusinfin119889119909
119892 119909 +infin
minusinfin119889119909
= Ɍ(x) 119891(119909)+infin
minusinfin119889119909
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 16
Lecture 3 Slide 16
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Example of Continuous Distribution Functions and Expectation Values Boltzmann Distribution Example from Kinetic Theory
Expected Values
The expectation value of a function
Ɍ(x) is
Ɍ x equiv Ɍ x 119892 119909
+infin
minusinfin119889119909
119892 119909 +infin
minusinfin119889119909
= Ɍ(x) 119891(119909)+infin
minusinfin119889119909
The Boltzmann distribution function for velocities of particles as
a function of temperature T is
119875 119907 119879 = 4120587 119872
2120587 119896119861119879
3 2
1199072119890119909119901
121198721199072
12119896119861119879
Then
v = 119907 119875(119907)+infin
minusinfin119889119907 =
8 119896119861119879120587 119872
1 2
v2 = v2 119875(119907)+infin
minusinfin119889119907 =
3 119896119861119879 119872
1 2
implies KE = 1
2119872 v2 =
3
2119896119861119879
vpeak=
2 119896119861119879 119872
1 2
= 2 3
1 2
v2
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 17
Lecture 3 Slide 17
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Example of Continuous Distribution Functions and Expectation Values Fermi-Dirac Distribution Example from Kinetic Theory
For a system of identical fermions the average number of fermions in a single-particle state is given
by the FermindashDirac (FndashD) distribution
where kB is Boltzmanns constant T is the absolute temperature is the energy of the single-particle
state and μ is the total chemical potential
Since the FndashD distribution was derived using the Pauli exclusion principle which allows at most one
electron to occupy each possible state a result is that
When a quasi-continuum of energies has an associated density of states (ie the number of
states per unit energy range per unit volume) the average number of fermions per unit energy range
per unit volume is
where is called the Fermi function
so that
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 18
Lecture 3 Slide 18
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Example of Continuous Distribution Functions and Expectation Values Finite Square Well Example from Quantum Mechanics
Expectation Values The expectation value of a QM operator O(x) is O x equiv Ψlowast x 119874 119909
+infin
minusinfinΨ x 119889119909
Ψlowast x Ψ x +infin
minusinfin119889119909
For a finite square well of width L Ψn x = 2L sin
n π x
L
Ψnlowast x |Ψn x equiv
Ψnlowast x 119874 119909
+infin
minusinfinΨn x 119889119909
Ψnlowast x Ψn x
+infin
minusinfin119889119909
= 1
119909 = Ψnlowast x |x|Ψn x equiv
Ψnlowast x 119909
+infin
minusinfinΨn x 119889119909
Ψnlowast x Ψn x
+infin
minusinfin119889119909
= 1198712
119901 = Ψnlowast x |
ℏ
i
part
partx|Ψn x equiv
Ψnlowast x
ℏi
partpartx
+infin
minusinfinΨn x 119889119909
Ψnlowast x Ψn x
+infin
minusinfin119889119909
= 0
119864119899 = Ψnlowast x |iℏ
part
partt|Ψn x equiv
Ψnlowast x iℏ
partpartt
+infin
minusinfinΨn x 119889119909
Ψnlowast x Ψn x
+infin
minusinfin119889119909
=11989921205872ℏ2
2 119898 1198712
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 19
Lecture 3 Slide 19
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Distribution Functions
Available
on web site
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 20
Lecture 3 Slide 20
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
The Gaussian Distribution
Function
References Taylor Ch 5
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 21
Lecture 3 Slide 21
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Gaussian Integrals
Factorial Approximations
119899 asymp 2120587119899 1 2 119899119899 119890119909119901 minus119899 + 1
12 119899+ 119874
1
1198992
119897119900119892 119899 asymp 1
2 119897119900119892 2120587 + 119899 +
1
2 log 119899 minus 119899 +
1
12 119899+ 119874
1
1198992
119897119900119892 119899 asymp 119899 log 119899 minus 119899 (for all terms decreasing faster than linearly with n)
Gaussian Integrals
119868119898 = 2 119909119898 exp(minus1199092)infin
0 119889119909 mgt-1
119868119898 = 2 119910119899 exp(minus119910)infin
0 119889119910 equiv Γ(119899 + 1) 1199092 equiv 119910 2 119889119909 = 1199101 2 119889119910 119899 equiv 1
2 (119898 minus 1)
1198680 = Γ 119899 =1
2 = 120587 m=0 119899 = minus1
2
1198682 119896 = Γ 119896 +1
2 = 119896 minus 1
2 119896 minus 32 3
2 12 120587 even m m=2 kgt0 119899 = 119896 minus 1
2
1198682 119896+1 = Γ 119896 + 1 = 119896 odd m m=2 k+1gt0 119899 = 119896 ge 0
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 22
Lecture 3 Slide 22
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Gaussian Distribution Function
Width of Distribution
(standard deviation)
Center of Distribution
(mean) Distribution
Function
Independent
Variable
Gaussian Distribution Function
Normalization
Constant
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 23
Lecture 3 Slide 23
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Effects of Increasing N on Gaussian Distribution Functions
Consider the limiting distribution function as N infin and dx0
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 24
Lecture 3 Slide 24
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Defining the Gaussian
distribution function
in Mathcad
I suggest you ldquoinvestigaterdquo
these with the Mathcad sheet
on the web site
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 25
Lecture 3 Slide 25
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Using Mathcad to
define other common
distribution functions
Consider the limiting
distribution function
as N ȸ and ∆k0
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 26
Lecture 3 Slide 26
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider the Gaussian distribution function
Use the normalization condition to
evaluate the normalization constant
(see Taylor p 132)
The mean Ẋ is the first moment
of the Gaussian distribution
function (see Taylor p 134)
The standard deviation σx is the standard
deviation of the mean of the Gaussian
distribution function (see Taylor p 143)
Gaussian Distribution Moments
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 27
Lecture 3 Slide 27
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
When is mean x not Xbest
Answer When the
distribution is not
symmetric about X
Example Cauchy
Distribution
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 28
Lecture 3 Slide 28
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
When is mean x not Xbest
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 29
Lecture 3 Slide 29
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
When is mean x not Xbest
Answer When the
distribution is has
more than one peak
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 30
Lecture 3 Slide 30
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
The Gaussian Distribution
Function and Its Relation to
Errors
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 31
Lecture 3 Slide 31
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
A Review of Probabilities in Combination
Probability of a data set of N like measurements (x1x2hellipxN)
P (x1x2hellipxN) = P(x)1P(x2)hellipP(xN)
1 head AND 1 Four
P(H4) = P(H) P(4)
1 head OR 1 Four
P(H4) = P(H) + P(4) - P(H and 4) (true for a ldquonon-mutually exclusiverdquo events)
NOT 1 Six
P(NOT 6) = 1 - P(6)
1 Six OR 1 Four
P(64) = P(6) + P(4) (true for a ldquomutually exclusiverdquo single role)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 32
Lecture 3 Slide 32
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
The Gaussian Distribution Function and Its Relation to Errors
We will use the Gaussian distribution as applied to random
variables to develop the mathematical basis for bull Mean
bull Standard deviation
bull Standard deviation of the mean (SDOM)
bull Moments and expectation values
bull Error propagation formulas
bull Addition of errors in quadrature (for independent and random
measurements)
bull Numerical values for confidence limits (t-test)
bull Principle of maximal likelihood
bull Central limit theorem
bull Weighted distributions and Chi squared
bull Schwartz inequality (ie the uncertainty principle) (next lecture)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 33
Lecture 3 Slide 33
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider the Gaussian distribution function
Use the normalization condition to
evaluate the normalization constant
(see Taylor p 132)
The mean Ẋ is the first moment
of the Gaussian distribution
function (see Taylor p 134)
The standard deviation σx is the standard
deviation of the mean of the Gaussian
distribution function (see Taylor p 143)
Gaussian Distribution Moments
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 34
Lecture 3 Slide 34
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Standard Deviation of Gaussian Distribution
Area under curve
(probability that
ndashσltxlt+σ) is 68
5 or ~2σ ldquoSignificantrdquo
1 or ~3σ ldquoHighly Significantrdquo
1σ ldquoWithin errorsrdquo
5 ppm or ~5σ ldquoValid for HEPrdquo
See Sec 106 Testing of
Hypotheses
More complete Table in App A and B
Ah thatrsquos
highly
significant
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 35
Lecture 3 Slide 35
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Function of Gaussian Distribution
Area under curve
(probability that
ndashtσltxlt+tσ) is given
in Table at right
Probable Error (probability that
ndash067σltxlt+067σ) is 50
Error Function (probability that ndashtσltxlt+tσ )
Error Function Tabulated values-see App A
Complementary Error Function
(probability that ndashxlt-tσ AND xgt+tσ )
Prob(x outside tσ) = 1 - Prob(x within tσ)
More complete Table in App A and B
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 36
Lecture 3 Slide 36
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Useful Points on Gaussian Distribution
Full Width at Half Maximum FWHM
(See Prob 512)
Points of Inflection Occur at plusmnσ
(See Prob 513)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 37
Lecture 3 Slide 37
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Analysis and Gaussian Distribution
Adding a Constant Multiplying by a Constant
XX+A XBX and σB σ
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 38
Lecture 3 Slide 38
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Sum of Two Variables
Consider the derived quantity
Z=X + Y
(with X=0 and Y=0)
Error Propagation Addition
Error in Z
Multiple two probabilities 119875119903119900119887 119909 119910 = 119875119903119900119887 119909 ∙ 119875119903119900119887 119909 prop 119890119909119901 minus
1
2
1199092
1205901199092+
1199102
1205901199102
prop 119890119909119901 minus1
2
(119909+119910)2
1205901199092+120590119910
2 minus1199112
prop 119890119909119901 minus1
2
(119909+119910)2
1205901199092+120590119910
2 119890119909119901 minus
1199112
2
ICBST (Eq 553)
X+YZ and σx2 + σy
2 σz
2
(addition in quadrature for random independent variables)
Integrates to radic2π
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 39
Lecture 3 Slide 39
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Error Propagation
General formula for error propagation see [Taylor Secs 35 and 39]
Uncertainty as a function of one variable [Taylor Sec 35]
1 Consider a graphical method of estimating error
a) Consider an arbitaray function q(x)
b) Plot q(x) vs x
c) On the graph label
(1) qbest = q(xbest)
(2) qhi = q(xbest + δx)
(3) qlow = q(xbest- δx)
d) Making a linear approximation
x
qqxslopeqq
x
qqxslopeqq
bestbestlow
bestbesthi
e) Therefore
xx
Note the absolute value
How do we determine the error of a derived quantity Z(XYhellip) from errors in XYhellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 40
Lecture 3 Slide 40
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Error Propagation
General formula for uncertainty of a function of one variable
xx
[Taylor Eq 323]
Can you now derive for specific rules of error propagation
1 Addition and Subtraction [Taylor p 49]
2 Multiplication and Division [Taylor p 51]
3 Multiplication by a constant (exact number) [Taylor p 54]
4 Exponentiation (powers) [Taylor p 56]
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 41
Lecture 3 Slide 41
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Multiple Variables
Uncertainty of a function of multiple variables [Taylor Sec 311]
1 It can easily (no really) be shown that (see Taylor Sec 311) for a
function of several variables
)(
z
z
qy
y
qx
x
qzyxq
[Taylor Eq 347]
2 More correctly it can be shown that (see Taylor Sec 311) for a
function of several variables
)(
z
z
qy
y
qx
x
qzyxq
[Taylor Eq 347]
where the equals sign represents an upper bound as discussed above
3 For a function of several independent and random variables
)(
222
z
z
qy
y
qx
x
qzyxq
[Taylor Eq 348]
Again the proof is left for Ch 5
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 42
Lecture 3 Slide 42
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Product of Two Variables
Consider the arbitrary derived quantity
q(xy) of two independent random variables x and y
Expand q(xy) in a Taylor series about the expected values of x and y
(ie at points near X and Y)
Error Propagation General Case
119902 119909 119910 = 119902 119883 119884 + 120597119902
120597119909
119883 119909 minus 119883 +
120597119902
120597119910
119884
(119910 minus 119884)
Fixed shifts peak of distribution
Fixed Distribution centered at X with width σX
120575119902 119909 119910 = 120590119902 = 119902 119883 119884 + 120597119902
120597119909
119883120590119909
2
+ 120597119902
120597119910
119884
120590119910
2
0
How do we determine the error of a derived quantity Z(XYhellip) from errors in XYhellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 43
Lecture 3 Slide 43
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
SDOM of Gaussian Distribution
Standard Deviation of the Mean
The SDOM decreases as the square root of the number of measurements
That is the relative width σẊ of the distribution gets narrower as more
measurements are made
Each measurement
has similar σXi= σẊ
and similar partial
derivatives
Thushellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 44
Lecture 3 Slide 44
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Two Key Theorems from Probability
Central Limit Theorem
For random independent measurements (each with a well-define
expectation value and well-defined variance) the arithmetic mean
(average) will be approximately normally distributed
Principle of Maximum Likelihood
Given the N observed measurements x1 x2hellipxN the best estimates
for Ẋ and σ are those values for which the observed x1 x2hellipxN are
most likely
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 45
Lecture 3 Slide 45
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Mean of Gaussian Distribution as ldquoBest Estimaterdquo
Principle of Maximum Likelihood
Best Estimate of X is from maximum probability or minimum summation
Consider a data set
Each randomly distributed with
x1 x2 x3 hellipxN
To find the most likely value of the mean (the best estimate of ẋ)
find X that yields the highest probability for the data set
The combined probability for the full data set is the product
Minimize
Sum
Solve for
derivative wrst
X set to 0
Best
estimate
of X
prop1
120590119890minus(1199091minus119883)2 2120590 times
1
120590119890minus(1199092minus119883)2 2120590 times hellip times
1
120590119890minus(119909119873minus119883)2 2120590 =
1
120590119873119890 minus(119909119894minus119883)2 2120590
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 46
Lecture 3 Slide 46
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Uncertainty of ldquoBest Estimatesrdquo of Gaussian Distribution Principle of Maximum Likelihood
Best Estimate of X is from maximum probability or minimum summation
Consider a data set x1 x2 x3 hellipxN
To find the most likely value of the standard deviation (the best estimate of
the width of the x distribution) find σx that yields the highest probability for
the data set
The combined probability for the full data set is the product
Minimize
Sum
Solve for
derivative
wrst X set to 0
Best
estimate
of X
Best Estimate of σ is from maximum probability or minimum summation
Minimize
Sum
Best
estimate
of σ
Solve for
derivative
wrst σ set to 0
See
Prob 526
prop1
120590119890minus(1199091minus119883)2 2120590 times
1
120590119890minus(1199092minus119883)2 2120590 times hellip times
1
120590119890minus(119909119873minus119883)2 2120590 =
1
120590119873119890 minus(119909119894minus119883)2 2120590
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 47
Lecture 3 Slide 47
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Combining Data Sets
Weighted Averages
References Taylor Ch 7
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 48
Lecture 3 Slide 48
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Weighted Averages
Question How can we properly combine two or more separate
independent measurements of the same randomly distributed
quantity to determine a best combined value with uncertainty
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 49
Lecture 3 Slide 49
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider two measurements of the same quantity described by a random Gaussian distribution
ltx1gt x1 and ltx2gt x2
The probability of measuring two such measurements is
119875119903119900119887119909 1199091 1199092 = 119875119903119900119887119909 1199091 119875119903119900119887119909 1199092
= 1
12059011205902119890minus12059422 119908ℎ119890119903119890 1205942 equiv
1199091 minus 119883
1205901 2 +
1199092 minus 119883
1205902 2
To find the best value for χ find the maximum Prob or minimum χ2
Weighted Averages
Assume negligible systematic errors
Note χ2 or Chi squared is the sum of the squares of the deviations from the mean divided by
the corresponding uncertainty
Such methods are called ldquoMethods of Least Squaresrdquo They follow directly from the Principle of
Maximum Likelihood
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 50
Lecture 3 Slide 50
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
The probability of measuring two such measurements is
119875119903119900119887119909 1199091 1199092 = 119875119903119900119887119909 1199091 119875119903119900119887119909 1199092
= 1
12059011205902119890minus12059422 119908ℎ119890119903119890 1205942 equiv
1199091 minus 119883
1205901 2 +
1199092 minus 119883
1205902 2
To find the best value for χ find the maximum Prob or minimum χ2
Weighted Averages
This leads to 119909119882_119886119907119892 = 11990811199091+11990821199092
1199081+1199082=
119908119894 119909119894
119908119894 119908ℎ119890119903119890 119908119894 = 1
120590119894 2
Best Estimate of χ is from maximum probibility or minimum summation
Minimize Sum Solve for best estimate of χ Solve for derivative wrst χ set to 0
Note If w1=w2 we recover the standard result Xwavg= (12) (x1+x2)
Finally the width of a weighted average distribution is
120590119908119894119890119892 ℎ119905119890119889 119886119907119892 = 1
119908119894119894
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 51
Lecture 3 Slide 51
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Weighted Averages on Steroids
A very powerful method for combining data from different sources
with different methods and uncertainties (or indeed data of
related measured and calculated quantities) is Kalman filtering
The Kalman filter also known as
linear quadratic estimation (LQE)
is an algorithm that uses a series
of measurements observed over
time containing noise (random
variations) and other
inaccuracies and produces
estimates of unknown variables
that tend to be more precise than
those based on a single
measurement alone
The Kalman filter keeps track of the estimated state of the system and the variance or uncertainty of the
estimate The estimate is updated using a state transition model and measurements xk|k-1 denotes the
estimate of the systems state at time step k before the kth measurement yk has been taken into account Pk|k-1
is the corresponding uncertainty --Wikipedia 2013
Ludger Scherliess of USU Physics is a world expert at using Kalman filtering for the assimilation
of satellite and ground-based data and the USU GAMES model to predict space weather
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 52
Lecture 3 Slide 52
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Rejecting Data
Chauvenetrsquos Criterion
References Taylor Ch 6
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 53
Lecture 3 Slide 53
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly
ldquounreasonablerdquo data point from a set of randomly distributed
measurements
bull Never
bull Whenever it makes things look better
bull Chauvenetrsquos criterion provides a (quantitative) compromise
Rejecting Data
What is a good criteria for
rejecting data
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 54
Lecture 3 Slide 54
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly
ldquounreasonablerdquo data point from a set of randomly distributed
measurements
bull Never
Rejecting Data
Zallenrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 55
Lecture 3 Slide 55
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly
ldquounreasonablerdquo data point from a set of randomly distributed
measurements
bull Whenever it makes things look better
Rejecting Data
Disneyrsquos Criterion
Disneyrsquos First Law
Wishing will make it so
Disneyrsquos Second Law
Dreams are more colorful than reality
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 56
Lecture 3 Slide 56
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Data may be rejected if the expected number of measurements at
least as deviant as the suspect measurement is less than 50
Consider a set of N measurements of a single quantity
x1x2 helliphellipxN
Calculate ltxgt and σx and then determine the fractional deviations from the mean of all
the points
For the suspect point(s) xsuspect find the probability of such a point occurring in N
measurements
n = (expected number as deviant as xsuspect)
= N Prob(outside xsuspectσx)
Rejecting Data
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 57
Lecture 3 Slide 57
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Function of Gaussian Distribution
Area under curve
(probability that
ndashtσltxlt+tσ) is given
in Table at right
Probable Error (probability that
ndash067σltxlt+067σ) is 50
Error Function (probability that ndashtσltxlt+tσ )
Error Function Tabulated values-see App A
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 58
Lecture 3 Slide 58
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 59
Lecture 3 Slide 59
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashExample 1
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 60
Lecture 3 Slide 60
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Details (1)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 61
Lecture 3 Slide 61
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashDetails (2)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 62
Lecture 3 Slide 62
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos
Criterionmdash
Details (3)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 63
Lecture 3 Slide 63
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos
Criterionmdash
Example 2
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 64
Lecture 3 Slide 64
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Summary of Probability Theory
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 65
Lecture 3 Slide 65
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of
Probability
Theory-I
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 66
Lecture 3 Slide 66
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of
Probability
Theory-II
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 67
Lecture 3 Slide 67
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-III
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 68
Lecture 3 Slide 68
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-IV
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 8
Lecture 3 Slide 8
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Two Practical Exercises in Probabilities
Roll a pair of dice 50 times and record the results
What is the mean value
The standard deviation
What is the asymmetry
(kurtosis)
What is the probability of
rolling a 4
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 9
Lecture 3 Slide 9
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Discrete Distribution Functions
A data set to play with Written in terms of ldquooccurrencerdquo F
The mean value In terms of fractional expectations
Fractional expectations
Normalization condition
Mean value
(This is just a weighted sum)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 10
Lecture 3 Slide 10
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Limit of Discrete Distribution Functions
ldquoNormalizingrdquo data sets
Binned data sets
fk equiv fractional occurrence
∆k equiv bin width
Mean value
Normalization
Expected value 119875119903119900119887 4 =1198654
119873= 1198914
119883 = 119865119896
119896
119909119896 = (119891119896119896
Δ119896)119909119896
1 = (119891119896119896 Δ119896)119909119896
Δ119896119896
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 11
Lecture 3 Slide 11
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Limits of Distribution Functions
Consider the limiting distribution function as N ȸ and ∆k0
Larger data sets
Mathcad Games
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 12
Lecture 3 Slide 12
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Continuous Distribution Functions
Central (mean) Value Width of Distribution
Meaning of Distribution Interval
Normalization of Distribution
Thus
and by extension
= fraction of measurements
that fall within altxltb
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 13
Lecture 3 Slide 13
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Moments of Distribution Functions
The first moment for a probability distribution function is
119909 equiv 119909 = 119891119894119903119904119905 119898119900119898119890119899119905 = 119909 119891(119909)+infin
minusinfin119889119909
For a general distribution function
119909 equiv 119909 = 119891119894119903119904119905 119898119900119898119890119899119905 = 119909 119892(119909)
+infinminusinfin
119889119909
119892(119909)+infinminusinfin
119889119909
Generalizing the n
th moment is
119909119899 equiv 119909119899 = 119899119905ℎ 119898119900119898119890119899119905 = 119909119899 119892(119909)
+infinminusinfin
119889119909
119892(119909)+infinminusinfin
119889119909= 119909119899 119891(119909)
+infin
minusinfin119889119909
Oth moment equiv N 2
nd moment equiv 119909 minus 119909 2 rarr 1199092
1st moment equiv 119909 3
rd moment equiv kurtosis
0 (for a centered distribution)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 14
Lecture 3 Slide 14
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Moments of Distribution Functions
Generalizing the nth
moment is
119909119899 equiv 119909119899 = 119899119905ℎ 119898119900119898119890119899119905 = 119909119899 119892(119909)
+infinminusinfin
119889119909
119892(119909)+infinminusinfin
119889119909= 119909119899 119891(119909)
+infin
minusinfin119889119909
Oth
moment equiv N 2nd
moment equiv 119909 minus 119909 2 rarr 1199092
1st moment equiv 119909 3
rd moment equiv kurtosis
The nth
moment about the mean is
120583119899 equiv 119909 minus 119909 119899 = 119899119905ℎ 119898119900119898119890119899119905 119886119887119900119906119905 119905ℎ119890 119898119890119886119899
= 119909minus119909 119899 119892(119909)
+infinminusinfin
119889119909
119892(119909)+infinminusinfin
119889119909= 119909 minus 119909 119899 119891(119909)
+infin
minusinfin119889119909
The standard deviation (or second moment about the mean) is
1205901199092 equiv 1205832 equiv 119909 minus 119909 2 = 2119899119889 119898119900119898119890119899119905 119886119887119900119906119905 119905ℎ119890 119898119890119886119899
= 119909minus119909 2 119892(119909)
+infinminusinfin
119889119909
119892(119909)+infinminusinfin
119889119909= 119909 minus 119909 2 119891(119909)
+infin
minusinfin119889119909
0
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 15
Lecture 3 Slide 15
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Example of Continuous Distribution Functions and Expectation Values Harmonic Oscillator Example from Mechanics
Expected Values
The expectation value of a function Ɍ(x) is
Ɍ x equiv Ɍ x 119892 119909
+infin
minusinfin119889119909
119892 119909 +infin
minusinfin119889119909
= Ɍ(x) 119891(119909)+infin
minusinfin119889119909
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 16
Lecture 3 Slide 16
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Example of Continuous Distribution Functions and Expectation Values Boltzmann Distribution Example from Kinetic Theory
Expected Values
The expectation value of a function
Ɍ(x) is
Ɍ x equiv Ɍ x 119892 119909
+infin
minusinfin119889119909
119892 119909 +infin
minusinfin119889119909
= Ɍ(x) 119891(119909)+infin
minusinfin119889119909
The Boltzmann distribution function for velocities of particles as
a function of temperature T is
119875 119907 119879 = 4120587 119872
2120587 119896119861119879
3 2
1199072119890119909119901
121198721199072
12119896119861119879
Then
v = 119907 119875(119907)+infin
minusinfin119889119907 =
8 119896119861119879120587 119872
1 2
v2 = v2 119875(119907)+infin
minusinfin119889119907 =
3 119896119861119879 119872
1 2
implies KE = 1
2119872 v2 =
3
2119896119861119879
vpeak=
2 119896119861119879 119872
1 2
= 2 3
1 2
v2
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 17
Lecture 3 Slide 17
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Example of Continuous Distribution Functions and Expectation Values Fermi-Dirac Distribution Example from Kinetic Theory
For a system of identical fermions the average number of fermions in a single-particle state is given
by the FermindashDirac (FndashD) distribution
where kB is Boltzmanns constant T is the absolute temperature is the energy of the single-particle
state and μ is the total chemical potential
Since the FndashD distribution was derived using the Pauli exclusion principle which allows at most one
electron to occupy each possible state a result is that
When a quasi-continuum of energies has an associated density of states (ie the number of
states per unit energy range per unit volume) the average number of fermions per unit energy range
per unit volume is
where is called the Fermi function
so that
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 18
Lecture 3 Slide 18
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Example of Continuous Distribution Functions and Expectation Values Finite Square Well Example from Quantum Mechanics
Expectation Values The expectation value of a QM operator O(x) is O x equiv Ψlowast x 119874 119909
+infin
minusinfinΨ x 119889119909
Ψlowast x Ψ x +infin
minusinfin119889119909
For a finite square well of width L Ψn x = 2L sin
n π x
L
Ψnlowast x |Ψn x equiv
Ψnlowast x 119874 119909
+infin
minusinfinΨn x 119889119909
Ψnlowast x Ψn x
+infin
minusinfin119889119909
= 1
119909 = Ψnlowast x |x|Ψn x equiv
Ψnlowast x 119909
+infin
minusinfinΨn x 119889119909
Ψnlowast x Ψn x
+infin
minusinfin119889119909
= 1198712
119901 = Ψnlowast x |
ℏ
i
part
partx|Ψn x equiv
Ψnlowast x
ℏi
partpartx
+infin
minusinfinΨn x 119889119909
Ψnlowast x Ψn x
+infin
minusinfin119889119909
= 0
119864119899 = Ψnlowast x |iℏ
part
partt|Ψn x equiv
Ψnlowast x iℏ
partpartt
+infin
minusinfinΨn x 119889119909
Ψnlowast x Ψn x
+infin
minusinfin119889119909
=11989921205872ℏ2
2 119898 1198712
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 19
Lecture 3 Slide 19
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Distribution Functions
Available
on web site
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 20
Lecture 3 Slide 20
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
The Gaussian Distribution
Function
References Taylor Ch 5
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 21
Lecture 3 Slide 21
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Gaussian Integrals
Factorial Approximations
119899 asymp 2120587119899 1 2 119899119899 119890119909119901 minus119899 + 1
12 119899+ 119874
1
1198992
119897119900119892 119899 asymp 1
2 119897119900119892 2120587 + 119899 +
1
2 log 119899 minus 119899 +
1
12 119899+ 119874
1
1198992
119897119900119892 119899 asymp 119899 log 119899 minus 119899 (for all terms decreasing faster than linearly with n)
Gaussian Integrals
119868119898 = 2 119909119898 exp(minus1199092)infin
0 119889119909 mgt-1
119868119898 = 2 119910119899 exp(minus119910)infin
0 119889119910 equiv Γ(119899 + 1) 1199092 equiv 119910 2 119889119909 = 1199101 2 119889119910 119899 equiv 1
2 (119898 minus 1)
1198680 = Γ 119899 =1
2 = 120587 m=0 119899 = minus1
2
1198682 119896 = Γ 119896 +1
2 = 119896 minus 1
2 119896 minus 32 3
2 12 120587 even m m=2 kgt0 119899 = 119896 minus 1
2
1198682 119896+1 = Γ 119896 + 1 = 119896 odd m m=2 k+1gt0 119899 = 119896 ge 0
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 22
Lecture 3 Slide 22
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Gaussian Distribution Function
Width of Distribution
(standard deviation)
Center of Distribution
(mean) Distribution
Function
Independent
Variable
Gaussian Distribution Function
Normalization
Constant
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 23
Lecture 3 Slide 23
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Effects of Increasing N on Gaussian Distribution Functions
Consider the limiting distribution function as N infin and dx0
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 24
Lecture 3 Slide 24
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Defining the Gaussian
distribution function
in Mathcad
I suggest you ldquoinvestigaterdquo
these with the Mathcad sheet
on the web site
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 25
Lecture 3 Slide 25
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Using Mathcad to
define other common
distribution functions
Consider the limiting
distribution function
as N ȸ and ∆k0
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 26
Lecture 3 Slide 26
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider the Gaussian distribution function
Use the normalization condition to
evaluate the normalization constant
(see Taylor p 132)
The mean Ẋ is the first moment
of the Gaussian distribution
function (see Taylor p 134)
The standard deviation σx is the standard
deviation of the mean of the Gaussian
distribution function (see Taylor p 143)
Gaussian Distribution Moments
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 27
Lecture 3 Slide 27
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
When is mean x not Xbest
Answer When the
distribution is not
symmetric about X
Example Cauchy
Distribution
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 28
Lecture 3 Slide 28
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
When is mean x not Xbest
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 29
Lecture 3 Slide 29
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
When is mean x not Xbest
Answer When the
distribution is has
more than one peak
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 30
Lecture 3 Slide 30
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
The Gaussian Distribution
Function and Its Relation to
Errors
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 31
Lecture 3 Slide 31
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
A Review of Probabilities in Combination
Probability of a data set of N like measurements (x1x2hellipxN)
P (x1x2hellipxN) = P(x)1P(x2)hellipP(xN)
1 head AND 1 Four
P(H4) = P(H) P(4)
1 head OR 1 Four
P(H4) = P(H) + P(4) - P(H and 4) (true for a ldquonon-mutually exclusiverdquo events)
NOT 1 Six
P(NOT 6) = 1 - P(6)
1 Six OR 1 Four
P(64) = P(6) + P(4) (true for a ldquomutually exclusiverdquo single role)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 32
Lecture 3 Slide 32
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
The Gaussian Distribution Function and Its Relation to Errors
We will use the Gaussian distribution as applied to random
variables to develop the mathematical basis for bull Mean
bull Standard deviation
bull Standard deviation of the mean (SDOM)
bull Moments and expectation values
bull Error propagation formulas
bull Addition of errors in quadrature (for independent and random
measurements)
bull Numerical values for confidence limits (t-test)
bull Principle of maximal likelihood
bull Central limit theorem
bull Weighted distributions and Chi squared
bull Schwartz inequality (ie the uncertainty principle) (next lecture)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 33
Lecture 3 Slide 33
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider the Gaussian distribution function
Use the normalization condition to
evaluate the normalization constant
(see Taylor p 132)
The mean Ẋ is the first moment
of the Gaussian distribution
function (see Taylor p 134)
The standard deviation σx is the standard
deviation of the mean of the Gaussian
distribution function (see Taylor p 143)
Gaussian Distribution Moments
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 34
Lecture 3 Slide 34
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Standard Deviation of Gaussian Distribution
Area under curve
(probability that
ndashσltxlt+σ) is 68
5 or ~2σ ldquoSignificantrdquo
1 or ~3σ ldquoHighly Significantrdquo
1σ ldquoWithin errorsrdquo
5 ppm or ~5σ ldquoValid for HEPrdquo
See Sec 106 Testing of
Hypotheses
More complete Table in App A and B
Ah thatrsquos
highly
significant
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 35
Lecture 3 Slide 35
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Function of Gaussian Distribution
Area under curve
(probability that
ndashtσltxlt+tσ) is given
in Table at right
Probable Error (probability that
ndash067σltxlt+067σ) is 50
Error Function (probability that ndashtσltxlt+tσ )
Error Function Tabulated values-see App A
Complementary Error Function
(probability that ndashxlt-tσ AND xgt+tσ )
Prob(x outside tσ) = 1 - Prob(x within tσ)
More complete Table in App A and B
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 36
Lecture 3 Slide 36
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Useful Points on Gaussian Distribution
Full Width at Half Maximum FWHM
(See Prob 512)
Points of Inflection Occur at plusmnσ
(See Prob 513)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 37
Lecture 3 Slide 37
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Analysis and Gaussian Distribution
Adding a Constant Multiplying by a Constant
XX+A XBX and σB σ
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 38
Lecture 3 Slide 38
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Sum of Two Variables
Consider the derived quantity
Z=X + Y
(with X=0 and Y=0)
Error Propagation Addition
Error in Z
Multiple two probabilities 119875119903119900119887 119909 119910 = 119875119903119900119887 119909 ∙ 119875119903119900119887 119909 prop 119890119909119901 minus
1
2
1199092
1205901199092+
1199102
1205901199102
prop 119890119909119901 minus1
2
(119909+119910)2
1205901199092+120590119910
2 minus1199112
prop 119890119909119901 minus1
2
(119909+119910)2
1205901199092+120590119910
2 119890119909119901 minus
1199112
2
ICBST (Eq 553)
X+YZ and σx2 + σy
2 σz
2
(addition in quadrature for random independent variables)
Integrates to radic2π
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 39
Lecture 3 Slide 39
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Error Propagation
General formula for error propagation see [Taylor Secs 35 and 39]
Uncertainty as a function of one variable [Taylor Sec 35]
1 Consider a graphical method of estimating error
a) Consider an arbitaray function q(x)
b) Plot q(x) vs x
c) On the graph label
(1) qbest = q(xbest)
(2) qhi = q(xbest + δx)
(3) qlow = q(xbest- δx)
d) Making a linear approximation
x
qqxslopeqq
x
qqxslopeqq
bestbestlow
bestbesthi
e) Therefore
xx
Note the absolute value
How do we determine the error of a derived quantity Z(XYhellip) from errors in XYhellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 40
Lecture 3 Slide 40
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Error Propagation
General formula for uncertainty of a function of one variable
xx
[Taylor Eq 323]
Can you now derive for specific rules of error propagation
1 Addition and Subtraction [Taylor p 49]
2 Multiplication and Division [Taylor p 51]
3 Multiplication by a constant (exact number) [Taylor p 54]
4 Exponentiation (powers) [Taylor p 56]
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 41
Lecture 3 Slide 41
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Multiple Variables
Uncertainty of a function of multiple variables [Taylor Sec 311]
1 It can easily (no really) be shown that (see Taylor Sec 311) for a
function of several variables
)(
z
z
qy
y
qx
x
qzyxq
[Taylor Eq 347]
2 More correctly it can be shown that (see Taylor Sec 311) for a
function of several variables
)(
z
z
qy
y
qx
x
qzyxq
[Taylor Eq 347]
where the equals sign represents an upper bound as discussed above
3 For a function of several independent and random variables
)(
222
z
z
qy
y
qx
x
qzyxq
[Taylor Eq 348]
Again the proof is left for Ch 5
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 42
Lecture 3 Slide 42
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Product of Two Variables
Consider the arbitrary derived quantity
q(xy) of two independent random variables x and y
Expand q(xy) in a Taylor series about the expected values of x and y
(ie at points near X and Y)
Error Propagation General Case
119902 119909 119910 = 119902 119883 119884 + 120597119902
120597119909
119883 119909 minus 119883 +
120597119902
120597119910
119884
(119910 minus 119884)
Fixed shifts peak of distribution
Fixed Distribution centered at X with width σX
120575119902 119909 119910 = 120590119902 = 119902 119883 119884 + 120597119902
120597119909
119883120590119909
2
+ 120597119902
120597119910
119884
120590119910
2
0
How do we determine the error of a derived quantity Z(XYhellip) from errors in XYhellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 43
Lecture 3 Slide 43
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
SDOM of Gaussian Distribution
Standard Deviation of the Mean
The SDOM decreases as the square root of the number of measurements
That is the relative width σẊ of the distribution gets narrower as more
measurements are made
Each measurement
has similar σXi= σẊ
and similar partial
derivatives
Thushellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 44
Lecture 3 Slide 44
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Two Key Theorems from Probability
Central Limit Theorem
For random independent measurements (each with a well-define
expectation value and well-defined variance) the arithmetic mean
(average) will be approximately normally distributed
Principle of Maximum Likelihood
Given the N observed measurements x1 x2hellipxN the best estimates
for Ẋ and σ are those values for which the observed x1 x2hellipxN are
most likely
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 45
Lecture 3 Slide 45
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Mean of Gaussian Distribution as ldquoBest Estimaterdquo
Principle of Maximum Likelihood
Best Estimate of X is from maximum probability or minimum summation
Consider a data set
Each randomly distributed with
x1 x2 x3 hellipxN
To find the most likely value of the mean (the best estimate of ẋ)
find X that yields the highest probability for the data set
The combined probability for the full data set is the product
Minimize
Sum
Solve for
derivative wrst
X set to 0
Best
estimate
of X
prop1
120590119890minus(1199091minus119883)2 2120590 times
1
120590119890minus(1199092minus119883)2 2120590 times hellip times
1
120590119890minus(119909119873minus119883)2 2120590 =
1
120590119873119890 minus(119909119894minus119883)2 2120590
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 46
Lecture 3 Slide 46
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Uncertainty of ldquoBest Estimatesrdquo of Gaussian Distribution Principle of Maximum Likelihood
Best Estimate of X is from maximum probability or minimum summation
Consider a data set x1 x2 x3 hellipxN
To find the most likely value of the standard deviation (the best estimate of
the width of the x distribution) find σx that yields the highest probability for
the data set
The combined probability for the full data set is the product
Minimize
Sum
Solve for
derivative
wrst X set to 0
Best
estimate
of X
Best Estimate of σ is from maximum probability or minimum summation
Minimize
Sum
Best
estimate
of σ
Solve for
derivative
wrst σ set to 0
See
Prob 526
prop1
120590119890minus(1199091minus119883)2 2120590 times
1
120590119890minus(1199092minus119883)2 2120590 times hellip times
1
120590119890minus(119909119873minus119883)2 2120590 =
1
120590119873119890 minus(119909119894minus119883)2 2120590
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 47
Lecture 3 Slide 47
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Combining Data Sets
Weighted Averages
References Taylor Ch 7
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 48
Lecture 3 Slide 48
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Weighted Averages
Question How can we properly combine two or more separate
independent measurements of the same randomly distributed
quantity to determine a best combined value with uncertainty
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 49
Lecture 3 Slide 49
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider two measurements of the same quantity described by a random Gaussian distribution
ltx1gt x1 and ltx2gt x2
The probability of measuring two such measurements is
119875119903119900119887119909 1199091 1199092 = 119875119903119900119887119909 1199091 119875119903119900119887119909 1199092
= 1
12059011205902119890minus12059422 119908ℎ119890119903119890 1205942 equiv
1199091 minus 119883
1205901 2 +
1199092 minus 119883
1205902 2
To find the best value for χ find the maximum Prob or minimum χ2
Weighted Averages
Assume negligible systematic errors
Note χ2 or Chi squared is the sum of the squares of the deviations from the mean divided by
the corresponding uncertainty
Such methods are called ldquoMethods of Least Squaresrdquo They follow directly from the Principle of
Maximum Likelihood
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 50
Lecture 3 Slide 50
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
The probability of measuring two such measurements is
119875119903119900119887119909 1199091 1199092 = 119875119903119900119887119909 1199091 119875119903119900119887119909 1199092
= 1
12059011205902119890minus12059422 119908ℎ119890119903119890 1205942 equiv
1199091 minus 119883
1205901 2 +
1199092 minus 119883
1205902 2
To find the best value for χ find the maximum Prob or minimum χ2
Weighted Averages
This leads to 119909119882_119886119907119892 = 11990811199091+11990821199092
1199081+1199082=
119908119894 119909119894
119908119894 119908ℎ119890119903119890 119908119894 = 1
120590119894 2
Best Estimate of χ is from maximum probibility or minimum summation
Minimize Sum Solve for best estimate of χ Solve for derivative wrst χ set to 0
Note If w1=w2 we recover the standard result Xwavg= (12) (x1+x2)
Finally the width of a weighted average distribution is
120590119908119894119890119892 ℎ119905119890119889 119886119907119892 = 1
119908119894119894
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 51
Lecture 3 Slide 51
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Weighted Averages on Steroids
A very powerful method for combining data from different sources
with different methods and uncertainties (or indeed data of
related measured and calculated quantities) is Kalman filtering
The Kalman filter also known as
linear quadratic estimation (LQE)
is an algorithm that uses a series
of measurements observed over
time containing noise (random
variations) and other
inaccuracies and produces
estimates of unknown variables
that tend to be more precise than
those based on a single
measurement alone
The Kalman filter keeps track of the estimated state of the system and the variance or uncertainty of the
estimate The estimate is updated using a state transition model and measurements xk|k-1 denotes the
estimate of the systems state at time step k before the kth measurement yk has been taken into account Pk|k-1
is the corresponding uncertainty --Wikipedia 2013
Ludger Scherliess of USU Physics is a world expert at using Kalman filtering for the assimilation
of satellite and ground-based data and the USU GAMES model to predict space weather
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 52
Lecture 3 Slide 52
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Rejecting Data
Chauvenetrsquos Criterion
References Taylor Ch 6
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 53
Lecture 3 Slide 53
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly
ldquounreasonablerdquo data point from a set of randomly distributed
measurements
bull Never
bull Whenever it makes things look better
bull Chauvenetrsquos criterion provides a (quantitative) compromise
Rejecting Data
What is a good criteria for
rejecting data
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 54
Lecture 3 Slide 54
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly
ldquounreasonablerdquo data point from a set of randomly distributed
measurements
bull Never
Rejecting Data
Zallenrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 55
Lecture 3 Slide 55
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly
ldquounreasonablerdquo data point from a set of randomly distributed
measurements
bull Whenever it makes things look better
Rejecting Data
Disneyrsquos Criterion
Disneyrsquos First Law
Wishing will make it so
Disneyrsquos Second Law
Dreams are more colorful than reality
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 56
Lecture 3 Slide 56
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Data may be rejected if the expected number of measurements at
least as deviant as the suspect measurement is less than 50
Consider a set of N measurements of a single quantity
x1x2 helliphellipxN
Calculate ltxgt and σx and then determine the fractional deviations from the mean of all
the points
For the suspect point(s) xsuspect find the probability of such a point occurring in N
measurements
n = (expected number as deviant as xsuspect)
= N Prob(outside xsuspectσx)
Rejecting Data
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 57
Lecture 3 Slide 57
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Function of Gaussian Distribution
Area under curve
(probability that
ndashtσltxlt+tσ) is given
in Table at right
Probable Error (probability that
ndash067σltxlt+067σ) is 50
Error Function (probability that ndashtσltxlt+tσ )
Error Function Tabulated values-see App A
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 58
Lecture 3 Slide 58
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 59
Lecture 3 Slide 59
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashExample 1
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 60
Lecture 3 Slide 60
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Details (1)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 61
Lecture 3 Slide 61
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashDetails (2)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 62
Lecture 3 Slide 62
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos
Criterionmdash
Details (3)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 63
Lecture 3 Slide 63
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos
Criterionmdash
Example 2
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 64
Lecture 3 Slide 64
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Summary of Probability Theory
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 65
Lecture 3 Slide 65
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of
Probability
Theory-I
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 66
Lecture 3 Slide 66
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of
Probability
Theory-II
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 67
Lecture 3 Slide 67
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-III
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 68
Lecture 3 Slide 68
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-IV
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 9
Lecture 3 Slide 9
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Discrete Distribution Functions
A data set to play with Written in terms of ldquooccurrencerdquo F
The mean value In terms of fractional expectations
Fractional expectations
Normalization condition
Mean value
(This is just a weighted sum)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 10
Lecture 3 Slide 10
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Limit of Discrete Distribution Functions
ldquoNormalizingrdquo data sets
Binned data sets
fk equiv fractional occurrence
∆k equiv bin width
Mean value
Normalization
Expected value 119875119903119900119887 4 =1198654
119873= 1198914
119883 = 119865119896
119896
119909119896 = (119891119896119896
Δ119896)119909119896
1 = (119891119896119896 Δ119896)119909119896
Δ119896119896
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 11
Lecture 3 Slide 11
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Limits of Distribution Functions
Consider the limiting distribution function as N ȸ and ∆k0
Larger data sets
Mathcad Games
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 12
Lecture 3 Slide 12
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Continuous Distribution Functions
Central (mean) Value Width of Distribution
Meaning of Distribution Interval
Normalization of Distribution
Thus
and by extension
= fraction of measurements
that fall within altxltb
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 13
Lecture 3 Slide 13
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Moments of Distribution Functions
The first moment for a probability distribution function is
119909 equiv 119909 = 119891119894119903119904119905 119898119900119898119890119899119905 = 119909 119891(119909)+infin
minusinfin119889119909
For a general distribution function
119909 equiv 119909 = 119891119894119903119904119905 119898119900119898119890119899119905 = 119909 119892(119909)
+infinminusinfin
119889119909
119892(119909)+infinminusinfin
119889119909
Generalizing the n
th moment is
119909119899 equiv 119909119899 = 119899119905ℎ 119898119900119898119890119899119905 = 119909119899 119892(119909)
+infinminusinfin
119889119909
119892(119909)+infinminusinfin
119889119909= 119909119899 119891(119909)
+infin
minusinfin119889119909
Oth moment equiv N 2
nd moment equiv 119909 minus 119909 2 rarr 1199092
1st moment equiv 119909 3
rd moment equiv kurtosis
0 (for a centered distribution)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 14
Lecture 3 Slide 14
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Moments of Distribution Functions
Generalizing the nth
moment is
119909119899 equiv 119909119899 = 119899119905ℎ 119898119900119898119890119899119905 = 119909119899 119892(119909)
+infinminusinfin
119889119909
119892(119909)+infinminusinfin
119889119909= 119909119899 119891(119909)
+infin
minusinfin119889119909
Oth
moment equiv N 2nd
moment equiv 119909 minus 119909 2 rarr 1199092
1st moment equiv 119909 3
rd moment equiv kurtosis
The nth
moment about the mean is
120583119899 equiv 119909 minus 119909 119899 = 119899119905ℎ 119898119900119898119890119899119905 119886119887119900119906119905 119905ℎ119890 119898119890119886119899
= 119909minus119909 119899 119892(119909)
+infinminusinfin
119889119909
119892(119909)+infinminusinfin
119889119909= 119909 minus 119909 119899 119891(119909)
+infin
minusinfin119889119909
The standard deviation (or second moment about the mean) is
1205901199092 equiv 1205832 equiv 119909 minus 119909 2 = 2119899119889 119898119900119898119890119899119905 119886119887119900119906119905 119905ℎ119890 119898119890119886119899
= 119909minus119909 2 119892(119909)
+infinminusinfin
119889119909
119892(119909)+infinminusinfin
119889119909= 119909 minus 119909 2 119891(119909)
+infin
minusinfin119889119909
0
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 15
Lecture 3 Slide 15
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Example of Continuous Distribution Functions and Expectation Values Harmonic Oscillator Example from Mechanics
Expected Values
The expectation value of a function Ɍ(x) is
Ɍ x equiv Ɍ x 119892 119909
+infin
minusinfin119889119909
119892 119909 +infin
minusinfin119889119909
= Ɍ(x) 119891(119909)+infin
minusinfin119889119909
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 16
Lecture 3 Slide 16
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Example of Continuous Distribution Functions and Expectation Values Boltzmann Distribution Example from Kinetic Theory
Expected Values
The expectation value of a function
Ɍ(x) is
Ɍ x equiv Ɍ x 119892 119909
+infin
minusinfin119889119909
119892 119909 +infin
minusinfin119889119909
= Ɍ(x) 119891(119909)+infin
minusinfin119889119909
The Boltzmann distribution function for velocities of particles as
a function of temperature T is
119875 119907 119879 = 4120587 119872
2120587 119896119861119879
3 2
1199072119890119909119901
121198721199072
12119896119861119879
Then
v = 119907 119875(119907)+infin
minusinfin119889119907 =
8 119896119861119879120587 119872
1 2
v2 = v2 119875(119907)+infin
minusinfin119889119907 =
3 119896119861119879 119872
1 2
implies KE = 1
2119872 v2 =
3
2119896119861119879
vpeak=
2 119896119861119879 119872
1 2
= 2 3
1 2
v2
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 17
Lecture 3 Slide 17
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Example of Continuous Distribution Functions and Expectation Values Fermi-Dirac Distribution Example from Kinetic Theory
For a system of identical fermions the average number of fermions in a single-particle state is given
by the FermindashDirac (FndashD) distribution
where kB is Boltzmanns constant T is the absolute temperature is the energy of the single-particle
state and μ is the total chemical potential
Since the FndashD distribution was derived using the Pauli exclusion principle which allows at most one
electron to occupy each possible state a result is that
When a quasi-continuum of energies has an associated density of states (ie the number of
states per unit energy range per unit volume) the average number of fermions per unit energy range
per unit volume is
where is called the Fermi function
so that
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 18
Lecture 3 Slide 18
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Example of Continuous Distribution Functions and Expectation Values Finite Square Well Example from Quantum Mechanics
Expectation Values The expectation value of a QM operator O(x) is O x equiv Ψlowast x 119874 119909
+infin
minusinfinΨ x 119889119909
Ψlowast x Ψ x +infin
minusinfin119889119909
For a finite square well of width L Ψn x = 2L sin
n π x
L
Ψnlowast x |Ψn x equiv
Ψnlowast x 119874 119909
+infin
minusinfinΨn x 119889119909
Ψnlowast x Ψn x
+infin
minusinfin119889119909
= 1
119909 = Ψnlowast x |x|Ψn x equiv
Ψnlowast x 119909
+infin
minusinfinΨn x 119889119909
Ψnlowast x Ψn x
+infin
minusinfin119889119909
= 1198712
119901 = Ψnlowast x |
ℏ
i
part
partx|Ψn x equiv
Ψnlowast x
ℏi
partpartx
+infin
minusinfinΨn x 119889119909
Ψnlowast x Ψn x
+infin
minusinfin119889119909
= 0
119864119899 = Ψnlowast x |iℏ
part
partt|Ψn x equiv
Ψnlowast x iℏ
partpartt
+infin
minusinfinΨn x 119889119909
Ψnlowast x Ψn x
+infin
minusinfin119889119909
=11989921205872ℏ2
2 119898 1198712
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 19
Lecture 3 Slide 19
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Distribution Functions
Available
on web site
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 20
Lecture 3 Slide 20
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
The Gaussian Distribution
Function
References Taylor Ch 5
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 21
Lecture 3 Slide 21
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Gaussian Integrals
Factorial Approximations
119899 asymp 2120587119899 1 2 119899119899 119890119909119901 minus119899 + 1
12 119899+ 119874
1
1198992
119897119900119892 119899 asymp 1
2 119897119900119892 2120587 + 119899 +
1
2 log 119899 minus 119899 +
1
12 119899+ 119874
1
1198992
119897119900119892 119899 asymp 119899 log 119899 minus 119899 (for all terms decreasing faster than linearly with n)
Gaussian Integrals
119868119898 = 2 119909119898 exp(minus1199092)infin
0 119889119909 mgt-1
119868119898 = 2 119910119899 exp(minus119910)infin
0 119889119910 equiv Γ(119899 + 1) 1199092 equiv 119910 2 119889119909 = 1199101 2 119889119910 119899 equiv 1
2 (119898 minus 1)
1198680 = Γ 119899 =1
2 = 120587 m=0 119899 = minus1
2
1198682 119896 = Γ 119896 +1
2 = 119896 minus 1
2 119896 minus 32 3
2 12 120587 even m m=2 kgt0 119899 = 119896 minus 1
2
1198682 119896+1 = Γ 119896 + 1 = 119896 odd m m=2 k+1gt0 119899 = 119896 ge 0
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 22
Lecture 3 Slide 22
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Gaussian Distribution Function
Width of Distribution
(standard deviation)
Center of Distribution
(mean) Distribution
Function
Independent
Variable
Gaussian Distribution Function
Normalization
Constant
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 23
Lecture 3 Slide 23
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Effects of Increasing N on Gaussian Distribution Functions
Consider the limiting distribution function as N infin and dx0
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 24
Lecture 3 Slide 24
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Defining the Gaussian
distribution function
in Mathcad
I suggest you ldquoinvestigaterdquo
these with the Mathcad sheet
on the web site
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 25
Lecture 3 Slide 25
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Using Mathcad to
define other common
distribution functions
Consider the limiting
distribution function
as N ȸ and ∆k0
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 26
Lecture 3 Slide 26
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider the Gaussian distribution function
Use the normalization condition to
evaluate the normalization constant
(see Taylor p 132)
The mean Ẋ is the first moment
of the Gaussian distribution
function (see Taylor p 134)
The standard deviation σx is the standard
deviation of the mean of the Gaussian
distribution function (see Taylor p 143)
Gaussian Distribution Moments
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 27
Lecture 3 Slide 27
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
When is mean x not Xbest
Answer When the
distribution is not
symmetric about X
Example Cauchy
Distribution
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 28
Lecture 3 Slide 28
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
When is mean x not Xbest
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 29
Lecture 3 Slide 29
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
When is mean x not Xbest
Answer When the
distribution is has
more than one peak
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 30
Lecture 3 Slide 30
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
The Gaussian Distribution
Function and Its Relation to
Errors
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 31
Lecture 3 Slide 31
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
A Review of Probabilities in Combination
Probability of a data set of N like measurements (x1x2hellipxN)
P (x1x2hellipxN) = P(x)1P(x2)hellipP(xN)
1 head AND 1 Four
P(H4) = P(H) P(4)
1 head OR 1 Four
P(H4) = P(H) + P(4) - P(H and 4) (true for a ldquonon-mutually exclusiverdquo events)
NOT 1 Six
P(NOT 6) = 1 - P(6)
1 Six OR 1 Four
P(64) = P(6) + P(4) (true for a ldquomutually exclusiverdquo single role)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 32
Lecture 3 Slide 32
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
The Gaussian Distribution Function and Its Relation to Errors
We will use the Gaussian distribution as applied to random
variables to develop the mathematical basis for bull Mean
bull Standard deviation
bull Standard deviation of the mean (SDOM)
bull Moments and expectation values
bull Error propagation formulas
bull Addition of errors in quadrature (for independent and random
measurements)
bull Numerical values for confidence limits (t-test)
bull Principle of maximal likelihood
bull Central limit theorem
bull Weighted distributions and Chi squared
bull Schwartz inequality (ie the uncertainty principle) (next lecture)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 33
Lecture 3 Slide 33
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider the Gaussian distribution function
Use the normalization condition to
evaluate the normalization constant
(see Taylor p 132)
The mean Ẋ is the first moment
of the Gaussian distribution
function (see Taylor p 134)
The standard deviation σx is the standard
deviation of the mean of the Gaussian
distribution function (see Taylor p 143)
Gaussian Distribution Moments
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 34
Lecture 3 Slide 34
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Standard Deviation of Gaussian Distribution
Area under curve
(probability that
ndashσltxlt+σ) is 68
5 or ~2σ ldquoSignificantrdquo
1 or ~3σ ldquoHighly Significantrdquo
1σ ldquoWithin errorsrdquo
5 ppm or ~5σ ldquoValid for HEPrdquo
See Sec 106 Testing of
Hypotheses
More complete Table in App A and B
Ah thatrsquos
highly
significant
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 35
Lecture 3 Slide 35
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Function of Gaussian Distribution
Area under curve
(probability that
ndashtσltxlt+tσ) is given
in Table at right
Probable Error (probability that
ndash067σltxlt+067σ) is 50
Error Function (probability that ndashtσltxlt+tσ )
Error Function Tabulated values-see App A
Complementary Error Function
(probability that ndashxlt-tσ AND xgt+tσ )
Prob(x outside tσ) = 1 - Prob(x within tσ)
More complete Table in App A and B
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 36
Lecture 3 Slide 36
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Useful Points on Gaussian Distribution
Full Width at Half Maximum FWHM
(See Prob 512)
Points of Inflection Occur at plusmnσ
(See Prob 513)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 37
Lecture 3 Slide 37
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Analysis and Gaussian Distribution
Adding a Constant Multiplying by a Constant
XX+A XBX and σB σ
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 38
Lecture 3 Slide 38
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Sum of Two Variables
Consider the derived quantity
Z=X + Y
(with X=0 and Y=0)
Error Propagation Addition
Error in Z
Multiple two probabilities 119875119903119900119887 119909 119910 = 119875119903119900119887 119909 ∙ 119875119903119900119887 119909 prop 119890119909119901 minus
1
2
1199092
1205901199092+
1199102
1205901199102
prop 119890119909119901 minus1
2
(119909+119910)2
1205901199092+120590119910
2 minus1199112
prop 119890119909119901 minus1
2
(119909+119910)2
1205901199092+120590119910
2 119890119909119901 minus
1199112
2
ICBST (Eq 553)
X+YZ and σx2 + σy
2 σz
2
(addition in quadrature for random independent variables)
Integrates to radic2π
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 39
Lecture 3 Slide 39
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Error Propagation
General formula for error propagation see [Taylor Secs 35 and 39]
Uncertainty as a function of one variable [Taylor Sec 35]
1 Consider a graphical method of estimating error
a) Consider an arbitaray function q(x)
b) Plot q(x) vs x
c) On the graph label
(1) qbest = q(xbest)
(2) qhi = q(xbest + δx)
(3) qlow = q(xbest- δx)
d) Making a linear approximation
x
qqxslopeqq
x
qqxslopeqq
bestbestlow
bestbesthi
e) Therefore
xx
Note the absolute value
How do we determine the error of a derived quantity Z(XYhellip) from errors in XYhellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 40
Lecture 3 Slide 40
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Error Propagation
General formula for uncertainty of a function of one variable
xx
[Taylor Eq 323]
Can you now derive for specific rules of error propagation
1 Addition and Subtraction [Taylor p 49]
2 Multiplication and Division [Taylor p 51]
3 Multiplication by a constant (exact number) [Taylor p 54]
4 Exponentiation (powers) [Taylor p 56]
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 41
Lecture 3 Slide 41
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Multiple Variables
Uncertainty of a function of multiple variables [Taylor Sec 311]
1 It can easily (no really) be shown that (see Taylor Sec 311) for a
function of several variables
)(
z
z
qy
y
qx
x
qzyxq
[Taylor Eq 347]
2 More correctly it can be shown that (see Taylor Sec 311) for a
function of several variables
)(
z
z
qy
y
qx
x
qzyxq
[Taylor Eq 347]
where the equals sign represents an upper bound as discussed above
3 For a function of several independent and random variables
)(
222
z
z
qy
y
qx
x
qzyxq
[Taylor Eq 348]
Again the proof is left for Ch 5
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 42
Lecture 3 Slide 42
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Product of Two Variables
Consider the arbitrary derived quantity
q(xy) of two independent random variables x and y
Expand q(xy) in a Taylor series about the expected values of x and y
(ie at points near X and Y)
Error Propagation General Case
119902 119909 119910 = 119902 119883 119884 + 120597119902
120597119909
119883 119909 minus 119883 +
120597119902
120597119910
119884
(119910 minus 119884)
Fixed shifts peak of distribution
Fixed Distribution centered at X with width σX
120575119902 119909 119910 = 120590119902 = 119902 119883 119884 + 120597119902
120597119909
119883120590119909
2
+ 120597119902
120597119910
119884
120590119910
2
0
How do we determine the error of a derived quantity Z(XYhellip) from errors in XYhellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 43
Lecture 3 Slide 43
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
SDOM of Gaussian Distribution
Standard Deviation of the Mean
The SDOM decreases as the square root of the number of measurements
That is the relative width σẊ of the distribution gets narrower as more
measurements are made
Each measurement
has similar σXi= σẊ
and similar partial
derivatives
Thushellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 44
Lecture 3 Slide 44
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Two Key Theorems from Probability
Central Limit Theorem
For random independent measurements (each with a well-define
expectation value and well-defined variance) the arithmetic mean
(average) will be approximately normally distributed
Principle of Maximum Likelihood
Given the N observed measurements x1 x2hellipxN the best estimates
for Ẋ and σ are those values for which the observed x1 x2hellipxN are
most likely
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 45
Lecture 3 Slide 45
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Mean of Gaussian Distribution as ldquoBest Estimaterdquo
Principle of Maximum Likelihood
Best Estimate of X is from maximum probability or minimum summation
Consider a data set
Each randomly distributed with
x1 x2 x3 hellipxN
To find the most likely value of the mean (the best estimate of ẋ)
find X that yields the highest probability for the data set
The combined probability for the full data set is the product
Minimize
Sum
Solve for
derivative wrst
X set to 0
Best
estimate
of X
prop1
120590119890minus(1199091minus119883)2 2120590 times
1
120590119890minus(1199092minus119883)2 2120590 times hellip times
1
120590119890minus(119909119873minus119883)2 2120590 =
1
120590119873119890 minus(119909119894minus119883)2 2120590
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 46
Lecture 3 Slide 46
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Uncertainty of ldquoBest Estimatesrdquo of Gaussian Distribution Principle of Maximum Likelihood
Best Estimate of X is from maximum probability or minimum summation
Consider a data set x1 x2 x3 hellipxN
To find the most likely value of the standard deviation (the best estimate of
the width of the x distribution) find σx that yields the highest probability for
the data set
The combined probability for the full data set is the product
Minimize
Sum
Solve for
derivative
wrst X set to 0
Best
estimate
of X
Best Estimate of σ is from maximum probability or minimum summation
Minimize
Sum
Best
estimate
of σ
Solve for
derivative
wrst σ set to 0
See
Prob 526
prop1
120590119890minus(1199091minus119883)2 2120590 times
1
120590119890minus(1199092minus119883)2 2120590 times hellip times
1
120590119890minus(119909119873minus119883)2 2120590 =
1
120590119873119890 minus(119909119894minus119883)2 2120590
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 47
Lecture 3 Slide 47
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Combining Data Sets
Weighted Averages
References Taylor Ch 7
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 48
Lecture 3 Slide 48
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Weighted Averages
Question How can we properly combine two or more separate
independent measurements of the same randomly distributed
quantity to determine a best combined value with uncertainty
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 49
Lecture 3 Slide 49
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider two measurements of the same quantity described by a random Gaussian distribution
ltx1gt x1 and ltx2gt x2
The probability of measuring two such measurements is
119875119903119900119887119909 1199091 1199092 = 119875119903119900119887119909 1199091 119875119903119900119887119909 1199092
= 1
12059011205902119890minus12059422 119908ℎ119890119903119890 1205942 equiv
1199091 minus 119883
1205901 2 +
1199092 minus 119883
1205902 2
To find the best value for χ find the maximum Prob or minimum χ2
Weighted Averages
Assume negligible systematic errors
Note χ2 or Chi squared is the sum of the squares of the deviations from the mean divided by
the corresponding uncertainty
Such methods are called ldquoMethods of Least Squaresrdquo They follow directly from the Principle of
Maximum Likelihood
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 50
Lecture 3 Slide 50
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
The probability of measuring two such measurements is
119875119903119900119887119909 1199091 1199092 = 119875119903119900119887119909 1199091 119875119903119900119887119909 1199092
= 1
12059011205902119890minus12059422 119908ℎ119890119903119890 1205942 equiv
1199091 minus 119883
1205901 2 +
1199092 minus 119883
1205902 2
To find the best value for χ find the maximum Prob or minimum χ2
Weighted Averages
This leads to 119909119882_119886119907119892 = 11990811199091+11990821199092
1199081+1199082=
119908119894 119909119894
119908119894 119908ℎ119890119903119890 119908119894 = 1
120590119894 2
Best Estimate of χ is from maximum probibility or minimum summation
Minimize Sum Solve for best estimate of χ Solve for derivative wrst χ set to 0
Note If w1=w2 we recover the standard result Xwavg= (12) (x1+x2)
Finally the width of a weighted average distribution is
120590119908119894119890119892 ℎ119905119890119889 119886119907119892 = 1
119908119894119894
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 51
Lecture 3 Slide 51
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Weighted Averages on Steroids
A very powerful method for combining data from different sources
with different methods and uncertainties (or indeed data of
related measured and calculated quantities) is Kalman filtering
The Kalman filter also known as
linear quadratic estimation (LQE)
is an algorithm that uses a series
of measurements observed over
time containing noise (random
variations) and other
inaccuracies and produces
estimates of unknown variables
that tend to be more precise than
those based on a single
measurement alone
The Kalman filter keeps track of the estimated state of the system and the variance or uncertainty of the
estimate The estimate is updated using a state transition model and measurements xk|k-1 denotes the
estimate of the systems state at time step k before the kth measurement yk has been taken into account Pk|k-1
is the corresponding uncertainty --Wikipedia 2013
Ludger Scherliess of USU Physics is a world expert at using Kalman filtering for the assimilation
of satellite and ground-based data and the USU GAMES model to predict space weather
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 52
Lecture 3 Slide 52
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Rejecting Data
Chauvenetrsquos Criterion
References Taylor Ch 6
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 53
Lecture 3 Slide 53
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly
ldquounreasonablerdquo data point from a set of randomly distributed
measurements
bull Never
bull Whenever it makes things look better
bull Chauvenetrsquos criterion provides a (quantitative) compromise
Rejecting Data
What is a good criteria for
rejecting data
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 54
Lecture 3 Slide 54
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly
ldquounreasonablerdquo data point from a set of randomly distributed
measurements
bull Never
Rejecting Data
Zallenrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 55
Lecture 3 Slide 55
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly
ldquounreasonablerdquo data point from a set of randomly distributed
measurements
bull Whenever it makes things look better
Rejecting Data
Disneyrsquos Criterion
Disneyrsquos First Law
Wishing will make it so
Disneyrsquos Second Law
Dreams are more colorful than reality
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 56
Lecture 3 Slide 56
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Data may be rejected if the expected number of measurements at
least as deviant as the suspect measurement is less than 50
Consider a set of N measurements of a single quantity
x1x2 helliphellipxN
Calculate ltxgt and σx and then determine the fractional deviations from the mean of all
the points
For the suspect point(s) xsuspect find the probability of such a point occurring in N
measurements
n = (expected number as deviant as xsuspect)
= N Prob(outside xsuspectσx)
Rejecting Data
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 57
Lecture 3 Slide 57
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Function of Gaussian Distribution
Area under curve
(probability that
ndashtσltxlt+tσ) is given
in Table at right
Probable Error (probability that
ndash067σltxlt+067σ) is 50
Error Function (probability that ndashtσltxlt+tσ )
Error Function Tabulated values-see App A
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 58
Lecture 3 Slide 58
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 59
Lecture 3 Slide 59
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashExample 1
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 60
Lecture 3 Slide 60
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Details (1)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 61
Lecture 3 Slide 61
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashDetails (2)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 62
Lecture 3 Slide 62
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos
Criterionmdash
Details (3)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 63
Lecture 3 Slide 63
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos
Criterionmdash
Example 2
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 64
Lecture 3 Slide 64
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Summary of Probability Theory
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 65
Lecture 3 Slide 65
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of
Probability
Theory-I
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 66
Lecture 3 Slide 66
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of
Probability
Theory-II
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 67
Lecture 3 Slide 67
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-III
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 68
Lecture 3 Slide 68
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-IV
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 10
Lecture 3 Slide 10
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Limit of Discrete Distribution Functions
ldquoNormalizingrdquo data sets
Binned data sets
fk equiv fractional occurrence
∆k equiv bin width
Mean value
Normalization
Expected value 119875119903119900119887 4 =1198654
119873= 1198914
119883 = 119865119896
119896
119909119896 = (119891119896119896
Δ119896)119909119896
1 = (119891119896119896 Δ119896)119909119896
Δ119896119896
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 11
Lecture 3 Slide 11
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Limits of Distribution Functions
Consider the limiting distribution function as N ȸ and ∆k0
Larger data sets
Mathcad Games
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 12
Lecture 3 Slide 12
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Continuous Distribution Functions
Central (mean) Value Width of Distribution
Meaning of Distribution Interval
Normalization of Distribution
Thus
and by extension
= fraction of measurements
that fall within altxltb
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 13
Lecture 3 Slide 13
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Moments of Distribution Functions
The first moment for a probability distribution function is
119909 equiv 119909 = 119891119894119903119904119905 119898119900119898119890119899119905 = 119909 119891(119909)+infin
minusinfin119889119909
For a general distribution function
119909 equiv 119909 = 119891119894119903119904119905 119898119900119898119890119899119905 = 119909 119892(119909)
+infinminusinfin
119889119909
119892(119909)+infinminusinfin
119889119909
Generalizing the n
th moment is
119909119899 equiv 119909119899 = 119899119905ℎ 119898119900119898119890119899119905 = 119909119899 119892(119909)
+infinminusinfin
119889119909
119892(119909)+infinminusinfin
119889119909= 119909119899 119891(119909)
+infin
minusinfin119889119909
Oth moment equiv N 2
nd moment equiv 119909 minus 119909 2 rarr 1199092
1st moment equiv 119909 3
rd moment equiv kurtosis
0 (for a centered distribution)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 14
Lecture 3 Slide 14
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Moments of Distribution Functions
Generalizing the nth
moment is
119909119899 equiv 119909119899 = 119899119905ℎ 119898119900119898119890119899119905 = 119909119899 119892(119909)
+infinminusinfin
119889119909
119892(119909)+infinminusinfin
119889119909= 119909119899 119891(119909)
+infin
minusinfin119889119909
Oth
moment equiv N 2nd
moment equiv 119909 minus 119909 2 rarr 1199092
1st moment equiv 119909 3
rd moment equiv kurtosis
The nth
moment about the mean is
120583119899 equiv 119909 minus 119909 119899 = 119899119905ℎ 119898119900119898119890119899119905 119886119887119900119906119905 119905ℎ119890 119898119890119886119899
= 119909minus119909 119899 119892(119909)
+infinminusinfin
119889119909
119892(119909)+infinminusinfin
119889119909= 119909 minus 119909 119899 119891(119909)
+infin
minusinfin119889119909
The standard deviation (or second moment about the mean) is
1205901199092 equiv 1205832 equiv 119909 minus 119909 2 = 2119899119889 119898119900119898119890119899119905 119886119887119900119906119905 119905ℎ119890 119898119890119886119899
= 119909minus119909 2 119892(119909)
+infinminusinfin
119889119909
119892(119909)+infinminusinfin
119889119909= 119909 minus 119909 2 119891(119909)
+infin
minusinfin119889119909
0
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 15
Lecture 3 Slide 15
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Example of Continuous Distribution Functions and Expectation Values Harmonic Oscillator Example from Mechanics
Expected Values
The expectation value of a function Ɍ(x) is
Ɍ x equiv Ɍ x 119892 119909
+infin
minusinfin119889119909
119892 119909 +infin
minusinfin119889119909
= Ɍ(x) 119891(119909)+infin
minusinfin119889119909
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 16
Lecture 3 Slide 16
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Example of Continuous Distribution Functions and Expectation Values Boltzmann Distribution Example from Kinetic Theory
Expected Values
The expectation value of a function
Ɍ(x) is
Ɍ x equiv Ɍ x 119892 119909
+infin
minusinfin119889119909
119892 119909 +infin
minusinfin119889119909
= Ɍ(x) 119891(119909)+infin
minusinfin119889119909
The Boltzmann distribution function for velocities of particles as
a function of temperature T is
119875 119907 119879 = 4120587 119872
2120587 119896119861119879
3 2
1199072119890119909119901
121198721199072
12119896119861119879
Then
v = 119907 119875(119907)+infin
minusinfin119889119907 =
8 119896119861119879120587 119872
1 2
v2 = v2 119875(119907)+infin
minusinfin119889119907 =
3 119896119861119879 119872
1 2
implies KE = 1
2119872 v2 =
3
2119896119861119879
vpeak=
2 119896119861119879 119872
1 2
= 2 3
1 2
v2
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 17
Lecture 3 Slide 17
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Example of Continuous Distribution Functions and Expectation Values Fermi-Dirac Distribution Example from Kinetic Theory
For a system of identical fermions the average number of fermions in a single-particle state is given
by the FermindashDirac (FndashD) distribution
where kB is Boltzmanns constant T is the absolute temperature is the energy of the single-particle
state and μ is the total chemical potential
Since the FndashD distribution was derived using the Pauli exclusion principle which allows at most one
electron to occupy each possible state a result is that
When a quasi-continuum of energies has an associated density of states (ie the number of
states per unit energy range per unit volume) the average number of fermions per unit energy range
per unit volume is
where is called the Fermi function
so that
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 18
Lecture 3 Slide 18
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Example of Continuous Distribution Functions and Expectation Values Finite Square Well Example from Quantum Mechanics
Expectation Values The expectation value of a QM operator O(x) is O x equiv Ψlowast x 119874 119909
+infin
minusinfinΨ x 119889119909
Ψlowast x Ψ x +infin
minusinfin119889119909
For a finite square well of width L Ψn x = 2L sin
n π x
L
Ψnlowast x |Ψn x equiv
Ψnlowast x 119874 119909
+infin
minusinfinΨn x 119889119909
Ψnlowast x Ψn x
+infin
minusinfin119889119909
= 1
119909 = Ψnlowast x |x|Ψn x equiv
Ψnlowast x 119909
+infin
minusinfinΨn x 119889119909
Ψnlowast x Ψn x
+infin
minusinfin119889119909
= 1198712
119901 = Ψnlowast x |
ℏ
i
part
partx|Ψn x equiv
Ψnlowast x
ℏi
partpartx
+infin
minusinfinΨn x 119889119909
Ψnlowast x Ψn x
+infin
minusinfin119889119909
= 0
119864119899 = Ψnlowast x |iℏ
part
partt|Ψn x equiv
Ψnlowast x iℏ
partpartt
+infin
minusinfinΨn x 119889119909
Ψnlowast x Ψn x
+infin
minusinfin119889119909
=11989921205872ℏ2
2 119898 1198712
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 19
Lecture 3 Slide 19
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Distribution Functions
Available
on web site
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 20
Lecture 3 Slide 20
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
The Gaussian Distribution
Function
References Taylor Ch 5
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 21
Lecture 3 Slide 21
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Gaussian Integrals
Factorial Approximations
119899 asymp 2120587119899 1 2 119899119899 119890119909119901 minus119899 + 1
12 119899+ 119874
1
1198992
119897119900119892 119899 asymp 1
2 119897119900119892 2120587 + 119899 +
1
2 log 119899 minus 119899 +
1
12 119899+ 119874
1
1198992
119897119900119892 119899 asymp 119899 log 119899 minus 119899 (for all terms decreasing faster than linearly with n)
Gaussian Integrals
119868119898 = 2 119909119898 exp(minus1199092)infin
0 119889119909 mgt-1
119868119898 = 2 119910119899 exp(minus119910)infin
0 119889119910 equiv Γ(119899 + 1) 1199092 equiv 119910 2 119889119909 = 1199101 2 119889119910 119899 equiv 1
2 (119898 minus 1)
1198680 = Γ 119899 =1
2 = 120587 m=0 119899 = minus1
2
1198682 119896 = Γ 119896 +1
2 = 119896 minus 1
2 119896 minus 32 3
2 12 120587 even m m=2 kgt0 119899 = 119896 minus 1
2
1198682 119896+1 = Γ 119896 + 1 = 119896 odd m m=2 k+1gt0 119899 = 119896 ge 0
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 22
Lecture 3 Slide 22
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Gaussian Distribution Function
Width of Distribution
(standard deviation)
Center of Distribution
(mean) Distribution
Function
Independent
Variable
Gaussian Distribution Function
Normalization
Constant
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 23
Lecture 3 Slide 23
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Effects of Increasing N on Gaussian Distribution Functions
Consider the limiting distribution function as N infin and dx0
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 24
Lecture 3 Slide 24
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Defining the Gaussian
distribution function
in Mathcad
I suggest you ldquoinvestigaterdquo
these with the Mathcad sheet
on the web site
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 25
Lecture 3 Slide 25
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Using Mathcad to
define other common
distribution functions
Consider the limiting
distribution function
as N ȸ and ∆k0
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 26
Lecture 3 Slide 26
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider the Gaussian distribution function
Use the normalization condition to
evaluate the normalization constant
(see Taylor p 132)
The mean Ẋ is the first moment
of the Gaussian distribution
function (see Taylor p 134)
The standard deviation σx is the standard
deviation of the mean of the Gaussian
distribution function (see Taylor p 143)
Gaussian Distribution Moments
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 27
Lecture 3 Slide 27
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
When is mean x not Xbest
Answer When the
distribution is not
symmetric about X
Example Cauchy
Distribution
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 28
Lecture 3 Slide 28
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
When is mean x not Xbest
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 29
Lecture 3 Slide 29
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
When is mean x not Xbest
Answer When the
distribution is has
more than one peak
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 30
Lecture 3 Slide 30
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
The Gaussian Distribution
Function and Its Relation to
Errors
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 31
Lecture 3 Slide 31
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
A Review of Probabilities in Combination
Probability of a data set of N like measurements (x1x2hellipxN)
P (x1x2hellipxN) = P(x)1P(x2)hellipP(xN)
1 head AND 1 Four
P(H4) = P(H) P(4)
1 head OR 1 Four
P(H4) = P(H) + P(4) - P(H and 4) (true for a ldquonon-mutually exclusiverdquo events)
NOT 1 Six
P(NOT 6) = 1 - P(6)
1 Six OR 1 Four
P(64) = P(6) + P(4) (true for a ldquomutually exclusiverdquo single role)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 32
Lecture 3 Slide 32
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
The Gaussian Distribution Function and Its Relation to Errors
We will use the Gaussian distribution as applied to random
variables to develop the mathematical basis for bull Mean
bull Standard deviation
bull Standard deviation of the mean (SDOM)
bull Moments and expectation values
bull Error propagation formulas
bull Addition of errors in quadrature (for independent and random
measurements)
bull Numerical values for confidence limits (t-test)
bull Principle of maximal likelihood
bull Central limit theorem
bull Weighted distributions and Chi squared
bull Schwartz inequality (ie the uncertainty principle) (next lecture)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 33
Lecture 3 Slide 33
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider the Gaussian distribution function
Use the normalization condition to
evaluate the normalization constant
(see Taylor p 132)
The mean Ẋ is the first moment
of the Gaussian distribution
function (see Taylor p 134)
The standard deviation σx is the standard
deviation of the mean of the Gaussian
distribution function (see Taylor p 143)
Gaussian Distribution Moments
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 34
Lecture 3 Slide 34
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Standard Deviation of Gaussian Distribution
Area under curve
(probability that
ndashσltxlt+σ) is 68
5 or ~2σ ldquoSignificantrdquo
1 or ~3σ ldquoHighly Significantrdquo
1σ ldquoWithin errorsrdquo
5 ppm or ~5σ ldquoValid for HEPrdquo
See Sec 106 Testing of
Hypotheses
More complete Table in App A and B
Ah thatrsquos
highly
significant
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 35
Lecture 3 Slide 35
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Function of Gaussian Distribution
Area under curve
(probability that
ndashtσltxlt+tσ) is given
in Table at right
Probable Error (probability that
ndash067σltxlt+067σ) is 50
Error Function (probability that ndashtσltxlt+tσ )
Error Function Tabulated values-see App A
Complementary Error Function
(probability that ndashxlt-tσ AND xgt+tσ )
Prob(x outside tσ) = 1 - Prob(x within tσ)
More complete Table in App A and B
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 36
Lecture 3 Slide 36
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Useful Points on Gaussian Distribution
Full Width at Half Maximum FWHM
(See Prob 512)
Points of Inflection Occur at plusmnσ
(See Prob 513)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 37
Lecture 3 Slide 37
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Analysis and Gaussian Distribution
Adding a Constant Multiplying by a Constant
XX+A XBX and σB σ
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 38
Lecture 3 Slide 38
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Sum of Two Variables
Consider the derived quantity
Z=X + Y
(with X=0 and Y=0)
Error Propagation Addition
Error in Z
Multiple two probabilities 119875119903119900119887 119909 119910 = 119875119903119900119887 119909 ∙ 119875119903119900119887 119909 prop 119890119909119901 minus
1
2
1199092
1205901199092+
1199102
1205901199102
prop 119890119909119901 minus1
2
(119909+119910)2
1205901199092+120590119910
2 minus1199112
prop 119890119909119901 minus1
2
(119909+119910)2
1205901199092+120590119910
2 119890119909119901 minus
1199112
2
ICBST (Eq 553)
X+YZ and σx2 + σy
2 σz
2
(addition in quadrature for random independent variables)
Integrates to radic2π
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 39
Lecture 3 Slide 39
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Error Propagation
General formula for error propagation see [Taylor Secs 35 and 39]
Uncertainty as a function of one variable [Taylor Sec 35]
1 Consider a graphical method of estimating error
a) Consider an arbitaray function q(x)
b) Plot q(x) vs x
c) On the graph label
(1) qbest = q(xbest)
(2) qhi = q(xbest + δx)
(3) qlow = q(xbest- δx)
d) Making a linear approximation
x
qqxslopeqq
x
qqxslopeqq
bestbestlow
bestbesthi
e) Therefore
xx
Note the absolute value
How do we determine the error of a derived quantity Z(XYhellip) from errors in XYhellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 40
Lecture 3 Slide 40
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Error Propagation
General formula for uncertainty of a function of one variable
xx
[Taylor Eq 323]
Can you now derive for specific rules of error propagation
1 Addition and Subtraction [Taylor p 49]
2 Multiplication and Division [Taylor p 51]
3 Multiplication by a constant (exact number) [Taylor p 54]
4 Exponentiation (powers) [Taylor p 56]
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 41
Lecture 3 Slide 41
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Multiple Variables
Uncertainty of a function of multiple variables [Taylor Sec 311]
1 It can easily (no really) be shown that (see Taylor Sec 311) for a
function of several variables
)(
z
z
qy
y
qx
x
qzyxq
[Taylor Eq 347]
2 More correctly it can be shown that (see Taylor Sec 311) for a
function of several variables
)(
z
z
qy
y
qx
x
qzyxq
[Taylor Eq 347]
where the equals sign represents an upper bound as discussed above
3 For a function of several independent and random variables
)(
222
z
z
qy
y
qx
x
qzyxq
[Taylor Eq 348]
Again the proof is left for Ch 5
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 42
Lecture 3 Slide 42
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Product of Two Variables
Consider the arbitrary derived quantity
q(xy) of two independent random variables x and y
Expand q(xy) in a Taylor series about the expected values of x and y
(ie at points near X and Y)
Error Propagation General Case
119902 119909 119910 = 119902 119883 119884 + 120597119902
120597119909
119883 119909 minus 119883 +
120597119902
120597119910
119884
(119910 minus 119884)
Fixed shifts peak of distribution
Fixed Distribution centered at X with width σX
120575119902 119909 119910 = 120590119902 = 119902 119883 119884 + 120597119902
120597119909
119883120590119909
2
+ 120597119902
120597119910
119884
120590119910
2
0
How do we determine the error of a derived quantity Z(XYhellip) from errors in XYhellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 43
Lecture 3 Slide 43
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
SDOM of Gaussian Distribution
Standard Deviation of the Mean
The SDOM decreases as the square root of the number of measurements
That is the relative width σẊ of the distribution gets narrower as more
measurements are made
Each measurement
has similar σXi= σẊ
and similar partial
derivatives
Thushellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 44
Lecture 3 Slide 44
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Two Key Theorems from Probability
Central Limit Theorem
For random independent measurements (each with a well-define
expectation value and well-defined variance) the arithmetic mean
(average) will be approximately normally distributed
Principle of Maximum Likelihood
Given the N observed measurements x1 x2hellipxN the best estimates
for Ẋ and σ are those values for which the observed x1 x2hellipxN are
most likely
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 45
Lecture 3 Slide 45
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Mean of Gaussian Distribution as ldquoBest Estimaterdquo
Principle of Maximum Likelihood
Best Estimate of X is from maximum probability or minimum summation
Consider a data set
Each randomly distributed with
x1 x2 x3 hellipxN
To find the most likely value of the mean (the best estimate of ẋ)
find X that yields the highest probability for the data set
The combined probability for the full data set is the product
Minimize
Sum
Solve for
derivative wrst
X set to 0
Best
estimate
of X
prop1
120590119890minus(1199091minus119883)2 2120590 times
1
120590119890minus(1199092minus119883)2 2120590 times hellip times
1
120590119890minus(119909119873minus119883)2 2120590 =
1
120590119873119890 minus(119909119894minus119883)2 2120590
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 46
Lecture 3 Slide 46
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Uncertainty of ldquoBest Estimatesrdquo of Gaussian Distribution Principle of Maximum Likelihood
Best Estimate of X is from maximum probability or minimum summation
Consider a data set x1 x2 x3 hellipxN
To find the most likely value of the standard deviation (the best estimate of
the width of the x distribution) find σx that yields the highest probability for
the data set
The combined probability for the full data set is the product
Minimize
Sum
Solve for
derivative
wrst X set to 0
Best
estimate
of X
Best Estimate of σ is from maximum probability or minimum summation
Minimize
Sum
Best
estimate
of σ
Solve for
derivative
wrst σ set to 0
See
Prob 526
prop1
120590119890minus(1199091minus119883)2 2120590 times
1
120590119890minus(1199092minus119883)2 2120590 times hellip times
1
120590119890minus(119909119873minus119883)2 2120590 =
1
120590119873119890 minus(119909119894minus119883)2 2120590
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 47
Lecture 3 Slide 47
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Combining Data Sets
Weighted Averages
References Taylor Ch 7
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 48
Lecture 3 Slide 48
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Weighted Averages
Question How can we properly combine two or more separate
independent measurements of the same randomly distributed
quantity to determine a best combined value with uncertainty
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 49
Lecture 3 Slide 49
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider two measurements of the same quantity described by a random Gaussian distribution
ltx1gt x1 and ltx2gt x2
The probability of measuring two such measurements is
119875119903119900119887119909 1199091 1199092 = 119875119903119900119887119909 1199091 119875119903119900119887119909 1199092
= 1
12059011205902119890minus12059422 119908ℎ119890119903119890 1205942 equiv
1199091 minus 119883
1205901 2 +
1199092 minus 119883
1205902 2
To find the best value for χ find the maximum Prob or minimum χ2
Weighted Averages
Assume negligible systematic errors
Note χ2 or Chi squared is the sum of the squares of the deviations from the mean divided by
the corresponding uncertainty
Such methods are called ldquoMethods of Least Squaresrdquo They follow directly from the Principle of
Maximum Likelihood
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 50
Lecture 3 Slide 50
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
The probability of measuring two such measurements is
119875119903119900119887119909 1199091 1199092 = 119875119903119900119887119909 1199091 119875119903119900119887119909 1199092
= 1
12059011205902119890minus12059422 119908ℎ119890119903119890 1205942 equiv
1199091 minus 119883
1205901 2 +
1199092 minus 119883
1205902 2
To find the best value for χ find the maximum Prob or minimum χ2
Weighted Averages
This leads to 119909119882_119886119907119892 = 11990811199091+11990821199092
1199081+1199082=
119908119894 119909119894
119908119894 119908ℎ119890119903119890 119908119894 = 1
120590119894 2
Best Estimate of χ is from maximum probibility or minimum summation
Minimize Sum Solve for best estimate of χ Solve for derivative wrst χ set to 0
Note If w1=w2 we recover the standard result Xwavg= (12) (x1+x2)
Finally the width of a weighted average distribution is
120590119908119894119890119892 ℎ119905119890119889 119886119907119892 = 1
119908119894119894
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 51
Lecture 3 Slide 51
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Weighted Averages on Steroids
A very powerful method for combining data from different sources
with different methods and uncertainties (or indeed data of
related measured and calculated quantities) is Kalman filtering
The Kalman filter also known as
linear quadratic estimation (LQE)
is an algorithm that uses a series
of measurements observed over
time containing noise (random
variations) and other
inaccuracies and produces
estimates of unknown variables
that tend to be more precise than
those based on a single
measurement alone
The Kalman filter keeps track of the estimated state of the system and the variance or uncertainty of the
estimate The estimate is updated using a state transition model and measurements xk|k-1 denotes the
estimate of the systems state at time step k before the kth measurement yk has been taken into account Pk|k-1
is the corresponding uncertainty --Wikipedia 2013
Ludger Scherliess of USU Physics is a world expert at using Kalman filtering for the assimilation
of satellite and ground-based data and the USU GAMES model to predict space weather
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 52
Lecture 3 Slide 52
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Rejecting Data
Chauvenetrsquos Criterion
References Taylor Ch 6
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 53
Lecture 3 Slide 53
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly
ldquounreasonablerdquo data point from a set of randomly distributed
measurements
bull Never
bull Whenever it makes things look better
bull Chauvenetrsquos criterion provides a (quantitative) compromise
Rejecting Data
What is a good criteria for
rejecting data
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 54
Lecture 3 Slide 54
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly
ldquounreasonablerdquo data point from a set of randomly distributed
measurements
bull Never
Rejecting Data
Zallenrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 55
Lecture 3 Slide 55
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly
ldquounreasonablerdquo data point from a set of randomly distributed
measurements
bull Whenever it makes things look better
Rejecting Data
Disneyrsquos Criterion
Disneyrsquos First Law
Wishing will make it so
Disneyrsquos Second Law
Dreams are more colorful than reality
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 56
Lecture 3 Slide 56
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Data may be rejected if the expected number of measurements at
least as deviant as the suspect measurement is less than 50
Consider a set of N measurements of a single quantity
x1x2 helliphellipxN
Calculate ltxgt and σx and then determine the fractional deviations from the mean of all
the points
For the suspect point(s) xsuspect find the probability of such a point occurring in N
measurements
n = (expected number as deviant as xsuspect)
= N Prob(outside xsuspectσx)
Rejecting Data
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 57
Lecture 3 Slide 57
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Function of Gaussian Distribution
Area under curve
(probability that
ndashtσltxlt+tσ) is given
in Table at right
Probable Error (probability that
ndash067σltxlt+067σ) is 50
Error Function (probability that ndashtσltxlt+tσ )
Error Function Tabulated values-see App A
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 58
Lecture 3 Slide 58
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 59
Lecture 3 Slide 59
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashExample 1
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 60
Lecture 3 Slide 60
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Details (1)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 61
Lecture 3 Slide 61
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashDetails (2)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 62
Lecture 3 Slide 62
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos
Criterionmdash
Details (3)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 63
Lecture 3 Slide 63
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos
Criterionmdash
Example 2
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 64
Lecture 3 Slide 64
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Summary of Probability Theory
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 65
Lecture 3 Slide 65
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of
Probability
Theory-I
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 66
Lecture 3 Slide 66
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of
Probability
Theory-II
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 67
Lecture 3 Slide 67
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-III
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 68
Lecture 3 Slide 68
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-IV
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 11
Lecture 3 Slide 11
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Limits of Distribution Functions
Consider the limiting distribution function as N ȸ and ∆k0
Larger data sets
Mathcad Games
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 12
Lecture 3 Slide 12
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Continuous Distribution Functions
Central (mean) Value Width of Distribution
Meaning of Distribution Interval
Normalization of Distribution
Thus
and by extension
= fraction of measurements
that fall within altxltb
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 13
Lecture 3 Slide 13
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Moments of Distribution Functions
The first moment for a probability distribution function is
119909 equiv 119909 = 119891119894119903119904119905 119898119900119898119890119899119905 = 119909 119891(119909)+infin
minusinfin119889119909
For a general distribution function
119909 equiv 119909 = 119891119894119903119904119905 119898119900119898119890119899119905 = 119909 119892(119909)
+infinminusinfin
119889119909
119892(119909)+infinminusinfin
119889119909
Generalizing the n
th moment is
119909119899 equiv 119909119899 = 119899119905ℎ 119898119900119898119890119899119905 = 119909119899 119892(119909)
+infinminusinfin
119889119909
119892(119909)+infinminusinfin
119889119909= 119909119899 119891(119909)
+infin
minusinfin119889119909
Oth moment equiv N 2
nd moment equiv 119909 minus 119909 2 rarr 1199092
1st moment equiv 119909 3
rd moment equiv kurtosis
0 (for a centered distribution)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 14
Lecture 3 Slide 14
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Moments of Distribution Functions
Generalizing the nth
moment is
119909119899 equiv 119909119899 = 119899119905ℎ 119898119900119898119890119899119905 = 119909119899 119892(119909)
+infinminusinfin
119889119909
119892(119909)+infinminusinfin
119889119909= 119909119899 119891(119909)
+infin
minusinfin119889119909
Oth
moment equiv N 2nd
moment equiv 119909 minus 119909 2 rarr 1199092
1st moment equiv 119909 3
rd moment equiv kurtosis
The nth
moment about the mean is
120583119899 equiv 119909 minus 119909 119899 = 119899119905ℎ 119898119900119898119890119899119905 119886119887119900119906119905 119905ℎ119890 119898119890119886119899
= 119909minus119909 119899 119892(119909)
+infinminusinfin
119889119909
119892(119909)+infinminusinfin
119889119909= 119909 minus 119909 119899 119891(119909)
+infin
minusinfin119889119909
The standard deviation (or second moment about the mean) is
1205901199092 equiv 1205832 equiv 119909 minus 119909 2 = 2119899119889 119898119900119898119890119899119905 119886119887119900119906119905 119905ℎ119890 119898119890119886119899
= 119909minus119909 2 119892(119909)
+infinminusinfin
119889119909
119892(119909)+infinminusinfin
119889119909= 119909 minus 119909 2 119891(119909)
+infin
minusinfin119889119909
0
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 15
Lecture 3 Slide 15
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Example of Continuous Distribution Functions and Expectation Values Harmonic Oscillator Example from Mechanics
Expected Values
The expectation value of a function Ɍ(x) is
Ɍ x equiv Ɍ x 119892 119909
+infin
minusinfin119889119909
119892 119909 +infin
minusinfin119889119909
= Ɍ(x) 119891(119909)+infin
minusinfin119889119909
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 16
Lecture 3 Slide 16
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Example of Continuous Distribution Functions and Expectation Values Boltzmann Distribution Example from Kinetic Theory
Expected Values
The expectation value of a function
Ɍ(x) is
Ɍ x equiv Ɍ x 119892 119909
+infin
minusinfin119889119909
119892 119909 +infin
minusinfin119889119909
= Ɍ(x) 119891(119909)+infin
minusinfin119889119909
The Boltzmann distribution function for velocities of particles as
a function of temperature T is
119875 119907 119879 = 4120587 119872
2120587 119896119861119879
3 2
1199072119890119909119901
121198721199072
12119896119861119879
Then
v = 119907 119875(119907)+infin
minusinfin119889119907 =
8 119896119861119879120587 119872
1 2
v2 = v2 119875(119907)+infin
minusinfin119889119907 =
3 119896119861119879 119872
1 2
implies KE = 1
2119872 v2 =
3
2119896119861119879
vpeak=
2 119896119861119879 119872
1 2
= 2 3
1 2
v2
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 17
Lecture 3 Slide 17
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Example of Continuous Distribution Functions and Expectation Values Fermi-Dirac Distribution Example from Kinetic Theory
For a system of identical fermions the average number of fermions in a single-particle state is given
by the FermindashDirac (FndashD) distribution
where kB is Boltzmanns constant T is the absolute temperature is the energy of the single-particle
state and μ is the total chemical potential
Since the FndashD distribution was derived using the Pauli exclusion principle which allows at most one
electron to occupy each possible state a result is that
When a quasi-continuum of energies has an associated density of states (ie the number of
states per unit energy range per unit volume) the average number of fermions per unit energy range
per unit volume is
where is called the Fermi function
so that
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 18
Lecture 3 Slide 18
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Example of Continuous Distribution Functions and Expectation Values Finite Square Well Example from Quantum Mechanics
Expectation Values The expectation value of a QM operator O(x) is O x equiv Ψlowast x 119874 119909
+infin
minusinfinΨ x 119889119909
Ψlowast x Ψ x +infin
minusinfin119889119909
For a finite square well of width L Ψn x = 2L sin
n π x
L
Ψnlowast x |Ψn x equiv
Ψnlowast x 119874 119909
+infin
minusinfinΨn x 119889119909
Ψnlowast x Ψn x
+infin
minusinfin119889119909
= 1
119909 = Ψnlowast x |x|Ψn x equiv
Ψnlowast x 119909
+infin
minusinfinΨn x 119889119909
Ψnlowast x Ψn x
+infin
minusinfin119889119909
= 1198712
119901 = Ψnlowast x |
ℏ
i
part
partx|Ψn x equiv
Ψnlowast x
ℏi
partpartx
+infin
minusinfinΨn x 119889119909
Ψnlowast x Ψn x
+infin
minusinfin119889119909
= 0
119864119899 = Ψnlowast x |iℏ
part
partt|Ψn x equiv
Ψnlowast x iℏ
partpartt
+infin
minusinfinΨn x 119889119909
Ψnlowast x Ψn x
+infin
minusinfin119889119909
=11989921205872ℏ2
2 119898 1198712
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 19
Lecture 3 Slide 19
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Distribution Functions
Available
on web site
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 20
Lecture 3 Slide 20
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
The Gaussian Distribution
Function
References Taylor Ch 5
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 21
Lecture 3 Slide 21
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Gaussian Integrals
Factorial Approximations
119899 asymp 2120587119899 1 2 119899119899 119890119909119901 minus119899 + 1
12 119899+ 119874
1
1198992
119897119900119892 119899 asymp 1
2 119897119900119892 2120587 + 119899 +
1
2 log 119899 minus 119899 +
1
12 119899+ 119874
1
1198992
119897119900119892 119899 asymp 119899 log 119899 minus 119899 (for all terms decreasing faster than linearly with n)
Gaussian Integrals
119868119898 = 2 119909119898 exp(minus1199092)infin
0 119889119909 mgt-1
119868119898 = 2 119910119899 exp(minus119910)infin
0 119889119910 equiv Γ(119899 + 1) 1199092 equiv 119910 2 119889119909 = 1199101 2 119889119910 119899 equiv 1
2 (119898 minus 1)
1198680 = Γ 119899 =1
2 = 120587 m=0 119899 = minus1
2
1198682 119896 = Γ 119896 +1
2 = 119896 minus 1
2 119896 minus 32 3
2 12 120587 even m m=2 kgt0 119899 = 119896 minus 1
2
1198682 119896+1 = Γ 119896 + 1 = 119896 odd m m=2 k+1gt0 119899 = 119896 ge 0
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 22
Lecture 3 Slide 22
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Gaussian Distribution Function
Width of Distribution
(standard deviation)
Center of Distribution
(mean) Distribution
Function
Independent
Variable
Gaussian Distribution Function
Normalization
Constant
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 23
Lecture 3 Slide 23
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Effects of Increasing N on Gaussian Distribution Functions
Consider the limiting distribution function as N infin and dx0
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 24
Lecture 3 Slide 24
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Defining the Gaussian
distribution function
in Mathcad
I suggest you ldquoinvestigaterdquo
these with the Mathcad sheet
on the web site
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 25
Lecture 3 Slide 25
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Using Mathcad to
define other common
distribution functions
Consider the limiting
distribution function
as N ȸ and ∆k0
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 26
Lecture 3 Slide 26
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider the Gaussian distribution function
Use the normalization condition to
evaluate the normalization constant
(see Taylor p 132)
The mean Ẋ is the first moment
of the Gaussian distribution
function (see Taylor p 134)
The standard deviation σx is the standard
deviation of the mean of the Gaussian
distribution function (see Taylor p 143)
Gaussian Distribution Moments
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 27
Lecture 3 Slide 27
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
When is mean x not Xbest
Answer When the
distribution is not
symmetric about X
Example Cauchy
Distribution
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 28
Lecture 3 Slide 28
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
When is mean x not Xbest
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 29
Lecture 3 Slide 29
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
When is mean x not Xbest
Answer When the
distribution is has
more than one peak
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 30
Lecture 3 Slide 30
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
The Gaussian Distribution
Function and Its Relation to
Errors
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 31
Lecture 3 Slide 31
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
A Review of Probabilities in Combination
Probability of a data set of N like measurements (x1x2hellipxN)
P (x1x2hellipxN) = P(x)1P(x2)hellipP(xN)
1 head AND 1 Four
P(H4) = P(H) P(4)
1 head OR 1 Four
P(H4) = P(H) + P(4) - P(H and 4) (true for a ldquonon-mutually exclusiverdquo events)
NOT 1 Six
P(NOT 6) = 1 - P(6)
1 Six OR 1 Four
P(64) = P(6) + P(4) (true for a ldquomutually exclusiverdquo single role)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 32
Lecture 3 Slide 32
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
The Gaussian Distribution Function and Its Relation to Errors
We will use the Gaussian distribution as applied to random
variables to develop the mathematical basis for bull Mean
bull Standard deviation
bull Standard deviation of the mean (SDOM)
bull Moments and expectation values
bull Error propagation formulas
bull Addition of errors in quadrature (for independent and random
measurements)
bull Numerical values for confidence limits (t-test)
bull Principle of maximal likelihood
bull Central limit theorem
bull Weighted distributions and Chi squared
bull Schwartz inequality (ie the uncertainty principle) (next lecture)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 33
Lecture 3 Slide 33
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider the Gaussian distribution function
Use the normalization condition to
evaluate the normalization constant
(see Taylor p 132)
The mean Ẋ is the first moment
of the Gaussian distribution
function (see Taylor p 134)
The standard deviation σx is the standard
deviation of the mean of the Gaussian
distribution function (see Taylor p 143)
Gaussian Distribution Moments
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 34
Lecture 3 Slide 34
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Standard Deviation of Gaussian Distribution
Area under curve
(probability that
ndashσltxlt+σ) is 68
5 or ~2σ ldquoSignificantrdquo
1 or ~3σ ldquoHighly Significantrdquo
1σ ldquoWithin errorsrdquo
5 ppm or ~5σ ldquoValid for HEPrdquo
See Sec 106 Testing of
Hypotheses
More complete Table in App A and B
Ah thatrsquos
highly
significant
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 35
Lecture 3 Slide 35
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Function of Gaussian Distribution
Area under curve
(probability that
ndashtσltxlt+tσ) is given
in Table at right
Probable Error (probability that
ndash067σltxlt+067σ) is 50
Error Function (probability that ndashtσltxlt+tσ )
Error Function Tabulated values-see App A
Complementary Error Function
(probability that ndashxlt-tσ AND xgt+tσ )
Prob(x outside tσ) = 1 - Prob(x within tσ)
More complete Table in App A and B
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 36
Lecture 3 Slide 36
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Useful Points on Gaussian Distribution
Full Width at Half Maximum FWHM
(See Prob 512)
Points of Inflection Occur at plusmnσ
(See Prob 513)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 37
Lecture 3 Slide 37
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Analysis and Gaussian Distribution
Adding a Constant Multiplying by a Constant
XX+A XBX and σB σ
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 38
Lecture 3 Slide 38
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Sum of Two Variables
Consider the derived quantity
Z=X + Y
(with X=0 and Y=0)
Error Propagation Addition
Error in Z
Multiple two probabilities 119875119903119900119887 119909 119910 = 119875119903119900119887 119909 ∙ 119875119903119900119887 119909 prop 119890119909119901 minus
1
2
1199092
1205901199092+
1199102
1205901199102
prop 119890119909119901 minus1
2
(119909+119910)2
1205901199092+120590119910
2 minus1199112
prop 119890119909119901 minus1
2
(119909+119910)2
1205901199092+120590119910
2 119890119909119901 minus
1199112
2
ICBST (Eq 553)
X+YZ and σx2 + σy
2 σz
2
(addition in quadrature for random independent variables)
Integrates to radic2π
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 39
Lecture 3 Slide 39
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Error Propagation
General formula for error propagation see [Taylor Secs 35 and 39]
Uncertainty as a function of one variable [Taylor Sec 35]
1 Consider a graphical method of estimating error
a) Consider an arbitaray function q(x)
b) Plot q(x) vs x
c) On the graph label
(1) qbest = q(xbest)
(2) qhi = q(xbest + δx)
(3) qlow = q(xbest- δx)
d) Making a linear approximation
x
qqxslopeqq
x
qqxslopeqq
bestbestlow
bestbesthi
e) Therefore
xx
Note the absolute value
How do we determine the error of a derived quantity Z(XYhellip) from errors in XYhellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 40
Lecture 3 Slide 40
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Error Propagation
General formula for uncertainty of a function of one variable
xx
[Taylor Eq 323]
Can you now derive for specific rules of error propagation
1 Addition and Subtraction [Taylor p 49]
2 Multiplication and Division [Taylor p 51]
3 Multiplication by a constant (exact number) [Taylor p 54]
4 Exponentiation (powers) [Taylor p 56]
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 41
Lecture 3 Slide 41
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Multiple Variables
Uncertainty of a function of multiple variables [Taylor Sec 311]
1 It can easily (no really) be shown that (see Taylor Sec 311) for a
function of several variables
)(
z
z
qy
y
qx
x
qzyxq
[Taylor Eq 347]
2 More correctly it can be shown that (see Taylor Sec 311) for a
function of several variables
)(
z
z
qy
y
qx
x
qzyxq
[Taylor Eq 347]
where the equals sign represents an upper bound as discussed above
3 For a function of several independent and random variables
)(
222
z
z
qy
y
qx
x
qzyxq
[Taylor Eq 348]
Again the proof is left for Ch 5
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 42
Lecture 3 Slide 42
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Product of Two Variables
Consider the arbitrary derived quantity
q(xy) of two independent random variables x and y
Expand q(xy) in a Taylor series about the expected values of x and y
(ie at points near X and Y)
Error Propagation General Case
119902 119909 119910 = 119902 119883 119884 + 120597119902
120597119909
119883 119909 minus 119883 +
120597119902
120597119910
119884
(119910 minus 119884)
Fixed shifts peak of distribution
Fixed Distribution centered at X with width σX
120575119902 119909 119910 = 120590119902 = 119902 119883 119884 + 120597119902
120597119909
119883120590119909
2
+ 120597119902
120597119910
119884
120590119910
2
0
How do we determine the error of a derived quantity Z(XYhellip) from errors in XYhellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 43
Lecture 3 Slide 43
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
SDOM of Gaussian Distribution
Standard Deviation of the Mean
The SDOM decreases as the square root of the number of measurements
That is the relative width σẊ of the distribution gets narrower as more
measurements are made
Each measurement
has similar σXi= σẊ
and similar partial
derivatives
Thushellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 44
Lecture 3 Slide 44
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Two Key Theorems from Probability
Central Limit Theorem
For random independent measurements (each with a well-define
expectation value and well-defined variance) the arithmetic mean
(average) will be approximately normally distributed
Principle of Maximum Likelihood
Given the N observed measurements x1 x2hellipxN the best estimates
for Ẋ and σ are those values for which the observed x1 x2hellipxN are
most likely
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 45
Lecture 3 Slide 45
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Mean of Gaussian Distribution as ldquoBest Estimaterdquo
Principle of Maximum Likelihood
Best Estimate of X is from maximum probability or minimum summation
Consider a data set
Each randomly distributed with
x1 x2 x3 hellipxN
To find the most likely value of the mean (the best estimate of ẋ)
find X that yields the highest probability for the data set
The combined probability for the full data set is the product
Minimize
Sum
Solve for
derivative wrst
X set to 0
Best
estimate
of X
prop1
120590119890minus(1199091minus119883)2 2120590 times
1
120590119890minus(1199092minus119883)2 2120590 times hellip times
1
120590119890minus(119909119873minus119883)2 2120590 =
1
120590119873119890 minus(119909119894minus119883)2 2120590
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 46
Lecture 3 Slide 46
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Uncertainty of ldquoBest Estimatesrdquo of Gaussian Distribution Principle of Maximum Likelihood
Best Estimate of X is from maximum probability or minimum summation
Consider a data set x1 x2 x3 hellipxN
To find the most likely value of the standard deviation (the best estimate of
the width of the x distribution) find σx that yields the highest probability for
the data set
The combined probability for the full data set is the product
Minimize
Sum
Solve for
derivative
wrst X set to 0
Best
estimate
of X
Best Estimate of σ is from maximum probability or minimum summation
Minimize
Sum
Best
estimate
of σ
Solve for
derivative
wrst σ set to 0
See
Prob 526
prop1
120590119890minus(1199091minus119883)2 2120590 times
1
120590119890minus(1199092minus119883)2 2120590 times hellip times
1
120590119890minus(119909119873minus119883)2 2120590 =
1
120590119873119890 minus(119909119894minus119883)2 2120590
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 47
Lecture 3 Slide 47
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Combining Data Sets
Weighted Averages
References Taylor Ch 7
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 48
Lecture 3 Slide 48
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Weighted Averages
Question How can we properly combine two or more separate
independent measurements of the same randomly distributed
quantity to determine a best combined value with uncertainty
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 49
Lecture 3 Slide 49
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider two measurements of the same quantity described by a random Gaussian distribution
ltx1gt x1 and ltx2gt x2
The probability of measuring two such measurements is
119875119903119900119887119909 1199091 1199092 = 119875119903119900119887119909 1199091 119875119903119900119887119909 1199092
= 1
12059011205902119890minus12059422 119908ℎ119890119903119890 1205942 equiv
1199091 minus 119883
1205901 2 +
1199092 minus 119883
1205902 2
To find the best value for χ find the maximum Prob or minimum χ2
Weighted Averages
Assume negligible systematic errors
Note χ2 or Chi squared is the sum of the squares of the deviations from the mean divided by
the corresponding uncertainty
Such methods are called ldquoMethods of Least Squaresrdquo They follow directly from the Principle of
Maximum Likelihood
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 50
Lecture 3 Slide 50
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
The probability of measuring two such measurements is
119875119903119900119887119909 1199091 1199092 = 119875119903119900119887119909 1199091 119875119903119900119887119909 1199092
= 1
12059011205902119890minus12059422 119908ℎ119890119903119890 1205942 equiv
1199091 minus 119883
1205901 2 +
1199092 minus 119883
1205902 2
To find the best value for χ find the maximum Prob or minimum χ2
Weighted Averages
This leads to 119909119882_119886119907119892 = 11990811199091+11990821199092
1199081+1199082=
119908119894 119909119894
119908119894 119908ℎ119890119903119890 119908119894 = 1
120590119894 2
Best Estimate of χ is from maximum probibility or minimum summation
Minimize Sum Solve for best estimate of χ Solve for derivative wrst χ set to 0
Note If w1=w2 we recover the standard result Xwavg= (12) (x1+x2)
Finally the width of a weighted average distribution is
120590119908119894119890119892 ℎ119905119890119889 119886119907119892 = 1
119908119894119894
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 51
Lecture 3 Slide 51
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Weighted Averages on Steroids
A very powerful method for combining data from different sources
with different methods and uncertainties (or indeed data of
related measured and calculated quantities) is Kalman filtering
The Kalman filter also known as
linear quadratic estimation (LQE)
is an algorithm that uses a series
of measurements observed over
time containing noise (random
variations) and other
inaccuracies and produces
estimates of unknown variables
that tend to be more precise than
those based on a single
measurement alone
The Kalman filter keeps track of the estimated state of the system and the variance or uncertainty of the
estimate The estimate is updated using a state transition model and measurements xk|k-1 denotes the
estimate of the systems state at time step k before the kth measurement yk has been taken into account Pk|k-1
is the corresponding uncertainty --Wikipedia 2013
Ludger Scherliess of USU Physics is a world expert at using Kalman filtering for the assimilation
of satellite and ground-based data and the USU GAMES model to predict space weather
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 52
Lecture 3 Slide 52
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Rejecting Data
Chauvenetrsquos Criterion
References Taylor Ch 6
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 53
Lecture 3 Slide 53
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly
ldquounreasonablerdquo data point from a set of randomly distributed
measurements
bull Never
bull Whenever it makes things look better
bull Chauvenetrsquos criterion provides a (quantitative) compromise
Rejecting Data
What is a good criteria for
rejecting data
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 54
Lecture 3 Slide 54
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly
ldquounreasonablerdquo data point from a set of randomly distributed
measurements
bull Never
Rejecting Data
Zallenrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 55
Lecture 3 Slide 55
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly
ldquounreasonablerdquo data point from a set of randomly distributed
measurements
bull Whenever it makes things look better
Rejecting Data
Disneyrsquos Criterion
Disneyrsquos First Law
Wishing will make it so
Disneyrsquos Second Law
Dreams are more colorful than reality
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 56
Lecture 3 Slide 56
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Data may be rejected if the expected number of measurements at
least as deviant as the suspect measurement is less than 50
Consider a set of N measurements of a single quantity
x1x2 helliphellipxN
Calculate ltxgt and σx and then determine the fractional deviations from the mean of all
the points
For the suspect point(s) xsuspect find the probability of such a point occurring in N
measurements
n = (expected number as deviant as xsuspect)
= N Prob(outside xsuspectσx)
Rejecting Data
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 57
Lecture 3 Slide 57
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Function of Gaussian Distribution
Area under curve
(probability that
ndashtσltxlt+tσ) is given
in Table at right
Probable Error (probability that
ndash067σltxlt+067σ) is 50
Error Function (probability that ndashtσltxlt+tσ )
Error Function Tabulated values-see App A
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 58
Lecture 3 Slide 58
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 59
Lecture 3 Slide 59
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashExample 1
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 60
Lecture 3 Slide 60
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Details (1)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 61
Lecture 3 Slide 61
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashDetails (2)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 62
Lecture 3 Slide 62
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos
Criterionmdash
Details (3)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 63
Lecture 3 Slide 63
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos
Criterionmdash
Example 2
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 64
Lecture 3 Slide 64
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Summary of Probability Theory
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 65
Lecture 3 Slide 65
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of
Probability
Theory-I
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 66
Lecture 3 Slide 66
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of
Probability
Theory-II
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 67
Lecture 3 Slide 67
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-III
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 68
Lecture 3 Slide 68
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-IV
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 12
Lecture 3 Slide 12
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Continuous Distribution Functions
Central (mean) Value Width of Distribution
Meaning of Distribution Interval
Normalization of Distribution
Thus
and by extension
= fraction of measurements
that fall within altxltb
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 13
Lecture 3 Slide 13
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Moments of Distribution Functions
The first moment for a probability distribution function is
119909 equiv 119909 = 119891119894119903119904119905 119898119900119898119890119899119905 = 119909 119891(119909)+infin
minusinfin119889119909
For a general distribution function
119909 equiv 119909 = 119891119894119903119904119905 119898119900119898119890119899119905 = 119909 119892(119909)
+infinminusinfin
119889119909
119892(119909)+infinminusinfin
119889119909
Generalizing the n
th moment is
119909119899 equiv 119909119899 = 119899119905ℎ 119898119900119898119890119899119905 = 119909119899 119892(119909)
+infinminusinfin
119889119909
119892(119909)+infinminusinfin
119889119909= 119909119899 119891(119909)
+infin
minusinfin119889119909
Oth moment equiv N 2
nd moment equiv 119909 minus 119909 2 rarr 1199092
1st moment equiv 119909 3
rd moment equiv kurtosis
0 (for a centered distribution)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 14
Lecture 3 Slide 14
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Moments of Distribution Functions
Generalizing the nth
moment is
119909119899 equiv 119909119899 = 119899119905ℎ 119898119900119898119890119899119905 = 119909119899 119892(119909)
+infinminusinfin
119889119909
119892(119909)+infinminusinfin
119889119909= 119909119899 119891(119909)
+infin
minusinfin119889119909
Oth
moment equiv N 2nd
moment equiv 119909 minus 119909 2 rarr 1199092
1st moment equiv 119909 3
rd moment equiv kurtosis
The nth
moment about the mean is
120583119899 equiv 119909 minus 119909 119899 = 119899119905ℎ 119898119900119898119890119899119905 119886119887119900119906119905 119905ℎ119890 119898119890119886119899
= 119909minus119909 119899 119892(119909)
+infinminusinfin
119889119909
119892(119909)+infinminusinfin
119889119909= 119909 minus 119909 119899 119891(119909)
+infin
minusinfin119889119909
The standard deviation (or second moment about the mean) is
1205901199092 equiv 1205832 equiv 119909 minus 119909 2 = 2119899119889 119898119900119898119890119899119905 119886119887119900119906119905 119905ℎ119890 119898119890119886119899
= 119909minus119909 2 119892(119909)
+infinminusinfin
119889119909
119892(119909)+infinminusinfin
119889119909= 119909 minus 119909 2 119891(119909)
+infin
minusinfin119889119909
0
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 15
Lecture 3 Slide 15
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Example of Continuous Distribution Functions and Expectation Values Harmonic Oscillator Example from Mechanics
Expected Values
The expectation value of a function Ɍ(x) is
Ɍ x equiv Ɍ x 119892 119909
+infin
minusinfin119889119909
119892 119909 +infin
minusinfin119889119909
= Ɍ(x) 119891(119909)+infin
minusinfin119889119909
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 16
Lecture 3 Slide 16
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Example of Continuous Distribution Functions and Expectation Values Boltzmann Distribution Example from Kinetic Theory
Expected Values
The expectation value of a function
Ɍ(x) is
Ɍ x equiv Ɍ x 119892 119909
+infin
minusinfin119889119909
119892 119909 +infin
minusinfin119889119909
= Ɍ(x) 119891(119909)+infin
minusinfin119889119909
The Boltzmann distribution function for velocities of particles as
a function of temperature T is
119875 119907 119879 = 4120587 119872
2120587 119896119861119879
3 2
1199072119890119909119901
121198721199072
12119896119861119879
Then
v = 119907 119875(119907)+infin
minusinfin119889119907 =
8 119896119861119879120587 119872
1 2
v2 = v2 119875(119907)+infin
minusinfin119889119907 =
3 119896119861119879 119872
1 2
implies KE = 1
2119872 v2 =
3
2119896119861119879
vpeak=
2 119896119861119879 119872
1 2
= 2 3
1 2
v2
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 17
Lecture 3 Slide 17
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Example of Continuous Distribution Functions and Expectation Values Fermi-Dirac Distribution Example from Kinetic Theory
For a system of identical fermions the average number of fermions in a single-particle state is given
by the FermindashDirac (FndashD) distribution
where kB is Boltzmanns constant T is the absolute temperature is the energy of the single-particle
state and μ is the total chemical potential
Since the FndashD distribution was derived using the Pauli exclusion principle which allows at most one
electron to occupy each possible state a result is that
When a quasi-continuum of energies has an associated density of states (ie the number of
states per unit energy range per unit volume) the average number of fermions per unit energy range
per unit volume is
where is called the Fermi function
so that
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 18
Lecture 3 Slide 18
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Example of Continuous Distribution Functions and Expectation Values Finite Square Well Example from Quantum Mechanics
Expectation Values The expectation value of a QM operator O(x) is O x equiv Ψlowast x 119874 119909
+infin
minusinfinΨ x 119889119909
Ψlowast x Ψ x +infin
minusinfin119889119909
For a finite square well of width L Ψn x = 2L sin
n π x
L
Ψnlowast x |Ψn x equiv
Ψnlowast x 119874 119909
+infin
minusinfinΨn x 119889119909
Ψnlowast x Ψn x
+infin
minusinfin119889119909
= 1
119909 = Ψnlowast x |x|Ψn x equiv
Ψnlowast x 119909
+infin
minusinfinΨn x 119889119909
Ψnlowast x Ψn x
+infin
minusinfin119889119909
= 1198712
119901 = Ψnlowast x |
ℏ
i
part
partx|Ψn x equiv
Ψnlowast x
ℏi
partpartx
+infin
minusinfinΨn x 119889119909
Ψnlowast x Ψn x
+infin
minusinfin119889119909
= 0
119864119899 = Ψnlowast x |iℏ
part
partt|Ψn x equiv
Ψnlowast x iℏ
partpartt
+infin
minusinfinΨn x 119889119909
Ψnlowast x Ψn x
+infin
minusinfin119889119909
=11989921205872ℏ2
2 119898 1198712
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 19
Lecture 3 Slide 19
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Distribution Functions
Available
on web site
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 20
Lecture 3 Slide 20
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
The Gaussian Distribution
Function
References Taylor Ch 5
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 21
Lecture 3 Slide 21
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Gaussian Integrals
Factorial Approximations
119899 asymp 2120587119899 1 2 119899119899 119890119909119901 minus119899 + 1
12 119899+ 119874
1
1198992
119897119900119892 119899 asymp 1
2 119897119900119892 2120587 + 119899 +
1
2 log 119899 minus 119899 +
1
12 119899+ 119874
1
1198992
119897119900119892 119899 asymp 119899 log 119899 minus 119899 (for all terms decreasing faster than linearly with n)
Gaussian Integrals
119868119898 = 2 119909119898 exp(minus1199092)infin
0 119889119909 mgt-1
119868119898 = 2 119910119899 exp(minus119910)infin
0 119889119910 equiv Γ(119899 + 1) 1199092 equiv 119910 2 119889119909 = 1199101 2 119889119910 119899 equiv 1
2 (119898 minus 1)
1198680 = Γ 119899 =1
2 = 120587 m=0 119899 = minus1
2
1198682 119896 = Γ 119896 +1
2 = 119896 minus 1
2 119896 minus 32 3
2 12 120587 even m m=2 kgt0 119899 = 119896 minus 1
2
1198682 119896+1 = Γ 119896 + 1 = 119896 odd m m=2 k+1gt0 119899 = 119896 ge 0
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 22
Lecture 3 Slide 22
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Gaussian Distribution Function
Width of Distribution
(standard deviation)
Center of Distribution
(mean) Distribution
Function
Independent
Variable
Gaussian Distribution Function
Normalization
Constant
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 23
Lecture 3 Slide 23
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Effects of Increasing N on Gaussian Distribution Functions
Consider the limiting distribution function as N infin and dx0
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 24
Lecture 3 Slide 24
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Defining the Gaussian
distribution function
in Mathcad
I suggest you ldquoinvestigaterdquo
these with the Mathcad sheet
on the web site
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 25
Lecture 3 Slide 25
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Using Mathcad to
define other common
distribution functions
Consider the limiting
distribution function
as N ȸ and ∆k0
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 26
Lecture 3 Slide 26
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider the Gaussian distribution function
Use the normalization condition to
evaluate the normalization constant
(see Taylor p 132)
The mean Ẋ is the first moment
of the Gaussian distribution
function (see Taylor p 134)
The standard deviation σx is the standard
deviation of the mean of the Gaussian
distribution function (see Taylor p 143)
Gaussian Distribution Moments
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 27
Lecture 3 Slide 27
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
When is mean x not Xbest
Answer When the
distribution is not
symmetric about X
Example Cauchy
Distribution
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 28
Lecture 3 Slide 28
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
When is mean x not Xbest
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 29
Lecture 3 Slide 29
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
When is mean x not Xbest
Answer When the
distribution is has
more than one peak
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 30
Lecture 3 Slide 30
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
The Gaussian Distribution
Function and Its Relation to
Errors
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 31
Lecture 3 Slide 31
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
A Review of Probabilities in Combination
Probability of a data set of N like measurements (x1x2hellipxN)
P (x1x2hellipxN) = P(x)1P(x2)hellipP(xN)
1 head AND 1 Four
P(H4) = P(H) P(4)
1 head OR 1 Four
P(H4) = P(H) + P(4) - P(H and 4) (true for a ldquonon-mutually exclusiverdquo events)
NOT 1 Six
P(NOT 6) = 1 - P(6)
1 Six OR 1 Four
P(64) = P(6) + P(4) (true for a ldquomutually exclusiverdquo single role)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 32
Lecture 3 Slide 32
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
The Gaussian Distribution Function and Its Relation to Errors
We will use the Gaussian distribution as applied to random
variables to develop the mathematical basis for bull Mean
bull Standard deviation
bull Standard deviation of the mean (SDOM)
bull Moments and expectation values
bull Error propagation formulas
bull Addition of errors in quadrature (for independent and random
measurements)
bull Numerical values for confidence limits (t-test)
bull Principle of maximal likelihood
bull Central limit theorem
bull Weighted distributions and Chi squared
bull Schwartz inequality (ie the uncertainty principle) (next lecture)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 33
Lecture 3 Slide 33
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider the Gaussian distribution function
Use the normalization condition to
evaluate the normalization constant
(see Taylor p 132)
The mean Ẋ is the first moment
of the Gaussian distribution
function (see Taylor p 134)
The standard deviation σx is the standard
deviation of the mean of the Gaussian
distribution function (see Taylor p 143)
Gaussian Distribution Moments
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 34
Lecture 3 Slide 34
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Standard Deviation of Gaussian Distribution
Area under curve
(probability that
ndashσltxlt+σ) is 68
5 or ~2σ ldquoSignificantrdquo
1 or ~3σ ldquoHighly Significantrdquo
1σ ldquoWithin errorsrdquo
5 ppm or ~5σ ldquoValid for HEPrdquo
See Sec 106 Testing of
Hypotheses
More complete Table in App A and B
Ah thatrsquos
highly
significant
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 35
Lecture 3 Slide 35
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Function of Gaussian Distribution
Area under curve
(probability that
ndashtσltxlt+tσ) is given
in Table at right
Probable Error (probability that
ndash067σltxlt+067σ) is 50
Error Function (probability that ndashtσltxlt+tσ )
Error Function Tabulated values-see App A
Complementary Error Function
(probability that ndashxlt-tσ AND xgt+tσ )
Prob(x outside tσ) = 1 - Prob(x within tσ)
More complete Table in App A and B
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 36
Lecture 3 Slide 36
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Useful Points on Gaussian Distribution
Full Width at Half Maximum FWHM
(See Prob 512)
Points of Inflection Occur at plusmnσ
(See Prob 513)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 37
Lecture 3 Slide 37
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Analysis and Gaussian Distribution
Adding a Constant Multiplying by a Constant
XX+A XBX and σB σ
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 38
Lecture 3 Slide 38
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Sum of Two Variables
Consider the derived quantity
Z=X + Y
(with X=0 and Y=0)
Error Propagation Addition
Error in Z
Multiple two probabilities 119875119903119900119887 119909 119910 = 119875119903119900119887 119909 ∙ 119875119903119900119887 119909 prop 119890119909119901 minus
1
2
1199092
1205901199092+
1199102
1205901199102
prop 119890119909119901 minus1
2
(119909+119910)2
1205901199092+120590119910
2 minus1199112
prop 119890119909119901 minus1
2
(119909+119910)2
1205901199092+120590119910
2 119890119909119901 minus
1199112
2
ICBST (Eq 553)
X+YZ and σx2 + σy
2 σz
2
(addition in quadrature for random independent variables)
Integrates to radic2π
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 39
Lecture 3 Slide 39
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Error Propagation
General formula for error propagation see [Taylor Secs 35 and 39]
Uncertainty as a function of one variable [Taylor Sec 35]
1 Consider a graphical method of estimating error
a) Consider an arbitaray function q(x)
b) Plot q(x) vs x
c) On the graph label
(1) qbest = q(xbest)
(2) qhi = q(xbest + δx)
(3) qlow = q(xbest- δx)
d) Making a linear approximation
x
qqxslopeqq
x
qqxslopeqq
bestbestlow
bestbesthi
e) Therefore
xx
Note the absolute value
How do we determine the error of a derived quantity Z(XYhellip) from errors in XYhellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 40
Lecture 3 Slide 40
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Error Propagation
General formula for uncertainty of a function of one variable
xx
[Taylor Eq 323]
Can you now derive for specific rules of error propagation
1 Addition and Subtraction [Taylor p 49]
2 Multiplication and Division [Taylor p 51]
3 Multiplication by a constant (exact number) [Taylor p 54]
4 Exponentiation (powers) [Taylor p 56]
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 41
Lecture 3 Slide 41
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Multiple Variables
Uncertainty of a function of multiple variables [Taylor Sec 311]
1 It can easily (no really) be shown that (see Taylor Sec 311) for a
function of several variables
)(
z
z
qy
y
qx
x
qzyxq
[Taylor Eq 347]
2 More correctly it can be shown that (see Taylor Sec 311) for a
function of several variables
)(
z
z
qy
y
qx
x
qzyxq
[Taylor Eq 347]
where the equals sign represents an upper bound as discussed above
3 For a function of several independent and random variables
)(
222
z
z
qy
y
qx
x
qzyxq
[Taylor Eq 348]
Again the proof is left for Ch 5
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 42
Lecture 3 Slide 42
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Product of Two Variables
Consider the arbitrary derived quantity
q(xy) of two independent random variables x and y
Expand q(xy) in a Taylor series about the expected values of x and y
(ie at points near X and Y)
Error Propagation General Case
119902 119909 119910 = 119902 119883 119884 + 120597119902
120597119909
119883 119909 minus 119883 +
120597119902
120597119910
119884
(119910 minus 119884)
Fixed shifts peak of distribution
Fixed Distribution centered at X with width σX
120575119902 119909 119910 = 120590119902 = 119902 119883 119884 + 120597119902
120597119909
119883120590119909
2
+ 120597119902
120597119910
119884
120590119910
2
0
How do we determine the error of a derived quantity Z(XYhellip) from errors in XYhellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 43
Lecture 3 Slide 43
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
SDOM of Gaussian Distribution
Standard Deviation of the Mean
The SDOM decreases as the square root of the number of measurements
That is the relative width σẊ of the distribution gets narrower as more
measurements are made
Each measurement
has similar σXi= σẊ
and similar partial
derivatives
Thushellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 44
Lecture 3 Slide 44
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Two Key Theorems from Probability
Central Limit Theorem
For random independent measurements (each with a well-define
expectation value and well-defined variance) the arithmetic mean
(average) will be approximately normally distributed
Principle of Maximum Likelihood
Given the N observed measurements x1 x2hellipxN the best estimates
for Ẋ and σ are those values for which the observed x1 x2hellipxN are
most likely
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 45
Lecture 3 Slide 45
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Mean of Gaussian Distribution as ldquoBest Estimaterdquo
Principle of Maximum Likelihood
Best Estimate of X is from maximum probability or minimum summation
Consider a data set
Each randomly distributed with
x1 x2 x3 hellipxN
To find the most likely value of the mean (the best estimate of ẋ)
find X that yields the highest probability for the data set
The combined probability for the full data set is the product
Minimize
Sum
Solve for
derivative wrst
X set to 0
Best
estimate
of X
prop1
120590119890minus(1199091minus119883)2 2120590 times
1
120590119890minus(1199092minus119883)2 2120590 times hellip times
1
120590119890minus(119909119873minus119883)2 2120590 =
1
120590119873119890 minus(119909119894minus119883)2 2120590
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 46
Lecture 3 Slide 46
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Uncertainty of ldquoBest Estimatesrdquo of Gaussian Distribution Principle of Maximum Likelihood
Best Estimate of X is from maximum probability or minimum summation
Consider a data set x1 x2 x3 hellipxN
To find the most likely value of the standard deviation (the best estimate of
the width of the x distribution) find σx that yields the highest probability for
the data set
The combined probability for the full data set is the product
Minimize
Sum
Solve for
derivative
wrst X set to 0
Best
estimate
of X
Best Estimate of σ is from maximum probability or minimum summation
Minimize
Sum
Best
estimate
of σ
Solve for
derivative
wrst σ set to 0
See
Prob 526
prop1
120590119890minus(1199091minus119883)2 2120590 times
1
120590119890minus(1199092minus119883)2 2120590 times hellip times
1
120590119890minus(119909119873minus119883)2 2120590 =
1
120590119873119890 minus(119909119894minus119883)2 2120590
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 47
Lecture 3 Slide 47
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Combining Data Sets
Weighted Averages
References Taylor Ch 7
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 48
Lecture 3 Slide 48
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Weighted Averages
Question How can we properly combine two or more separate
independent measurements of the same randomly distributed
quantity to determine a best combined value with uncertainty
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 49
Lecture 3 Slide 49
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider two measurements of the same quantity described by a random Gaussian distribution
ltx1gt x1 and ltx2gt x2
The probability of measuring two such measurements is
119875119903119900119887119909 1199091 1199092 = 119875119903119900119887119909 1199091 119875119903119900119887119909 1199092
= 1
12059011205902119890minus12059422 119908ℎ119890119903119890 1205942 equiv
1199091 minus 119883
1205901 2 +
1199092 minus 119883
1205902 2
To find the best value for χ find the maximum Prob or minimum χ2
Weighted Averages
Assume negligible systematic errors
Note χ2 or Chi squared is the sum of the squares of the deviations from the mean divided by
the corresponding uncertainty
Such methods are called ldquoMethods of Least Squaresrdquo They follow directly from the Principle of
Maximum Likelihood
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 50
Lecture 3 Slide 50
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
The probability of measuring two such measurements is
119875119903119900119887119909 1199091 1199092 = 119875119903119900119887119909 1199091 119875119903119900119887119909 1199092
= 1
12059011205902119890minus12059422 119908ℎ119890119903119890 1205942 equiv
1199091 minus 119883
1205901 2 +
1199092 minus 119883
1205902 2
To find the best value for χ find the maximum Prob or minimum χ2
Weighted Averages
This leads to 119909119882_119886119907119892 = 11990811199091+11990821199092
1199081+1199082=
119908119894 119909119894
119908119894 119908ℎ119890119903119890 119908119894 = 1
120590119894 2
Best Estimate of χ is from maximum probibility or minimum summation
Minimize Sum Solve for best estimate of χ Solve for derivative wrst χ set to 0
Note If w1=w2 we recover the standard result Xwavg= (12) (x1+x2)
Finally the width of a weighted average distribution is
120590119908119894119890119892 ℎ119905119890119889 119886119907119892 = 1
119908119894119894
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 51
Lecture 3 Slide 51
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Weighted Averages on Steroids
A very powerful method for combining data from different sources
with different methods and uncertainties (or indeed data of
related measured and calculated quantities) is Kalman filtering
The Kalman filter also known as
linear quadratic estimation (LQE)
is an algorithm that uses a series
of measurements observed over
time containing noise (random
variations) and other
inaccuracies and produces
estimates of unknown variables
that tend to be more precise than
those based on a single
measurement alone
The Kalman filter keeps track of the estimated state of the system and the variance or uncertainty of the
estimate The estimate is updated using a state transition model and measurements xk|k-1 denotes the
estimate of the systems state at time step k before the kth measurement yk has been taken into account Pk|k-1
is the corresponding uncertainty --Wikipedia 2013
Ludger Scherliess of USU Physics is a world expert at using Kalman filtering for the assimilation
of satellite and ground-based data and the USU GAMES model to predict space weather
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 52
Lecture 3 Slide 52
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Rejecting Data
Chauvenetrsquos Criterion
References Taylor Ch 6
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 53
Lecture 3 Slide 53
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly
ldquounreasonablerdquo data point from a set of randomly distributed
measurements
bull Never
bull Whenever it makes things look better
bull Chauvenetrsquos criterion provides a (quantitative) compromise
Rejecting Data
What is a good criteria for
rejecting data
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 54
Lecture 3 Slide 54
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly
ldquounreasonablerdquo data point from a set of randomly distributed
measurements
bull Never
Rejecting Data
Zallenrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 55
Lecture 3 Slide 55
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly
ldquounreasonablerdquo data point from a set of randomly distributed
measurements
bull Whenever it makes things look better
Rejecting Data
Disneyrsquos Criterion
Disneyrsquos First Law
Wishing will make it so
Disneyrsquos Second Law
Dreams are more colorful than reality
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 56
Lecture 3 Slide 56
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Data may be rejected if the expected number of measurements at
least as deviant as the suspect measurement is less than 50
Consider a set of N measurements of a single quantity
x1x2 helliphellipxN
Calculate ltxgt and σx and then determine the fractional deviations from the mean of all
the points
For the suspect point(s) xsuspect find the probability of such a point occurring in N
measurements
n = (expected number as deviant as xsuspect)
= N Prob(outside xsuspectσx)
Rejecting Data
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 57
Lecture 3 Slide 57
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Function of Gaussian Distribution
Area under curve
(probability that
ndashtσltxlt+tσ) is given
in Table at right
Probable Error (probability that
ndash067σltxlt+067σ) is 50
Error Function (probability that ndashtσltxlt+tσ )
Error Function Tabulated values-see App A
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 58
Lecture 3 Slide 58
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 59
Lecture 3 Slide 59
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashExample 1
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 60
Lecture 3 Slide 60
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Details (1)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 61
Lecture 3 Slide 61
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashDetails (2)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 62
Lecture 3 Slide 62
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos
Criterionmdash
Details (3)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 63
Lecture 3 Slide 63
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos
Criterionmdash
Example 2
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 64
Lecture 3 Slide 64
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Summary of Probability Theory
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 65
Lecture 3 Slide 65
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of
Probability
Theory-I
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 66
Lecture 3 Slide 66
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of
Probability
Theory-II
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 67
Lecture 3 Slide 67
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-III
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 68
Lecture 3 Slide 68
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-IV
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 13
Lecture 3 Slide 13
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Moments of Distribution Functions
The first moment for a probability distribution function is
119909 equiv 119909 = 119891119894119903119904119905 119898119900119898119890119899119905 = 119909 119891(119909)+infin
minusinfin119889119909
For a general distribution function
119909 equiv 119909 = 119891119894119903119904119905 119898119900119898119890119899119905 = 119909 119892(119909)
+infinminusinfin
119889119909
119892(119909)+infinminusinfin
119889119909
Generalizing the n
th moment is
119909119899 equiv 119909119899 = 119899119905ℎ 119898119900119898119890119899119905 = 119909119899 119892(119909)
+infinminusinfin
119889119909
119892(119909)+infinminusinfin
119889119909= 119909119899 119891(119909)
+infin
minusinfin119889119909
Oth moment equiv N 2
nd moment equiv 119909 minus 119909 2 rarr 1199092
1st moment equiv 119909 3
rd moment equiv kurtosis
0 (for a centered distribution)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 14
Lecture 3 Slide 14
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Moments of Distribution Functions
Generalizing the nth
moment is
119909119899 equiv 119909119899 = 119899119905ℎ 119898119900119898119890119899119905 = 119909119899 119892(119909)
+infinminusinfin
119889119909
119892(119909)+infinminusinfin
119889119909= 119909119899 119891(119909)
+infin
minusinfin119889119909
Oth
moment equiv N 2nd
moment equiv 119909 minus 119909 2 rarr 1199092
1st moment equiv 119909 3
rd moment equiv kurtosis
The nth
moment about the mean is
120583119899 equiv 119909 minus 119909 119899 = 119899119905ℎ 119898119900119898119890119899119905 119886119887119900119906119905 119905ℎ119890 119898119890119886119899
= 119909minus119909 119899 119892(119909)
+infinminusinfin
119889119909
119892(119909)+infinminusinfin
119889119909= 119909 minus 119909 119899 119891(119909)
+infin
minusinfin119889119909
The standard deviation (or second moment about the mean) is
1205901199092 equiv 1205832 equiv 119909 minus 119909 2 = 2119899119889 119898119900119898119890119899119905 119886119887119900119906119905 119905ℎ119890 119898119890119886119899
= 119909minus119909 2 119892(119909)
+infinminusinfin
119889119909
119892(119909)+infinminusinfin
119889119909= 119909 minus 119909 2 119891(119909)
+infin
minusinfin119889119909
0
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 15
Lecture 3 Slide 15
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Example of Continuous Distribution Functions and Expectation Values Harmonic Oscillator Example from Mechanics
Expected Values
The expectation value of a function Ɍ(x) is
Ɍ x equiv Ɍ x 119892 119909
+infin
minusinfin119889119909
119892 119909 +infin
minusinfin119889119909
= Ɍ(x) 119891(119909)+infin
minusinfin119889119909
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 16
Lecture 3 Slide 16
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Example of Continuous Distribution Functions and Expectation Values Boltzmann Distribution Example from Kinetic Theory
Expected Values
The expectation value of a function
Ɍ(x) is
Ɍ x equiv Ɍ x 119892 119909
+infin
minusinfin119889119909
119892 119909 +infin
minusinfin119889119909
= Ɍ(x) 119891(119909)+infin
minusinfin119889119909
The Boltzmann distribution function for velocities of particles as
a function of temperature T is
119875 119907 119879 = 4120587 119872
2120587 119896119861119879
3 2
1199072119890119909119901
121198721199072
12119896119861119879
Then
v = 119907 119875(119907)+infin
minusinfin119889119907 =
8 119896119861119879120587 119872
1 2
v2 = v2 119875(119907)+infin
minusinfin119889119907 =
3 119896119861119879 119872
1 2
implies KE = 1
2119872 v2 =
3
2119896119861119879
vpeak=
2 119896119861119879 119872
1 2
= 2 3
1 2
v2
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 17
Lecture 3 Slide 17
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Example of Continuous Distribution Functions and Expectation Values Fermi-Dirac Distribution Example from Kinetic Theory
For a system of identical fermions the average number of fermions in a single-particle state is given
by the FermindashDirac (FndashD) distribution
where kB is Boltzmanns constant T is the absolute temperature is the energy of the single-particle
state and μ is the total chemical potential
Since the FndashD distribution was derived using the Pauli exclusion principle which allows at most one
electron to occupy each possible state a result is that
When a quasi-continuum of energies has an associated density of states (ie the number of
states per unit energy range per unit volume) the average number of fermions per unit energy range
per unit volume is
where is called the Fermi function
so that
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 18
Lecture 3 Slide 18
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Example of Continuous Distribution Functions and Expectation Values Finite Square Well Example from Quantum Mechanics
Expectation Values The expectation value of a QM operator O(x) is O x equiv Ψlowast x 119874 119909
+infin
minusinfinΨ x 119889119909
Ψlowast x Ψ x +infin
minusinfin119889119909
For a finite square well of width L Ψn x = 2L sin
n π x
L
Ψnlowast x |Ψn x equiv
Ψnlowast x 119874 119909
+infin
minusinfinΨn x 119889119909
Ψnlowast x Ψn x
+infin
minusinfin119889119909
= 1
119909 = Ψnlowast x |x|Ψn x equiv
Ψnlowast x 119909
+infin
minusinfinΨn x 119889119909
Ψnlowast x Ψn x
+infin
minusinfin119889119909
= 1198712
119901 = Ψnlowast x |
ℏ
i
part
partx|Ψn x equiv
Ψnlowast x
ℏi
partpartx
+infin
minusinfinΨn x 119889119909
Ψnlowast x Ψn x
+infin
minusinfin119889119909
= 0
119864119899 = Ψnlowast x |iℏ
part
partt|Ψn x equiv
Ψnlowast x iℏ
partpartt
+infin
minusinfinΨn x 119889119909
Ψnlowast x Ψn x
+infin
minusinfin119889119909
=11989921205872ℏ2
2 119898 1198712
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 19
Lecture 3 Slide 19
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Distribution Functions
Available
on web site
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 20
Lecture 3 Slide 20
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
The Gaussian Distribution
Function
References Taylor Ch 5
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 21
Lecture 3 Slide 21
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Gaussian Integrals
Factorial Approximations
119899 asymp 2120587119899 1 2 119899119899 119890119909119901 minus119899 + 1
12 119899+ 119874
1
1198992
119897119900119892 119899 asymp 1
2 119897119900119892 2120587 + 119899 +
1
2 log 119899 minus 119899 +
1
12 119899+ 119874
1
1198992
119897119900119892 119899 asymp 119899 log 119899 minus 119899 (for all terms decreasing faster than linearly with n)
Gaussian Integrals
119868119898 = 2 119909119898 exp(minus1199092)infin
0 119889119909 mgt-1
119868119898 = 2 119910119899 exp(minus119910)infin
0 119889119910 equiv Γ(119899 + 1) 1199092 equiv 119910 2 119889119909 = 1199101 2 119889119910 119899 equiv 1
2 (119898 minus 1)
1198680 = Γ 119899 =1
2 = 120587 m=0 119899 = minus1
2
1198682 119896 = Γ 119896 +1
2 = 119896 minus 1
2 119896 minus 32 3
2 12 120587 even m m=2 kgt0 119899 = 119896 minus 1
2
1198682 119896+1 = Γ 119896 + 1 = 119896 odd m m=2 k+1gt0 119899 = 119896 ge 0
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 22
Lecture 3 Slide 22
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Gaussian Distribution Function
Width of Distribution
(standard deviation)
Center of Distribution
(mean) Distribution
Function
Independent
Variable
Gaussian Distribution Function
Normalization
Constant
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 23
Lecture 3 Slide 23
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Effects of Increasing N on Gaussian Distribution Functions
Consider the limiting distribution function as N infin and dx0
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 24
Lecture 3 Slide 24
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Defining the Gaussian
distribution function
in Mathcad
I suggest you ldquoinvestigaterdquo
these with the Mathcad sheet
on the web site
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 25
Lecture 3 Slide 25
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Using Mathcad to
define other common
distribution functions
Consider the limiting
distribution function
as N ȸ and ∆k0
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 26
Lecture 3 Slide 26
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider the Gaussian distribution function
Use the normalization condition to
evaluate the normalization constant
(see Taylor p 132)
The mean Ẋ is the first moment
of the Gaussian distribution
function (see Taylor p 134)
The standard deviation σx is the standard
deviation of the mean of the Gaussian
distribution function (see Taylor p 143)
Gaussian Distribution Moments
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 27
Lecture 3 Slide 27
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
When is mean x not Xbest
Answer When the
distribution is not
symmetric about X
Example Cauchy
Distribution
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 28
Lecture 3 Slide 28
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
When is mean x not Xbest
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 29
Lecture 3 Slide 29
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
When is mean x not Xbest
Answer When the
distribution is has
more than one peak
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 30
Lecture 3 Slide 30
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
The Gaussian Distribution
Function and Its Relation to
Errors
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 31
Lecture 3 Slide 31
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
A Review of Probabilities in Combination
Probability of a data set of N like measurements (x1x2hellipxN)
P (x1x2hellipxN) = P(x)1P(x2)hellipP(xN)
1 head AND 1 Four
P(H4) = P(H) P(4)
1 head OR 1 Four
P(H4) = P(H) + P(4) - P(H and 4) (true for a ldquonon-mutually exclusiverdquo events)
NOT 1 Six
P(NOT 6) = 1 - P(6)
1 Six OR 1 Four
P(64) = P(6) + P(4) (true for a ldquomutually exclusiverdquo single role)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 32
Lecture 3 Slide 32
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
The Gaussian Distribution Function and Its Relation to Errors
We will use the Gaussian distribution as applied to random
variables to develop the mathematical basis for bull Mean
bull Standard deviation
bull Standard deviation of the mean (SDOM)
bull Moments and expectation values
bull Error propagation formulas
bull Addition of errors in quadrature (for independent and random
measurements)
bull Numerical values for confidence limits (t-test)
bull Principle of maximal likelihood
bull Central limit theorem
bull Weighted distributions and Chi squared
bull Schwartz inequality (ie the uncertainty principle) (next lecture)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 33
Lecture 3 Slide 33
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider the Gaussian distribution function
Use the normalization condition to
evaluate the normalization constant
(see Taylor p 132)
The mean Ẋ is the first moment
of the Gaussian distribution
function (see Taylor p 134)
The standard deviation σx is the standard
deviation of the mean of the Gaussian
distribution function (see Taylor p 143)
Gaussian Distribution Moments
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 34
Lecture 3 Slide 34
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Standard Deviation of Gaussian Distribution
Area under curve
(probability that
ndashσltxlt+σ) is 68
5 or ~2σ ldquoSignificantrdquo
1 or ~3σ ldquoHighly Significantrdquo
1σ ldquoWithin errorsrdquo
5 ppm or ~5σ ldquoValid for HEPrdquo
See Sec 106 Testing of
Hypotheses
More complete Table in App A and B
Ah thatrsquos
highly
significant
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 35
Lecture 3 Slide 35
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Function of Gaussian Distribution
Area under curve
(probability that
ndashtσltxlt+tσ) is given
in Table at right
Probable Error (probability that
ndash067σltxlt+067σ) is 50
Error Function (probability that ndashtσltxlt+tσ )
Error Function Tabulated values-see App A
Complementary Error Function
(probability that ndashxlt-tσ AND xgt+tσ )
Prob(x outside tσ) = 1 - Prob(x within tσ)
More complete Table in App A and B
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 36
Lecture 3 Slide 36
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Useful Points on Gaussian Distribution
Full Width at Half Maximum FWHM
(See Prob 512)
Points of Inflection Occur at plusmnσ
(See Prob 513)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 37
Lecture 3 Slide 37
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Analysis and Gaussian Distribution
Adding a Constant Multiplying by a Constant
XX+A XBX and σB σ
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 38
Lecture 3 Slide 38
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Sum of Two Variables
Consider the derived quantity
Z=X + Y
(with X=0 and Y=0)
Error Propagation Addition
Error in Z
Multiple two probabilities 119875119903119900119887 119909 119910 = 119875119903119900119887 119909 ∙ 119875119903119900119887 119909 prop 119890119909119901 minus
1
2
1199092
1205901199092+
1199102
1205901199102
prop 119890119909119901 minus1
2
(119909+119910)2
1205901199092+120590119910
2 minus1199112
prop 119890119909119901 minus1
2
(119909+119910)2
1205901199092+120590119910
2 119890119909119901 minus
1199112
2
ICBST (Eq 553)
X+YZ and σx2 + σy
2 σz
2
(addition in quadrature for random independent variables)
Integrates to radic2π
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 39
Lecture 3 Slide 39
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Error Propagation
General formula for error propagation see [Taylor Secs 35 and 39]
Uncertainty as a function of one variable [Taylor Sec 35]
1 Consider a graphical method of estimating error
a) Consider an arbitaray function q(x)
b) Plot q(x) vs x
c) On the graph label
(1) qbest = q(xbest)
(2) qhi = q(xbest + δx)
(3) qlow = q(xbest- δx)
d) Making a linear approximation
x
qqxslopeqq
x
qqxslopeqq
bestbestlow
bestbesthi
e) Therefore
xx
Note the absolute value
How do we determine the error of a derived quantity Z(XYhellip) from errors in XYhellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 40
Lecture 3 Slide 40
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Error Propagation
General formula for uncertainty of a function of one variable
xx
[Taylor Eq 323]
Can you now derive for specific rules of error propagation
1 Addition and Subtraction [Taylor p 49]
2 Multiplication and Division [Taylor p 51]
3 Multiplication by a constant (exact number) [Taylor p 54]
4 Exponentiation (powers) [Taylor p 56]
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 41
Lecture 3 Slide 41
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Multiple Variables
Uncertainty of a function of multiple variables [Taylor Sec 311]
1 It can easily (no really) be shown that (see Taylor Sec 311) for a
function of several variables
)(
z
z
qy
y
qx
x
qzyxq
[Taylor Eq 347]
2 More correctly it can be shown that (see Taylor Sec 311) for a
function of several variables
)(
z
z
qy
y
qx
x
qzyxq
[Taylor Eq 347]
where the equals sign represents an upper bound as discussed above
3 For a function of several independent and random variables
)(
222
z
z
qy
y
qx
x
qzyxq
[Taylor Eq 348]
Again the proof is left for Ch 5
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 42
Lecture 3 Slide 42
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Product of Two Variables
Consider the arbitrary derived quantity
q(xy) of two independent random variables x and y
Expand q(xy) in a Taylor series about the expected values of x and y
(ie at points near X and Y)
Error Propagation General Case
119902 119909 119910 = 119902 119883 119884 + 120597119902
120597119909
119883 119909 minus 119883 +
120597119902
120597119910
119884
(119910 minus 119884)
Fixed shifts peak of distribution
Fixed Distribution centered at X with width σX
120575119902 119909 119910 = 120590119902 = 119902 119883 119884 + 120597119902
120597119909
119883120590119909
2
+ 120597119902
120597119910
119884
120590119910
2
0
How do we determine the error of a derived quantity Z(XYhellip) from errors in XYhellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 43
Lecture 3 Slide 43
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
SDOM of Gaussian Distribution
Standard Deviation of the Mean
The SDOM decreases as the square root of the number of measurements
That is the relative width σẊ of the distribution gets narrower as more
measurements are made
Each measurement
has similar σXi= σẊ
and similar partial
derivatives
Thushellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 44
Lecture 3 Slide 44
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Two Key Theorems from Probability
Central Limit Theorem
For random independent measurements (each with a well-define
expectation value and well-defined variance) the arithmetic mean
(average) will be approximately normally distributed
Principle of Maximum Likelihood
Given the N observed measurements x1 x2hellipxN the best estimates
for Ẋ and σ are those values for which the observed x1 x2hellipxN are
most likely
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 45
Lecture 3 Slide 45
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Mean of Gaussian Distribution as ldquoBest Estimaterdquo
Principle of Maximum Likelihood
Best Estimate of X is from maximum probability or minimum summation
Consider a data set
Each randomly distributed with
x1 x2 x3 hellipxN
To find the most likely value of the mean (the best estimate of ẋ)
find X that yields the highest probability for the data set
The combined probability for the full data set is the product
Minimize
Sum
Solve for
derivative wrst
X set to 0
Best
estimate
of X
prop1
120590119890minus(1199091minus119883)2 2120590 times
1
120590119890minus(1199092minus119883)2 2120590 times hellip times
1
120590119890minus(119909119873minus119883)2 2120590 =
1
120590119873119890 minus(119909119894minus119883)2 2120590
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 46
Lecture 3 Slide 46
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Uncertainty of ldquoBest Estimatesrdquo of Gaussian Distribution Principle of Maximum Likelihood
Best Estimate of X is from maximum probability or minimum summation
Consider a data set x1 x2 x3 hellipxN
To find the most likely value of the standard deviation (the best estimate of
the width of the x distribution) find σx that yields the highest probability for
the data set
The combined probability for the full data set is the product
Minimize
Sum
Solve for
derivative
wrst X set to 0
Best
estimate
of X
Best Estimate of σ is from maximum probability or minimum summation
Minimize
Sum
Best
estimate
of σ
Solve for
derivative
wrst σ set to 0
See
Prob 526
prop1
120590119890minus(1199091minus119883)2 2120590 times
1
120590119890minus(1199092minus119883)2 2120590 times hellip times
1
120590119890minus(119909119873minus119883)2 2120590 =
1
120590119873119890 minus(119909119894minus119883)2 2120590
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 47
Lecture 3 Slide 47
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Combining Data Sets
Weighted Averages
References Taylor Ch 7
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 48
Lecture 3 Slide 48
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Weighted Averages
Question How can we properly combine two or more separate
independent measurements of the same randomly distributed
quantity to determine a best combined value with uncertainty
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 49
Lecture 3 Slide 49
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider two measurements of the same quantity described by a random Gaussian distribution
ltx1gt x1 and ltx2gt x2
The probability of measuring two such measurements is
119875119903119900119887119909 1199091 1199092 = 119875119903119900119887119909 1199091 119875119903119900119887119909 1199092
= 1
12059011205902119890minus12059422 119908ℎ119890119903119890 1205942 equiv
1199091 minus 119883
1205901 2 +
1199092 minus 119883
1205902 2
To find the best value for χ find the maximum Prob or minimum χ2
Weighted Averages
Assume negligible systematic errors
Note χ2 or Chi squared is the sum of the squares of the deviations from the mean divided by
the corresponding uncertainty
Such methods are called ldquoMethods of Least Squaresrdquo They follow directly from the Principle of
Maximum Likelihood
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 50
Lecture 3 Slide 50
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
The probability of measuring two such measurements is
119875119903119900119887119909 1199091 1199092 = 119875119903119900119887119909 1199091 119875119903119900119887119909 1199092
= 1
12059011205902119890minus12059422 119908ℎ119890119903119890 1205942 equiv
1199091 minus 119883
1205901 2 +
1199092 minus 119883
1205902 2
To find the best value for χ find the maximum Prob or minimum χ2
Weighted Averages
This leads to 119909119882_119886119907119892 = 11990811199091+11990821199092
1199081+1199082=
119908119894 119909119894
119908119894 119908ℎ119890119903119890 119908119894 = 1
120590119894 2
Best Estimate of χ is from maximum probibility or minimum summation
Minimize Sum Solve for best estimate of χ Solve for derivative wrst χ set to 0
Note If w1=w2 we recover the standard result Xwavg= (12) (x1+x2)
Finally the width of a weighted average distribution is
120590119908119894119890119892 ℎ119905119890119889 119886119907119892 = 1
119908119894119894
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 51
Lecture 3 Slide 51
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Weighted Averages on Steroids
A very powerful method for combining data from different sources
with different methods and uncertainties (or indeed data of
related measured and calculated quantities) is Kalman filtering
The Kalman filter also known as
linear quadratic estimation (LQE)
is an algorithm that uses a series
of measurements observed over
time containing noise (random
variations) and other
inaccuracies and produces
estimates of unknown variables
that tend to be more precise than
those based on a single
measurement alone
The Kalman filter keeps track of the estimated state of the system and the variance or uncertainty of the
estimate The estimate is updated using a state transition model and measurements xk|k-1 denotes the
estimate of the systems state at time step k before the kth measurement yk has been taken into account Pk|k-1
is the corresponding uncertainty --Wikipedia 2013
Ludger Scherliess of USU Physics is a world expert at using Kalman filtering for the assimilation
of satellite and ground-based data and the USU GAMES model to predict space weather
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 52
Lecture 3 Slide 52
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Rejecting Data
Chauvenetrsquos Criterion
References Taylor Ch 6
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 53
Lecture 3 Slide 53
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly
ldquounreasonablerdquo data point from a set of randomly distributed
measurements
bull Never
bull Whenever it makes things look better
bull Chauvenetrsquos criterion provides a (quantitative) compromise
Rejecting Data
What is a good criteria for
rejecting data
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 54
Lecture 3 Slide 54
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly
ldquounreasonablerdquo data point from a set of randomly distributed
measurements
bull Never
Rejecting Data
Zallenrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 55
Lecture 3 Slide 55
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly
ldquounreasonablerdquo data point from a set of randomly distributed
measurements
bull Whenever it makes things look better
Rejecting Data
Disneyrsquos Criterion
Disneyrsquos First Law
Wishing will make it so
Disneyrsquos Second Law
Dreams are more colorful than reality
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 56
Lecture 3 Slide 56
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Data may be rejected if the expected number of measurements at
least as deviant as the suspect measurement is less than 50
Consider a set of N measurements of a single quantity
x1x2 helliphellipxN
Calculate ltxgt and σx and then determine the fractional deviations from the mean of all
the points
For the suspect point(s) xsuspect find the probability of such a point occurring in N
measurements
n = (expected number as deviant as xsuspect)
= N Prob(outside xsuspectσx)
Rejecting Data
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 57
Lecture 3 Slide 57
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Function of Gaussian Distribution
Area under curve
(probability that
ndashtσltxlt+tσ) is given
in Table at right
Probable Error (probability that
ndash067σltxlt+067σ) is 50
Error Function (probability that ndashtσltxlt+tσ )
Error Function Tabulated values-see App A
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 58
Lecture 3 Slide 58
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 59
Lecture 3 Slide 59
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashExample 1
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 60
Lecture 3 Slide 60
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Details (1)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 61
Lecture 3 Slide 61
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashDetails (2)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 62
Lecture 3 Slide 62
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos
Criterionmdash
Details (3)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 63
Lecture 3 Slide 63
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos
Criterionmdash
Example 2
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 64
Lecture 3 Slide 64
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Summary of Probability Theory
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 65
Lecture 3 Slide 65
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of
Probability
Theory-I
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 66
Lecture 3 Slide 66
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of
Probability
Theory-II
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 67
Lecture 3 Slide 67
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-III
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 68
Lecture 3 Slide 68
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-IV
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 14
Lecture 3 Slide 14
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Moments of Distribution Functions
Generalizing the nth
moment is
119909119899 equiv 119909119899 = 119899119905ℎ 119898119900119898119890119899119905 = 119909119899 119892(119909)
+infinminusinfin
119889119909
119892(119909)+infinminusinfin
119889119909= 119909119899 119891(119909)
+infin
minusinfin119889119909
Oth
moment equiv N 2nd
moment equiv 119909 minus 119909 2 rarr 1199092
1st moment equiv 119909 3
rd moment equiv kurtosis
The nth
moment about the mean is
120583119899 equiv 119909 minus 119909 119899 = 119899119905ℎ 119898119900119898119890119899119905 119886119887119900119906119905 119905ℎ119890 119898119890119886119899
= 119909minus119909 119899 119892(119909)
+infinminusinfin
119889119909
119892(119909)+infinminusinfin
119889119909= 119909 minus 119909 119899 119891(119909)
+infin
minusinfin119889119909
The standard deviation (or second moment about the mean) is
1205901199092 equiv 1205832 equiv 119909 minus 119909 2 = 2119899119889 119898119900119898119890119899119905 119886119887119900119906119905 119905ℎ119890 119898119890119886119899
= 119909minus119909 2 119892(119909)
+infinminusinfin
119889119909
119892(119909)+infinminusinfin
119889119909= 119909 minus 119909 2 119891(119909)
+infin
minusinfin119889119909
0
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 15
Lecture 3 Slide 15
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Example of Continuous Distribution Functions and Expectation Values Harmonic Oscillator Example from Mechanics
Expected Values
The expectation value of a function Ɍ(x) is
Ɍ x equiv Ɍ x 119892 119909
+infin
minusinfin119889119909
119892 119909 +infin
minusinfin119889119909
= Ɍ(x) 119891(119909)+infin
minusinfin119889119909
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 16
Lecture 3 Slide 16
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Example of Continuous Distribution Functions and Expectation Values Boltzmann Distribution Example from Kinetic Theory
Expected Values
The expectation value of a function
Ɍ(x) is
Ɍ x equiv Ɍ x 119892 119909
+infin
minusinfin119889119909
119892 119909 +infin
minusinfin119889119909
= Ɍ(x) 119891(119909)+infin
minusinfin119889119909
The Boltzmann distribution function for velocities of particles as
a function of temperature T is
119875 119907 119879 = 4120587 119872
2120587 119896119861119879
3 2
1199072119890119909119901
121198721199072
12119896119861119879
Then
v = 119907 119875(119907)+infin
minusinfin119889119907 =
8 119896119861119879120587 119872
1 2
v2 = v2 119875(119907)+infin
minusinfin119889119907 =
3 119896119861119879 119872
1 2
implies KE = 1
2119872 v2 =
3
2119896119861119879
vpeak=
2 119896119861119879 119872
1 2
= 2 3
1 2
v2
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 17
Lecture 3 Slide 17
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Example of Continuous Distribution Functions and Expectation Values Fermi-Dirac Distribution Example from Kinetic Theory
For a system of identical fermions the average number of fermions in a single-particle state is given
by the FermindashDirac (FndashD) distribution
where kB is Boltzmanns constant T is the absolute temperature is the energy of the single-particle
state and μ is the total chemical potential
Since the FndashD distribution was derived using the Pauli exclusion principle which allows at most one
electron to occupy each possible state a result is that
When a quasi-continuum of energies has an associated density of states (ie the number of
states per unit energy range per unit volume) the average number of fermions per unit energy range
per unit volume is
where is called the Fermi function
so that
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 18
Lecture 3 Slide 18
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Example of Continuous Distribution Functions and Expectation Values Finite Square Well Example from Quantum Mechanics
Expectation Values The expectation value of a QM operator O(x) is O x equiv Ψlowast x 119874 119909
+infin
minusinfinΨ x 119889119909
Ψlowast x Ψ x +infin
minusinfin119889119909
For a finite square well of width L Ψn x = 2L sin
n π x
L
Ψnlowast x |Ψn x equiv
Ψnlowast x 119874 119909
+infin
minusinfinΨn x 119889119909
Ψnlowast x Ψn x
+infin
minusinfin119889119909
= 1
119909 = Ψnlowast x |x|Ψn x equiv
Ψnlowast x 119909
+infin
minusinfinΨn x 119889119909
Ψnlowast x Ψn x
+infin
minusinfin119889119909
= 1198712
119901 = Ψnlowast x |
ℏ
i
part
partx|Ψn x equiv
Ψnlowast x
ℏi
partpartx
+infin
minusinfinΨn x 119889119909
Ψnlowast x Ψn x
+infin
minusinfin119889119909
= 0
119864119899 = Ψnlowast x |iℏ
part
partt|Ψn x equiv
Ψnlowast x iℏ
partpartt
+infin
minusinfinΨn x 119889119909
Ψnlowast x Ψn x
+infin
minusinfin119889119909
=11989921205872ℏ2
2 119898 1198712
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 19
Lecture 3 Slide 19
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Distribution Functions
Available
on web site
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 20
Lecture 3 Slide 20
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
The Gaussian Distribution
Function
References Taylor Ch 5
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 21
Lecture 3 Slide 21
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Gaussian Integrals
Factorial Approximations
119899 asymp 2120587119899 1 2 119899119899 119890119909119901 minus119899 + 1
12 119899+ 119874
1
1198992
119897119900119892 119899 asymp 1
2 119897119900119892 2120587 + 119899 +
1
2 log 119899 minus 119899 +
1
12 119899+ 119874
1
1198992
119897119900119892 119899 asymp 119899 log 119899 minus 119899 (for all terms decreasing faster than linearly with n)
Gaussian Integrals
119868119898 = 2 119909119898 exp(minus1199092)infin
0 119889119909 mgt-1
119868119898 = 2 119910119899 exp(minus119910)infin
0 119889119910 equiv Γ(119899 + 1) 1199092 equiv 119910 2 119889119909 = 1199101 2 119889119910 119899 equiv 1
2 (119898 minus 1)
1198680 = Γ 119899 =1
2 = 120587 m=0 119899 = minus1
2
1198682 119896 = Γ 119896 +1
2 = 119896 minus 1
2 119896 minus 32 3
2 12 120587 even m m=2 kgt0 119899 = 119896 minus 1
2
1198682 119896+1 = Γ 119896 + 1 = 119896 odd m m=2 k+1gt0 119899 = 119896 ge 0
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 22
Lecture 3 Slide 22
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Gaussian Distribution Function
Width of Distribution
(standard deviation)
Center of Distribution
(mean) Distribution
Function
Independent
Variable
Gaussian Distribution Function
Normalization
Constant
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 23
Lecture 3 Slide 23
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Effects of Increasing N on Gaussian Distribution Functions
Consider the limiting distribution function as N infin and dx0
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 24
Lecture 3 Slide 24
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Defining the Gaussian
distribution function
in Mathcad
I suggest you ldquoinvestigaterdquo
these with the Mathcad sheet
on the web site
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 25
Lecture 3 Slide 25
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Using Mathcad to
define other common
distribution functions
Consider the limiting
distribution function
as N ȸ and ∆k0
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 26
Lecture 3 Slide 26
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider the Gaussian distribution function
Use the normalization condition to
evaluate the normalization constant
(see Taylor p 132)
The mean Ẋ is the first moment
of the Gaussian distribution
function (see Taylor p 134)
The standard deviation σx is the standard
deviation of the mean of the Gaussian
distribution function (see Taylor p 143)
Gaussian Distribution Moments
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 27
Lecture 3 Slide 27
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
When is mean x not Xbest
Answer When the
distribution is not
symmetric about X
Example Cauchy
Distribution
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 28
Lecture 3 Slide 28
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
When is mean x not Xbest
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 29
Lecture 3 Slide 29
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
When is mean x not Xbest
Answer When the
distribution is has
more than one peak
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 30
Lecture 3 Slide 30
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
The Gaussian Distribution
Function and Its Relation to
Errors
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 31
Lecture 3 Slide 31
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
A Review of Probabilities in Combination
Probability of a data set of N like measurements (x1x2hellipxN)
P (x1x2hellipxN) = P(x)1P(x2)hellipP(xN)
1 head AND 1 Four
P(H4) = P(H) P(4)
1 head OR 1 Four
P(H4) = P(H) + P(4) - P(H and 4) (true for a ldquonon-mutually exclusiverdquo events)
NOT 1 Six
P(NOT 6) = 1 - P(6)
1 Six OR 1 Four
P(64) = P(6) + P(4) (true for a ldquomutually exclusiverdquo single role)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 32
Lecture 3 Slide 32
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
The Gaussian Distribution Function and Its Relation to Errors
We will use the Gaussian distribution as applied to random
variables to develop the mathematical basis for bull Mean
bull Standard deviation
bull Standard deviation of the mean (SDOM)
bull Moments and expectation values
bull Error propagation formulas
bull Addition of errors in quadrature (for independent and random
measurements)
bull Numerical values for confidence limits (t-test)
bull Principle of maximal likelihood
bull Central limit theorem
bull Weighted distributions and Chi squared
bull Schwartz inequality (ie the uncertainty principle) (next lecture)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 33
Lecture 3 Slide 33
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider the Gaussian distribution function
Use the normalization condition to
evaluate the normalization constant
(see Taylor p 132)
The mean Ẋ is the first moment
of the Gaussian distribution
function (see Taylor p 134)
The standard deviation σx is the standard
deviation of the mean of the Gaussian
distribution function (see Taylor p 143)
Gaussian Distribution Moments
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 34
Lecture 3 Slide 34
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Standard Deviation of Gaussian Distribution
Area under curve
(probability that
ndashσltxlt+σ) is 68
5 or ~2σ ldquoSignificantrdquo
1 or ~3σ ldquoHighly Significantrdquo
1σ ldquoWithin errorsrdquo
5 ppm or ~5σ ldquoValid for HEPrdquo
See Sec 106 Testing of
Hypotheses
More complete Table in App A and B
Ah thatrsquos
highly
significant
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 35
Lecture 3 Slide 35
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Function of Gaussian Distribution
Area under curve
(probability that
ndashtσltxlt+tσ) is given
in Table at right
Probable Error (probability that
ndash067σltxlt+067σ) is 50
Error Function (probability that ndashtσltxlt+tσ )
Error Function Tabulated values-see App A
Complementary Error Function
(probability that ndashxlt-tσ AND xgt+tσ )
Prob(x outside tσ) = 1 - Prob(x within tσ)
More complete Table in App A and B
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 36
Lecture 3 Slide 36
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Useful Points on Gaussian Distribution
Full Width at Half Maximum FWHM
(See Prob 512)
Points of Inflection Occur at plusmnσ
(See Prob 513)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 37
Lecture 3 Slide 37
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Analysis and Gaussian Distribution
Adding a Constant Multiplying by a Constant
XX+A XBX and σB σ
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 38
Lecture 3 Slide 38
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Sum of Two Variables
Consider the derived quantity
Z=X + Y
(with X=0 and Y=0)
Error Propagation Addition
Error in Z
Multiple two probabilities 119875119903119900119887 119909 119910 = 119875119903119900119887 119909 ∙ 119875119903119900119887 119909 prop 119890119909119901 minus
1
2
1199092
1205901199092+
1199102
1205901199102
prop 119890119909119901 minus1
2
(119909+119910)2
1205901199092+120590119910
2 minus1199112
prop 119890119909119901 minus1
2
(119909+119910)2
1205901199092+120590119910
2 119890119909119901 minus
1199112
2
ICBST (Eq 553)
X+YZ and σx2 + σy
2 σz
2
(addition in quadrature for random independent variables)
Integrates to radic2π
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 39
Lecture 3 Slide 39
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Error Propagation
General formula for error propagation see [Taylor Secs 35 and 39]
Uncertainty as a function of one variable [Taylor Sec 35]
1 Consider a graphical method of estimating error
a) Consider an arbitaray function q(x)
b) Plot q(x) vs x
c) On the graph label
(1) qbest = q(xbest)
(2) qhi = q(xbest + δx)
(3) qlow = q(xbest- δx)
d) Making a linear approximation
x
qqxslopeqq
x
qqxslopeqq
bestbestlow
bestbesthi
e) Therefore
xx
Note the absolute value
How do we determine the error of a derived quantity Z(XYhellip) from errors in XYhellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 40
Lecture 3 Slide 40
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Error Propagation
General formula for uncertainty of a function of one variable
xx
[Taylor Eq 323]
Can you now derive for specific rules of error propagation
1 Addition and Subtraction [Taylor p 49]
2 Multiplication and Division [Taylor p 51]
3 Multiplication by a constant (exact number) [Taylor p 54]
4 Exponentiation (powers) [Taylor p 56]
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 41
Lecture 3 Slide 41
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Multiple Variables
Uncertainty of a function of multiple variables [Taylor Sec 311]
1 It can easily (no really) be shown that (see Taylor Sec 311) for a
function of several variables
)(
z
z
qy
y
qx
x
qzyxq
[Taylor Eq 347]
2 More correctly it can be shown that (see Taylor Sec 311) for a
function of several variables
)(
z
z
qy
y
qx
x
qzyxq
[Taylor Eq 347]
where the equals sign represents an upper bound as discussed above
3 For a function of several independent and random variables
)(
222
z
z
qy
y
qx
x
qzyxq
[Taylor Eq 348]
Again the proof is left for Ch 5
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 42
Lecture 3 Slide 42
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Product of Two Variables
Consider the arbitrary derived quantity
q(xy) of two independent random variables x and y
Expand q(xy) in a Taylor series about the expected values of x and y
(ie at points near X and Y)
Error Propagation General Case
119902 119909 119910 = 119902 119883 119884 + 120597119902
120597119909
119883 119909 minus 119883 +
120597119902
120597119910
119884
(119910 minus 119884)
Fixed shifts peak of distribution
Fixed Distribution centered at X with width σX
120575119902 119909 119910 = 120590119902 = 119902 119883 119884 + 120597119902
120597119909
119883120590119909
2
+ 120597119902
120597119910
119884
120590119910
2
0
How do we determine the error of a derived quantity Z(XYhellip) from errors in XYhellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 43
Lecture 3 Slide 43
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
SDOM of Gaussian Distribution
Standard Deviation of the Mean
The SDOM decreases as the square root of the number of measurements
That is the relative width σẊ of the distribution gets narrower as more
measurements are made
Each measurement
has similar σXi= σẊ
and similar partial
derivatives
Thushellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 44
Lecture 3 Slide 44
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Two Key Theorems from Probability
Central Limit Theorem
For random independent measurements (each with a well-define
expectation value and well-defined variance) the arithmetic mean
(average) will be approximately normally distributed
Principle of Maximum Likelihood
Given the N observed measurements x1 x2hellipxN the best estimates
for Ẋ and σ are those values for which the observed x1 x2hellipxN are
most likely
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 45
Lecture 3 Slide 45
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Mean of Gaussian Distribution as ldquoBest Estimaterdquo
Principle of Maximum Likelihood
Best Estimate of X is from maximum probability or minimum summation
Consider a data set
Each randomly distributed with
x1 x2 x3 hellipxN
To find the most likely value of the mean (the best estimate of ẋ)
find X that yields the highest probability for the data set
The combined probability for the full data set is the product
Minimize
Sum
Solve for
derivative wrst
X set to 0
Best
estimate
of X
prop1
120590119890minus(1199091minus119883)2 2120590 times
1
120590119890minus(1199092minus119883)2 2120590 times hellip times
1
120590119890minus(119909119873minus119883)2 2120590 =
1
120590119873119890 minus(119909119894minus119883)2 2120590
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 46
Lecture 3 Slide 46
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Uncertainty of ldquoBest Estimatesrdquo of Gaussian Distribution Principle of Maximum Likelihood
Best Estimate of X is from maximum probability or minimum summation
Consider a data set x1 x2 x3 hellipxN
To find the most likely value of the standard deviation (the best estimate of
the width of the x distribution) find σx that yields the highest probability for
the data set
The combined probability for the full data set is the product
Minimize
Sum
Solve for
derivative
wrst X set to 0
Best
estimate
of X
Best Estimate of σ is from maximum probability or minimum summation
Minimize
Sum
Best
estimate
of σ
Solve for
derivative
wrst σ set to 0
See
Prob 526
prop1
120590119890minus(1199091minus119883)2 2120590 times
1
120590119890minus(1199092minus119883)2 2120590 times hellip times
1
120590119890minus(119909119873minus119883)2 2120590 =
1
120590119873119890 minus(119909119894minus119883)2 2120590
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 47
Lecture 3 Slide 47
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Combining Data Sets
Weighted Averages
References Taylor Ch 7
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 48
Lecture 3 Slide 48
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Weighted Averages
Question How can we properly combine two or more separate
independent measurements of the same randomly distributed
quantity to determine a best combined value with uncertainty
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 49
Lecture 3 Slide 49
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider two measurements of the same quantity described by a random Gaussian distribution
ltx1gt x1 and ltx2gt x2
The probability of measuring two such measurements is
119875119903119900119887119909 1199091 1199092 = 119875119903119900119887119909 1199091 119875119903119900119887119909 1199092
= 1
12059011205902119890minus12059422 119908ℎ119890119903119890 1205942 equiv
1199091 minus 119883
1205901 2 +
1199092 minus 119883
1205902 2
To find the best value for χ find the maximum Prob or minimum χ2
Weighted Averages
Assume negligible systematic errors
Note χ2 or Chi squared is the sum of the squares of the deviations from the mean divided by
the corresponding uncertainty
Such methods are called ldquoMethods of Least Squaresrdquo They follow directly from the Principle of
Maximum Likelihood
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 50
Lecture 3 Slide 50
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
The probability of measuring two such measurements is
119875119903119900119887119909 1199091 1199092 = 119875119903119900119887119909 1199091 119875119903119900119887119909 1199092
= 1
12059011205902119890minus12059422 119908ℎ119890119903119890 1205942 equiv
1199091 minus 119883
1205901 2 +
1199092 minus 119883
1205902 2
To find the best value for χ find the maximum Prob or minimum χ2
Weighted Averages
This leads to 119909119882_119886119907119892 = 11990811199091+11990821199092
1199081+1199082=
119908119894 119909119894
119908119894 119908ℎ119890119903119890 119908119894 = 1
120590119894 2
Best Estimate of χ is from maximum probibility or minimum summation
Minimize Sum Solve for best estimate of χ Solve for derivative wrst χ set to 0
Note If w1=w2 we recover the standard result Xwavg= (12) (x1+x2)
Finally the width of a weighted average distribution is
120590119908119894119890119892 ℎ119905119890119889 119886119907119892 = 1
119908119894119894
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 51
Lecture 3 Slide 51
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Weighted Averages on Steroids
A very powerful method for combining data from different sources
with different methods and uncertainties (or indeed data of
related measured and calculated quantities) is Kalman filtering
The Kalman filter also known as
linear quadratic estimation (LQE)
is an algorithm that uses a series
of measurements observed over
time containing noise (random
variations) and other
inaccuracies and produces
estimates of unknown variables
that tend to be more precise than
those based on a single
measurement alone
The Kalman filter keeps track of the estimated state of the system and the variance or uncertainty of the
estimate The estimate is updated using a state transition model and measurements xk|k-1 denotes the
estimate of the systems state at time step k before the kth measurement yk has been taken into account Pk|k-1
is the corresponding uncertainty --Wikipedia 2013
Ludger Scherliess of USU Physics is a world expert at using Kalman filtering for the assimilation
of satellite and ground-based data and the USU GAMES model to predict space weather
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 52
Lecture 3 Slide 52
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Rejecting Data
Chauvenetrsquos Criterion
References Taylor Ch 6
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 53
Lecture 3 Slide 53
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly
ldquounreasonablerdquo data point from a set of randomly distributed
measurements
bull Never
bull Whenever it makes things look better
bull Chauvenetrsquos criterion provides a (quantitative) compromise
Rejecting Data
What is a good criteria for
rejecting data
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 54
Lecture 3 Slide 54
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly
ldquounreasonablerdquo data point from a set of randomly distributed
measurements
bull Never
Rejecting Data
Zallenrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 55
Lecture 3 Slide 55
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly
ldquounreasonablerdquo data point from a set of randomly distributed
measurements
bull Whenever it makes things look better
Rejecting Data
Disneyrsquos Criterion
Disneyrsquos First Law
Wishing will make it so
Disneyrsquos Second Law
Dreams are more colorful than reality
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 56
Lecture 3 Slide 56
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Data may be rejected if the expected number of measurements at
least as deviant as the suspect measurement is less than 50
Consider a set of N measurements of a single quantity
x1x2 helliphellipxN
Calculate ltxgt and σx and then determine the fractional deviations from the mean of all
the points
For the suspect point(s) xsuspect find the probability of such a point occurring in N
measurements
n = (expected number as deviant as xsuspect)
= N Prob(outside xsuspectσx)
Rejecting Data
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 57
Lecture 3 Slide 57
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Function of Gaussian Distribution
Area under curve
(probability that
ndashtσltxlt+tσ) is given
in Table at right
Probable Error (probability that
ndash067σltxlt+067σ) is 50
Error Function (probability that ndashtσltxlt+tσ )
Error Function Tabulated values-see App A
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 58
Lecture 3 Slide 58
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 59
Lecture 3 Slide 59
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashExample 1
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 60
Lecture 3 Slide 60
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Details (1)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 61
Lecture 3 Slide 61
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashDetails (2)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 62
Lecture 3 Slide 62
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos
Criterionmdash
Details (3)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 63
Lecture 3 Slide 63
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos
Criterionmdash
Example 2
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 64
Lecture 3 Slide 64
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Summary of Probability Theory
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 65
Lecture 3 Slide 65
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of
Probability
Theory-I
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 66
Lecture 3 Slide 66
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of
Probability
Theory-II
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 67
Lecture 3 Slide 67
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-III
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 68
Lecture 3 Slide 68
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-IV
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 15
Lecture 3 Slide 15
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Example of Continuous Distribution Functions and Expectation Values Harmonic Oscillator Example from Mechanics
Expected Values
The expectation value of a function Ɍ(x) is
Ɍ x equiv Ɍ x 119892 119909
+infin
minusinfin119889119909
119892 119909 +infin
minusinfin119889119909
= Ɍ(x) 119891(119909)+infin
minusinfin119889119909
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 16
Lecture 3 Slide 16
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Example of Continuous Distribution Functions and Expectation Values Boltzmann Distribution Example from Kinetic Theory
Expected Values
The expectation value of a function
Ɍ(x) is
Ɍ x equiv Ɍ x 119892 119909
+infin
minusinfin119889119909
119892 119909 +infin
minusinfin119889119909
= Ɍ(x) 119891(119909)+infin
minusinfin119889119909
The Boltzmann distribution function for velocities of particles as
a function of temperature T is
119875 119907 119879 = 4120587 119872
2120587 119896119861119879
3 2
1199072119890119909119901
121198721199072
12119896119861119879
Then
v = 119907 119875(119907)+infin
minusinfin119889119907 =
8 119896119861119879120587 119872
1 2
v2 = v2 119875(119907)+infin
minusinfin119889119907 =
3 119896119861119879 119872
1 2
implies KE = 1
2119872 v2 =
3
2119896119861119879
vpeak=
2 119896119861119879 119872
1 2
= 2 3
1 2
v2
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 17
Lecture 3 Slide 17
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Example of Continuous Distribution Functions and Expectation Values Fermi-Dirac Distribution Example from Kinetic Theory
For a system of identical fermions the average number of fermions in a single-particle state is given
by the FermindashDirac (FndashD) distribution
where kB is Boltzmanns constant T is the absolute temperature is the energy of the single-particle
state and μ is the total chemical potential
Since the FndashD distribution was derived using the Pauli exclusion principle which allows at most one
electron to occupy each possible state a result is that
When a quasi-continuum of energies has an associated density of states (ie the number of
states per unit energy range per unit volume) the average number of fermions per unit energy range
per unit volume is
where is called the Fermi function
so that
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 18
Lecture 3 Slide 18
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Example of Continuous Distribution Functions and Expectation Values Finite Square Well Example from Quantum Mechanics
Expectation Values The expectation value of a QM operator O(x) is O x equiv Ψlowast x 119874 119909
+infin
minusinfinΨ x 119889119909
Ψlowast x Ψ x +infin
minusinfin119889119909
For a finite square well of width L Ψn x = 2L sin
n π x
L
Ψnlowast x |Ψn x equiv
Ψnlowast x 119874 119909
+infin
minusinfinΨn x 119889119909
Ψnlowast x Ψn x
+infin
minusinfin119889119909
= 1
119909 = Ψnlowast x |x|Ψn x equiv
Ψnlowast x 119909
+infin
minusinfinΨn x 119889119909
Ψnlowast x Ψn x
+infin
minusinfin119889119909
= 1198712
119901 = Ψnlowast x |
ℏ
i
part
partx|Ψn x equiv
Ψnlowast x
ℏi
partpartx
+infin
minusinfinΨn x 119889119909
Ψnlowast x Ψn x
+infin
minusinfin119889119909
= 0
119864119899 = Ψnlowast x |iℏ
part
partt|Ψn x equiv
Ψnlowast x iℏ
partpartt
+infin
minusinfinΨn x 119889119909
Ψnlowast x Ψn x
+infin
minusinfin119889119909
=11989921205872ℏ2
2 119898 1198712
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 19
Lecture 3 Slide 19
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Distribution Functions
Available
on web site
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 20
Lecture 3 Slide 20
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
The Gaussian Distribution
Function
References Taylor Ch 5
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 21
Lecture 3 Slide 21
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Gaussian Integrals
Factorial Approximations
119899 asymp 2120587119899 1 2 119899119899 119890119909119901 minus119899 + 1
12 119899+ 119874
1
1198992
119897119900119892 119899 asymp 1
2 119897119900119892 2120587 + 119899 +
1
2 log 119899 minus 119899 +
1
12 119899+ 119874
1
1198992
119897119900119892 119899 asymp 119899 log 119899 minus 119899 (for all terms decreasing faster than linearly with n)
Gaussian Integrals
119868119898 = 2 119909119898 exp(minus1199092)infin
0 119889119909 mgt-1
119868119898 = 2 119910119899 exp(minus119910)infin
0 119889119910 equiv Γ(119899 + 1) 1199092 equiv 119910 2 119889119909 = 1199101 2 119889119910 119899 equiv 1
2 (119898 minus 1)
1198680 = Γ 119899 =1
2 = 120587 m=0 119899 = minus1
2
1198682 119896 = Γ 119896 +1
2 = 119896 minus 1
2 119896 minus 32 3
2 12 120587 even m m=2 kgt0 119899 = 119896 minus 1
2
1198682 119896+1 = Γ 119896 + 1 = 119896 odd m m=2 k+1gt0 119899 = 119896 ge 0
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 22
Lecture 3 Slide 22
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Gaussian Distribution Function
Width of Distribution
(standard deviation)
Center of Distribution
(mean) Distribution
Function
Independent
Variable
Gaussian Distribution Function
Normalization
Constant
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 23
Lecture 3 Slide 23
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Effects of Increasing N on Gaussian Distribution Functions
Consider the limiting distribution function as N infin and dx0
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 24
Lecture 3 Slide 24
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Defining the Gaussian
distribution function
in Mathcad
I suggest you ldquoinvestigaterdquo
these with the Mathcad sheet
on the web site
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 25
Lecture 3 Slide 25
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Using Mathcad to
define other common
distribution functions
Consider the limiting
distribution function
as N ȸ and ∆k0
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 26
Lecture 3 Slide 26
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider the Gaussian distribution function
Use the normalization condition to
evaluate the normalization constant
(see Taylor p 132)
The mean Ẋ is the first moment
of the Gaussian distribution
function (see Taylor p 134)
The standard deviation σx is the standard
deviation of the mean of the Gaussian
distribution function (see Taylor p 143)
Gaussian Distribution Moments
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 27
Lecture 3 Slide 27
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
When is mean x not Xbest
Answer When the
distribution is not
symmetric about X
Example Cauchy
Distribution
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 28
Lecture 3 Slide 28
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
When is mean x not Xbest
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 29
Lecture 3 Slide 29
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
When is mean x not Xbest
Answer When the
distribution is has
more than one peak
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 30
Lecture 3 Slide 30
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
The Gaussian Distribution
Function and Its Relation to
Errors
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 31
Lecture 3 Slide 31
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
A Review of Probabilities in Combination
Probability of a data set of N like measurements (x1x2hellipxN)
P (x1x2hellipxN) = P(x)1P(x2)hellipP(xN)
1 head AND 1 Four
P(H4) = P(H) P(4)
1 head OR 1 Four
P(H4) = P(H) + P(4) - P(H and 4) (true for a ldquonon-mutually exclusiverdquo events)
NOT 1 Six
P(NOT 6) = 1 - P(6)
1 Six OR 1 Four
P(64) = P(6) + P(4) (true for a ldquomutually exclusiverdquo single role)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 32
Lecture 3 Slide 32
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
The Gaussian Distribution Function and Its Relation to Errors
We will use the Gaussian distribution as applied to random
variables to develop the mathematical basis for bull Mean
bull Standard deviation
bull Standard deviation of the mean (SDOM)
bull Moments and expectation values
bull Error propagation formulas
bull Addition of errors in quadrature (for independent and random
measurements)
bull Numerical values for confidence limits (t-test)
bull Principle of maximal likelihood
bull Central limit theorem
bull Weighted distributions and Chi squared
bull Schwartz inequality (ie the uncertainty principle) (next lecture)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 33
Lecture 3 Slide 33
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider the Gaussian distribution function
Use the normalization condition to
evaluate the normalization constant
(see Taylor p 132)
The mean Ẋ is the first moment
of the Gaussian distribution
function (see Taylor p 134)
The standard deviation σx is the standard
deviation of the mean of the Gaussian
distribution function (see Taylor p 143)
Gaussian Distribution Moments
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 34
Lecture 3 Slide 34
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Standard Deviation of Gaussian Distribution
Area under curve
(probability that
ndashσltxlt+σ) is 68
5 or ~2σ ldquoSignificantrdquo
1 or ~3σ ldquoHighly Significantrdquo
1σ ldquoWithin errorsrdquo
5 ppm or ~5σ ldquoValid for HEPrdquo
See Sec 106 Testing of
Hypotheses
More complete Table in App A and B
Ah thatrsquos
highly
significant
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 35
Lecture 3 Slide 35
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Function of Gaussian Distribution
Area under curve
(probability that
ndashtσltxlt+tσ) is given
in Table at right
Probable Error (probability that
ndash067σltxlt+067σ) is 50
Error Function (probability that ndashtσltxlt+tσ )
Error Function Tabulated values-see App A
Complementary Error Function
(probability that ndashxlt-tσ AND xgt+tσ )
Prob(x outside tσ) = 1 - Prob(x within tσ)
More complete Table in App A and B
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 36
Lecture 3 Slide 36
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Useful Points on Gaussian Distribution
Full Width at Half Maximum FWHM
(See Prob 512)
Points of Inflection Occur at plusmnσ
(See Prob 513)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 37
Lecture 3 Slide 37
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Analysis and Gaussian Distribution
Adding a Constant Multiplying by a Constant
XX+A XBX and σB σ
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 38
Lecture 3 Slide 38
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Sum of Two Variables
Consider the derived quantity
Z=X + Y
(with X=0 and Y=0)
Error Propagation Addition
Error in Z
Multiple two probabilities 119875119903119900119887 119909 119910 = 119875119903119900119887 119909 ∙ 119875119903119900119887 119909 prop 119890119909119901 minus
1
2
1199092
1205901199092+
1199102
1205901199102
prop 119890119909119901 minus1
2
(119909+119910)2
1205901199092+120590119910
2 minus1199112
prop 119890119909119901 minus1
2
(119909+119910)2
1205901199092+120590119910
2 119890119909119901 minus
1199112
2
ICBST (Eq 553)
X+YZ and σx2 + σy
2 σz
2
(addition in quadrature for random independent variables)
Integrates to radic2π
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 39
Lecture 3 Slide 39
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Error Propagation
General formula for error propagation see [Taylor Secs 35 and 39]
Uncertainty as a function of one variable [Taylor Sec 35]
1 Consider a graphical method of estimating error
a) Consider an arbitaray function q(x)
b) Plot q(x) vs x
c) On the graph label
(1) qbest = q(xbest)
(2) qhi = q(xbest + δx)
(3) qlow = q(xbest- δx)
d) Making a linear approximation
x
qqxslopeqq
x
qqxslopeqq
bestbestlow
bestbesthi
e) Therefore
xx
Note the absolute value
How do we determine the error of a derived quantity Z(XYhellip) from errors in XYhellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 40
Lecture 3 Slide 40
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Error Propagation
General formula for uncertainty of a function of one variable
xx
[Taylor Eq 323]
Can you now derive for specific rules of error propagation
1 Addition and Subtraction [Taylor p 49]
2 Multiplication and Division [Taylor p 51]
3 Multiplication by a constant (exact number) [Taylor p 54]
4 Exponentiation (powers) [Taylor p 56]
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 41
Lecture 3 Slide 41
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Multiple Variables
Uncertainty of a function of multiple variables [Taylor Sec 311]
1 It can easily (no really) be shown that (see Taylor Sec 311) for a
function of several variables
)(
z
z
qy
y
qx
x
qzyxq
[Taylor Eq 347]
2 More correctly it can be shown that (see Taylor Sec 311) for a
function of several variables
)(
z
z
qy
y
qx
x
qzyxq
[Taylor Eq 347]
where the equals sign represents an upper bound as discussed above
3 For a function of several independent and random variables
)(
222
z
z
qy
y
qx
x
qzyxq
[Taylor Eq 348]
Again the proof is left for Ch 5
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 42
Lecture 3 Slide 42
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Product of Two Variables
Consider the arbitrary derived quantity
q(xy) of two independent random variables x and y
Expand q(xy) in a Taylor series about the expected values of x and y
(ie at points near X and Y)
Error Propagation General Case
119902 119909 119910 = 119902 119883 119884 + 120597119902
120597119909
119883 119909 minus 119883 +
120597119902
120597119910
119884
(119910 minus 119884)
Fixed shifts peak of distribution
Fixed Distribution centered at X with width σX
120575119902 119909 119910 = 120590119902 = 119902 119883 119884 + 120597119902
120597119909
119883120590119909
2
+ 120597119902
120597119910
119884
120590119910
2
0
How do we determine the error of a derived quantity Z(XYhellip) from errors in XYhellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 43
Lecture 3 Slide 43
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
SDOM of Gaussian Distribution
Standard Deviation of the Mean
The SDOM decreases as the square root of the number of measurements
That is the relative width σẊ of the distribution gets narrower as more
measurements are made
Each measurement
has similar σXi= σẊ
and similar partial
derivatives
Thushellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 44
Lecture 3 Slide 44
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Two Key Theorems from Probability
Central Limit Theorem
For random independent measurements (each with a well-define
expectation value and well-defined variance) the arithmetic mean
(average) will be approximately normally distributed
Principle of Maximum Likelihood
Given the N observed measurements x1 x2hellipxN the best estimates
for Ẋ and σ are those values for which the observed x1 x2hellipxN are
most likely
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 45
Lecture 3 Slide 45
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Mean of Gaussian Distribution as ldquoBest Estimaterdquo
Principle of Maximum Likelihood
Best Estimate of X is from maximum probability or minimum summation
Consider a data set
Each randomly distributed with
x1 x2 x3 hellipxN
To find the most likely value of the mean (the best estimate of ẋ)
find X that yields the highest probability for the data set
The combined probability for the full data set is the product
Minimize
Sum
Solve for
derivative wrst
X set to 0
Best
estimate
of X
prop1
120590119890minus(1199091minus119883)2 2120590 times
1
120590119890minus(1199092minus119883)2 2120590 times hellip times
1
120590119890minus(119909119873minus119883)2 2120590 =
1
120590119873119890 minus(119909119894minus119883)2 2120590
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 46
Lecture 3 Slide 46
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Uncertainty of ldquoBest Estimatesrdquo of Gaussian Distribution Principle of Maximum Likelihood
Best Estimate of X is from maximum probability or minimum summation
Consider a data set x1 x2 x3 hellipxN
To find the most likely value of the standard deviation (the best estimate of
the width of the x distribution) find σx that yields the highest probability for
the data set
The combined probability for the full data set is the product
Minimize
Sum
Solve for
derivative
wrst X set to 0
Best
estimate
of X
Best Estimate of σ is from maximum probability or minimum summation
Minimize
Sum
Best
estimate
of σ
Solve for
derivative
wrst σ set to 0
See
Prob 526
prop1
120590119890minus(1199091minus119883)2 2120590 times
1
120590119890minus(1199092minus119883)2 2120590 times hellip times
1
120590119890minus(119909119873minus119883)2 2120590 =
1
120590119873119890 minus(119909119894minus119883)2 2120590
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 47
Lecture 3 Slide 47
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Combining Data Sets
Weighted Averages
References Taylor Ch 7
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 48
Lecture 3 Slide 48
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Weighted Averages
Question How can we properly combine two or more separate
independent measurements of the same randomly distributed
quantity to determine a best combined value with uncertainty
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 49
Lecture 3 Slide 49
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider two measurements of the same quantity described by a random Gaussian distribution
ltx1gt x1 and ltx2gt x2
The probability of measuring two such measurements is
119875119903119900119887119909 1199091 1199092 = 119875119903119900119887119909 1199091 119875119903119900119887119909 1199092
= 1
12059011205902119890minus12059422 119908ℎ119890119903119890 1205942 equiv
1199091 minus 119883
1205901 2 +
1199092 minus 119883
1205902 2
To find the best value for χ find the maximum Prob or minimum χ2
Weighted Averages
Assume negligible systematic errors
Note χ2 or Chi squared is the sum of the squares of the deviations from the mean divided by
the corresponding uncertainty
Such methods are called ldquoMethods of Least Squaresrdquo They follow directly from the Principle of
Maximum Likelihood
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 50
Lecture 3 Slide 50
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
The probability of measuring two such measurements is
119875119903119900119887119909 1199091 1199092 = 119875119903119900119887119909 1199091 119875119903119900119887119909 1199092
= 1
12059011205902119890minus12059422 119908ℎ119890119903119890 1205942 equiv
1199091 minus 119883
1205901 2 +
1199092 minus 119883
1205902 2
To find the best value for χ find the maximum Prob or minimum χ2
Weighted Averages
This leads to 119909119882_119886119907119892 = 11990811199091+11990821199092
1199081+1199082=
119908119894 119909119894
119908119894 119908ℎ119890119903119890 119908119894 = 1
120590119894 2
Best Estimate of χ is from maximum probibility or minimum summation
Minimize Sum Solve for best estimate of χ Solve for derivative wrst χ set to 0
Note If w1=w2 we recover the standard result Xwavg= (12) (x1+x2)
Finally the width of a weighted average distribution is
120590119908119894119890119892 ℎ119905119890119889 119886119907119892 = 1
119908119894119894
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 51
Lecture 3 Slide 51
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Weighted Averages on Steroids
A very powerful method for combining data from different sources
with different methods and uncertainties (or indeed data of
related measured and calculated quantities) is Kalman filtering
The Kalman filter also known as
linear quadratic estimation (LQE)
is an algorithm that uses a series
of measurements observed over
time containing noise (random
variations) and other
inaccuracies and produces
estimates of unknown variables
that tend to be more precise than
those based on a single
measurement alone
The Kalman filter keeps track of the estimated state of the system and the variance or uncertainty of the
estimate The estimate is updated using a state transition model and measurements xk|k-1 denotes the
estimate of the systems state at time step k before the kth measurement yk has been taken into account Pk|k-1
is the corresponding uncertainty --Wikipedia 2013
Ludger Scherliess of USU Physics is a world expert at using Kalman filtering for the assimilation
of satellite and ground-based data and the USU GAMES model to predict space weather
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 52
Lecture 3 Slide 52
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Rejecting Data
Chauvenetrsquos Criterion
References Taylor Ch 6
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 53
Lecture 3 Slide 53
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly
ldquounreasonablerdquo data point from a set of randomly distributed
measurements
bull Never
bull Whenever it makes things look better
bull Chauvenetrsquos criterion provides a (quantitative) compromise
Rejecting Data
What is a good criteria for
rejecting data
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 54
Lecture 3 Slide 54
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly
ldquounreasonablerdquo data point from a set of randomly distributed
measurements
bull Never
Rejecting Data
Zallenrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 55
Lecture 3 Slide 55
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly
ldquounreasonablerdquo data point from a set of randomly distributed
measurements
bull Whenever it makes things look better
Rejecting Data
Disneyrsquos Criterion
Disneyrsquos First Law
Wishing will make it so
Disneyrsquos Second Law
Dreams are more colorful than reality
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 56
Lecture 3 Slide 56
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Data may be rejected if the expected number of measurements at
least as deviant as the suspect measurement is less than 50
Consider a set of N measurements of a single quantity
x1x2 helliphellipxN
Calculate ltxgt and σx and then determine the fractional deviations from the mean of all
the points
For the suspect point(s) xsuspect find the probability of such a point occurring in N
measurements
n = (expected number as deviant as xsuspect)
= N Prob(outside xsuspectσx)
Rejecting Data
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 57
Lecture 3 Slide 57
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Function of Gaussian Distribution
Area under curve
(probability that
ndashtσltxlt+tσ) is given
in Table at right
Probable Error (probability that
ndash067σltxlt+067σ) is 50
Error Function (probability that ndashtσltxlt+tσ )
Error Function Tabulated values-see App A
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 58
Lecture 3 Slide 58
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 59
Lecture 3 Slide 59
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashExample 1
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 60
Lecture 3 Slide 60
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Details (1)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 61
Lecture 3 Slide 61
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashDetails (2)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 62
Lecture 3 Slide 62
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos
Criterionmdash
Details (3)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 63
Lecture 3 Slide 63
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos
Criterionmdash
Example 2
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 64
Lecture 3 Slide 64
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Summary of Probability Theory
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 65
Lecture 3 Slide 65
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of
Probability
Theory-I
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 66
Lecture 3 Slide 66
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of
Probability
Theory-II
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 67
Lecture 3 Slide 67
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-III
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 68
Lecture 3 Slide 68
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-IV
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 16
Lecture 3 Slide 16
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Example of Continuous Distribution Functions and Expectation Values Boltzmann Distribution Example from Kinetic Theory
Expected Values
The expectation value of a function
Ɍ(x) is
Ɍ x equiv Ɍ x 119892 119909
+infin
minusinfin119889119909
119892 119909 +infin
minusinfin119889119909
= Ɍ(x) 119891(119909)+infin
minusinfin119889119909
The Boltzmann distribution function for velocities of particles as
a function of temperature T is
119875 119907 119879 = 4120587 119872
2120587 119896119861119879
3 2
1199072119890119909119901
121198721199072
12119896119861119879
Then
v = 119907 119875(119907)+infin
minusinfin119889119907 =
8 119896119861119879120587 119872
1 2
v2 = v2 119875(119907)+infin
minusinfin119889119907 =
3 119896119861119879 119872
1 2
implies KE = 1
2119872 v2 =
3
2119896119861119879
vpeak=
2 119896119861119879 119872
1 2
= 2 3
1 2
v2
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 17
Lecture 3 Slide 17
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Example of Continuous Distribution Functions and Expectation Values Fermi-Dirac Distribution Example from Kinetic Theory
For a system of identical fermions the average number of fermions in a single-particle state is given
by the FermindashDirac (FndashD) distribution
where kB is Boltzmanns constant T is the absolute temperature is the energy of the single-particle
state and μ is the total chemical potential
Since the FndashD distribution was derived using the Pauli exclusion principle which allows at most one
electron to occupy each possible state a result is that
When a quasi-continuum of energies has an associated density of states (ie the number of
states per unit energy range per unit volume) the average number of fermions per unit energy range
per unit volume is
where is called the Fermi function
so that
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 18
Lecture 3 Slide 18
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Example of Continuous Distribution Functions and Expectation Values Finite Square Well Example from Quantum Mechanics
Expectation Values The expectation value of a QM operator O(x) is O x equiv Ψlowast x 119874 119909
+infin
minusinfinΨ x 119889119909
Ψlowast x Ψ x +infin
minusinfin119889119909
For a finite square well of width L Ψn x = 2L sin
n π x
L
Ψnlowast x |Ψn x equiv
Ψnlowast x 119874 119909
+infin
minusinfinΨn x 119889119909
Ψnlowast x Ψn x
+infin
minusinfin119889119909
= 1
119909 = Ψnlowast x |x|Ψn x equiv
Ψnlowast x 119909
+infin
minusinfinΨn x 119889119909
Ψnlowast x Ψn x
+infin
minusinfin119889119909
= 1198712
119901 = Ψnlowast x |
ℏ
i
part
partx|Ψn x equiv
Ψnlowast x
ℏi
partpartx
+infin
minusinfinΨn x 119889119909
Ψnlowast x Ψn x
+infin
minusinfin119889119909
= 0
119864119899 = Ψnlowast x |iℏ
part
partt|Ψn x equiv
Ψnlowast x iℏ
partpartt
+infin
minusinfinΨn x 119889119909
Ψnlowast x Ψn x
+infin
minusinfin119889119909
=11989921205872ℏ2
2 119898 1198712
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 19
Lecture 3 Slide 19
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Distribution Functions
Available
on web site
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 20
Lecture 3 Slide 20
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
The Gaussian Distribution
Function
References Taylor Ch 5
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 21
Lecture 3 Slide 21
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Gaussian Integrals
Factorial Approximations
119899 asymp 2120587119899 1 2 119899119899 119890119909119901 minus119899 + 1
12 119899+ 119874
1
1198992
119897119900119892 119899 asymp 1
2 119897119900119892 2120587 + 119899 +
1
2 log 119899 minus 119899 +
1
12 119899+ 119874
1
1198992
119897119900119892 119899 asymp 119899 log 119899 minus 119899 (for all terms decreasing faster than linearly with n)
Gaussian Integrals
119868119898 = 2 119909119898 exp(minus1199092)infin
0 119889119909 mgt-1
119868119898 = 2 119910119899 exp(minus119910)infin
0 119889119910 equiv Γ(119899 + 1) 1199092 equiv 119910 2 119889119909 = 1199101 2 119889119910 119899 equiv 1
2 (119898 minus 1)
1198680 = Γ 119899 =1
2 = 120587 m=0 119899 = minus1
2
1198682 119896 = Γ 119896 +1
2 = 119896 minus 1
2 119896 minus 32 3
2 12 120587 even m m=2 kgt0 119899 = 119896 minus 1
2
1198682 119896+1 = Γ 119896 + 1 = 119896 odd m m=2 k+1gt0 119899 = 119896 ge 0
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 22
Lecture 3 Slide 22
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Gaussian Distribution Function
Width of Distribution
(standard deviation)
Center of Distribution
(mean) Distribution
Function
Independent
Variable
Gaussian Distribution Function
Normalization
Constant
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 23
Lecture 3 Slide 23
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Effects of Increasing N on Gaussian Distribution Functions
Consider the limiting distribution function as N infin and dx0
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 24
Lecture 3 Slide 24
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Defining the Gaussian
distribution function
in Mathcad
I suggest you ldquoinvestigaterdquo
these with the Mathcad sheet
on the web site
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 25
Lecture 3 Slide 25
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Using Mathcad to
define other common
distribution functions
Consider the limiting
distribution function
as N ȸ and ∆k0
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 26
Lecture 3 Slide 26
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider the Gaussian distribution function
Use the normalization condition to
evaluate the normalization constant
(see Taylor p 132)
The mean Ẋ is the first moment
of the Gaussian distribution
function (see Taylor p 134)
The standard deviation σx is the standard
deviation of the mean of the Gaussian
distribution function (see Taylor p 143)
Gaussian Distribution Moments
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 27
Lecture 3 Slide 27
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
When is mean x not Xbest
Answer When the
distribution is not
symmetric about X
Example Cauchy
Distribution
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 28
Lecture 3 Slide 28
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
When is mean x not Xbest
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 29
Lecture 3 Slide 29
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
When is mean x not Xbest
Answer When the
distribution is has
more than one peak
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 30
Lecture 3 Slide 30
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
The Gaussian Distribution
Function and Its Relation to
Errors
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 31
Lecture 3 Slide 31
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
A Review of Probabilities in Combination
Probability of a data set of N like measurements (x1x2hellipxN)
P (x1x2hellipxN) = P(x)1P(x2)hellipP(xN)
1 head AND 1 Four
P(H4) = P(H) P(4)
1 head OR 1 Four
P(H4) = P(H) + P(4) - P(H and 4) (true for a ldquonon-mutually exclusiverdquo events)
NOT 1 Six
P(NOT 6) = 1 - P(6)
1 Six OR 1 Four
P(64) = P(6) + P(4) (true for a ldquomutually exclusiverdquo single role)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 32
Lecture 3 Slide 32
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
The Gaussian Distribution Function and Its Relation to Errors
We will use the Gaussian distribution as applied to random
variables to develop the mathematical basis for bull Mean
bull Standard deviation
bull Standard deviation of the mean (SDOM)
bull Moments and expectation values
bull Error propagation formulas
bull Addition of errors in quadrature (for independent and random
measurements)
bull Numerical values for confidence limits (t-test)
bull Principle of maximal likelihood
bull Central limit theorem
bull Weighted distributions and Chi squared
bull Schwartz inequality (ie the uncertainty principle) (next lecture)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 33
Lecture 3 Slide 33
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider the Gaussian distribution function
Use the normalization condition to
evaluate the normalization constant
(see Taylor p 132)
The mean Ẋ is the first moment
of the Gaussian distribution
function (see Taylor p 134)
The standard deviation σx is the standard
deviation of the mean of the Gaussian
distribution function (see Taylor p 143)
Gaussian Distribution Moments
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 34
Lecture 3 Slide 34
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Standard Deviation of Gaussian Distribution
Area under curve
(probability that
ndashσltxlt+σ) is 68
5 or ~2σ ldquoSignificantrdquo
1 or ~3σ ldquoHighly Significantrdquo
1σ ldquoWithin errorsrdquo
5 ppm or ~5σ ldquoValid for HEPrdquo
See Sec 106 Testing of
Hypotheses
More complete Table in App A and B
Ah thatrsquos
highly
significant
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 35
Lecture 3 Slide 35
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Function of Gaussian Distribution
Area under curve
(probability that
ndashtσltxlt+tσ) is given
in Table at right
Probable Error (probability that
ndash067σltxlt+067σ) is 50
Error Function (probability that ndashtσltxlt+tσ )
Error Function Tabulated values-see App A
Complementary Error Function
(probability that ndashxlt-tσ AND xgt+tσ )
Prob(x outside tσ) = 1 - Prob(x within tσ)
More complete Table in App A and B
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 36
Lecture 3 Slide 36
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Useful Points on Gaussian Distribution
Full Width at Half Maximum FWHM
(See Prob 512)
Points of Inflection Occur at plusmnσ
(See Prob 513)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 37
Lecture 3 Slide 37
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Analysis and Gaussian Distribution
Adding a Constant Multiplying by a Constant
XX+A XBX and σB σ
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 38
Lecture 3 Slide 38
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Sum of Two Variables
Consider the derived quantity
Z=X + Y
(with X=0 and Y=0)
Error Propagation Addition
Error in Z
Multiple two probabilities 119875119903119900119887 119909 119910 = 119875119903119900119887 119909 ∙ 119875119903119900119887 119909 prop 119890119909119901 minus
1
2
1199092
1205901199092+
1199102
1205901199102
prop 119890119909119901 minus1
2
(119909+119910)2
1205901199092+120590119910
2 minus1199112
prop 119890119909119901 minus1
2
(119909+119910)2
1205901199092+120590119910
2 119890119909119901 minus
1199112
2
ICBST (Eq 553)
X+YZ and σx2 + σy
2 σz
2
(addition in quadrature for random independent variables)
Integrates to radic2π
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 39
Lecture 3 Slide 39
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Error Propagation
General formula for error propagation see [Taylor Secs 35 and 39]
Uncertainty as a function of one variable [Taylor Sec 35]
1 Consider a graphical method of estimating error
a) Consider an arbitaray function q(x)
b) Plot q(x) vs x
c) On the graph label
(1) qbest = q(xbest)
(2) qhi = q(xbest + δx)
(3) qlow = q(xbest- δx)
d) Making a linear approximation
x
qqxslopeqq
x
qqxslopeqq
bestbestlow
bestbesthi
e) Therefore
xx
Note the absolute value
How do we determine the error of a derived quantity Z(XYhellip) from errors in XYhellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 40
Lecture 3 Slide 40
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Error Propagation
General formula for uncertainty of a function of one variable
xx
[Taylor Eq 323]
Can you now derive for specific rules of error propagation
1 Addition and Subtraction [Taylor p 49]
2 Multiplication and Division [Taylor p 51]
3 Multiplication by a constant (exact number) [Taylor p 54]
4 Exponentiation (powers) [Taylor p 56]
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 41
Lecture 3 Slide 41
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Multiple Variables
Uncertainty of a function of multiple variables [Taylor Sec 311]
1 It can easily (no really) be shown that (see Taylor Sec 311) for a
function of several variables
)(
z
z
qy
y
qx
x
qzyxq
[Taylor Eq 347]
2 More correctly it can be shown that (see Taylor Sec 311) for a
function of several variables
)(
z
z
qy
y
qx
x
qzyxq
[Taylor Eq 347]
where the equals sign represents an upper bound as discussed above
3 For a function of several independent and random variables
)(
222
z
z
qy
y
qx
x
qzyxq
[Taylor Eq 348]
Again the proof is left for Ch 5
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 42
Lecture 3 Slide 42
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Product of Two Variables
Consider the arbitrary derived quantity
q(xy) of two independent random variables x and y
Expand q(xy) in a Taylor series about the expected values of x and y
(ie at points near X and Y)
Error Propagation General Case
119902 119909 119910 = 119902 119883 119884 + 120597119902
120597119909
119883 119909 minus 119883 +
120597119902
120597119910
119884
(119910 minus 119884)
Fixed shifts peak of distribution
Fixed Distribution centered at X with width σX
120575119902 119909 119910 = 120590119902 = 119902 119883 119884 + 120597119902
120597119909
119883120590119909
2
+ 120597119902
120597119910
119884
120590119910
2
0
How do we determine the error of a derived quantity Z(XYhellip) from errors in XYhellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 43
Lecture 3 Slide 43
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
SDOM of Gaussian Distribution
Standard Deviation of the Mean
The SDOM decreases as the square root of the number of measurements
That is the relative width σẊ of the distribution gets narrower as more
measurements are made
Each measurement
has similar σXi= σẊ
and similar partial
derivatives
Thushellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 44
Lecture 3 Slide 44
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Two Key Theorems from Probability
Central Limit Theorem
For random independent measurements (each with a well-define
expectation value and well-defined variance) the arithmetic mean
(average) will be approximately normally distributed
Principle of Maximum Likelihood
Given the N observed measurements x1 x2hellipxN the best estimates
for Ẋ and σ are those values for which the observed x1 x2hellipxN are
most likely
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 45
Lecture 3 Slide 45
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Mean of Gaussian Distribution as ldquoBest Estimaterdquo
Principle of Maximum Likelihood
Best Estimate of X is from maximum probability or minimum summation
Consider a data set
Each randomly distributed with
x1 x2 x3 hellipxN
To find the most likely value of the mean (the best estimate of ẋ)
find X that yields the highest probability for the data set
The combined probability for the full data set is the product
Minimize
Sum
Solve for
derivative wrst
X set to 0
Best
estimate
of X
prop1
120590119890minus(1199091minus119883)2 2120590 times
1
120590119890minus(1199092minus119883)2 2120590 times hellip times
1
120590119890minus(119909119873minus119883)2 2120590 =
1
120590119873119890 minus(119909119894minus119883)2 2120590
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 46
Lecture 3 Slide 46
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Uncertainty of ldquoBest Estimatesrdquo of Gaussian Distribution Principle of Maximum Likelihood
Best Estimate of X is from maximum probability or minimum summation
Consider a data set x1 x2 x3 hellipxN
To find the most likely value of the standard deviation (the best estimate of
the width of the x distribution) find σx that yields the highest probability for
the data set
The combined probability for the full data set is the product
Minimize
Sum
Solve for
derivative
wrst X set to 0
Best
estimate
of X
Best Estimate of σ is from maximum probability or minimum summation
Minimize
Sum
Best
estimate
of σ
Solve for
derivative
wrst σ set to 0
See
Prob 526
prop1
120590119890minus(1199091minus119883)2 2120590 times
1
120590119890minus(1199092minus119883)2 2120590 times hellip times
1
120590119890minus(119909119873minus119883)2 2120590 =
1
120590119873119890 minus(119909119894minus119883)2 2120590
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 47
Lecture 3 Slide 47
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Combining Data Sets
Weighted Averages
References Taylor Ch 7
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 48
Lecture 3 Slide 48
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Weighted Averages
Question How can we properly combine two or more separate
independent measurements of the same randomly distributed
quantity to determine a best combined value with uncertainty
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 49
Lecture 3 Slide 49
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider two measurements of the same quantity described by a random Gaussian distribution
ltx1gt x1 and ltx2gt x2
The probability of measuring two such measurements is
119875119903119900119887119909 1199091 1199092 = 119875119903119900119887119909 1199091 119875119903119900119887119909 1199092
= 1
12059011205902119890minus12059422 119908ℎ119890119903119890 1205942 equiv
1199091 minus 119883
1205901 2 +
1199092 minus 119883
1205902 2
To find the best value for χ find the maximum Prob or minimum χ2
Weighted Averages
Assume negligible systematic errors
Note χ2 or Chi squared is the sum of the squares of the deviations from the mean divided by
the corresponding uncertainty
Such methods are called ldquoMethods of Least Squaresrdquo They follow directly from the Principle of
Maximum Likelihood
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 50
Lecture 3 Slide 50
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
The probability of measuring two such measurements is
119875119903119900119887119909 1199091 1199092 = 119875119903119900119887119909 1199091 119875119903119900119887119909 1199092
= 1
12059011205902119890minus12059422 119908ℎ119890119903119890 1205942 equiv
1199091 minus 119883
1205901 2 +
1199092 minus 119883
1205902 2
To find the best value for χ find the maximum Prob or minimum χ2
Weighted Averages
This leads to 119909119882_119886119907119892 = 11990811199091+11990821199092
1199081+1199082=
119908119894 119909119894
119908119894 119908ℎ119890119903119890 119908119894 = 1
120590119894 2
Best Estimate of χ is from maximum probibility or minimum summation
Minimize Sum Solve for best estimate of χ Solve for derivative wrst χ set to 0
Note If w1=w2 we recover the standard result Xwavg= (12) (x1+x2)
Finally the width of a weighted average distribution is
120590119908119894119890119892 ℎ119905119890119889 119886119907119892 = 1
119908119894119894
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 51
Lecture 3 Slide 51
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Weighted Averages on Steroids
A very powerful method for combining data from different sources
with different methods and uncertainties (or indeed data of
related measured and calculated quantities) is Kalman filtering
The Kalman filter also known as
linear quadratic estimation (LQE)
is an algorithm that uses a series
of measurements observed over
time containing noise (random
variations) and other
inaccuracies and produces
estimates of unknown variables
that tend to be more precise than
those based on a single
measurement alone
The Kalman filter keeps track of the estimated state of the system and the variance or uncertainty of the
estimate The estimate is updated using a state transition model and measurements xk|k-1 denotes the
estimate of the systems state at time step k before the kth measurement yk has been taken into account Pk|k-1
is the corresponding uncertainty --Wikipedia 2013
Ludger Scherliess of USU Physics is a world expert at using Kalman filtering for the assimilation
of satellite and ground-based data and the USU GAMES model to predict space weather
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 52
Lecture 3 Slide 52
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Rejecting Data
Chauvenetrsquos Criterion
References Taylor Ch 6
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 53
Lecture 3 Slide 53
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly
ldquounreasonablerdquo data point from a set of randomly distributed
measurements
bull Never
bull Whenever it makes things look better
bull Chauvenetrsquos criterion provides a (quantitative) compromise
Rejecting Data
What is a good criteria for
rejecting data
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 54
Lecture 3 Slide 54
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly
ldquounreasonablerdquo data point from a set of randomly distributed
measurements
bull Never
Rejecting Data
Zallenrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 55
Lecture 3 Slide 55
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly
ldquounreasonablerdquo data point from a set of randomly distributed
measurements
bull Whenever it makes things look better
Rejecting Data
Disneyrsquos Criterion
Disneyrsquos First Law
Wishing will make it so
Disneyrsquos Second Law
Dreams are more colorful than reality
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 56
Lecture 3 Slide 56
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Data may be rejected if the expected number of measurements at
least as deviant as the suspect measurement is less than 50
Consider a set of N measurements of a single quantity
x1x2 helliphellipxN
Calculate ltxgt and σx and then determine the fractional deviations from the mean of all
the points
For the suspect point(s) xsuspect find the probability of such a point occurring in N
measurements
n = (expected number as deviant as xsuspect)
= N Prob(outside xsuspectσx)
Rejecting Data
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 57
Lecture 3 Slide 57
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Function of Gaussian Distribution
Area under curve
(probability that
ndashtσltxlt+tσ) is given
in Table at right
Probable Error (probability that
ndash067σltxlt+067σ) is 50
Error Function (probability that ndashtσltxlt+tσ )
Error Function Tabulated values-see App A
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 58
Lecture 3 Slide 58
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 59
Lecture 3 Slide 59
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashExample 1
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 60
Lecture 3 Slide 60
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Details (1)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 61
Lecture 3 Slide 61
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashDetails (2)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 62
Lecture 3 Slide 62
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos
Criterionmdash
Details (3)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 63
Lecture 3 Slide 63
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos
Criterionmdash
Example 2
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 64
Lecture 3 Slide 64
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Summary of Probability Theory
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 65
Lecture 3 Slide 65
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of
Probability
Theory-I
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 66
Lecture 3 Slide 66
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of
Probability
Theory-II
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 67
Lecture 3 Slide 67
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-III
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 68
Lecture 3 Slide 68
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-IV
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 17
Lecture 3 Slide 17
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Example of Continuous Distribution Functions and Expectation Values Fermi-Dirac Distribution Example from Kinetic Theory
For a system of identical fermions the average number of fermions in a single-particle state is given
by the FermindashDirac (FndashD) distribution
where kB is Boltzmanns constant T is the absolute temperature is the energy of the single-particle
state and μ is the total chemical potential
Since the FndashD distribution was derived using the Pauli exclusion principle which allows at most one
electron to occupy each possible state a result is that
When a quasi-continuum of energies has an associated density of states (ie the number of
states per unit energy range per unit volume) the average number of fermions per unit energy range
per unit volume is
where is called the Fermi function
so that
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 18
Lecture 3 Slide 18
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Example of Continuous Distribution Functions and Expectation Values Finite Square Well Example from Quantum Mechanics
Expectation Values The expectation value of a QM operator O(x) is O x equiv Ψlowast x 119874 119909
+infin
minusinfinΨ x 119889119909
Ψlowast x Ψ x +infin
minusinfin119889119909
For a finite square well of width L Ψn x = 2L sin
n π x
L
Ψnlowast x |Ψn x equiv
Ψnlowast x 119874 119909
+infin
minusinfinΨn x 119889119909
Ψnlowast x Ψn x
+infin
minusinfin119889119909
= 1
119909 = Ψnlowast x |x|Ψn x equiv
Ψnlowast x 119909
+infin
minusinfinΨn x 119889119909
Ψnlowast x Ψn x
+infin
minusinfin119889119909
= 1198712
119901 = Ψnlowast x |
ℏ
i
part
partx|Ψn x equiv
Ψnlowast x
ℏi
partpartx
+infin
minusinfinΨn x 119889119909
Ψnlowast x Ψn x
+infin
minusinfin119889119909
= 0
119864119899 = Ψnlowast x |iℏ
part
partt|Ψn x equiv
Ψnlowast x iℏ
partpartt
+infin
minusinfinΨn x 119889119909
Ψnlowast x Ψn x
+infin
minusinfin119889119909
=11989921205872ℏ2
2 119898 1198712
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 19
Lecture 3 Slide 19
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Distribution Functions
Available
on web site
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 20
Lecture 3 Slide 20
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
The Gaussian Distribution
Function
References Taylor Ch 5
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 21
Lecture 3 Slide 21
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Gaussian Integrals
Factorial Approximations
119899 asymp 2120587119899 1 2 119899119899 119890119909119901 minus119899 + 1
12 119899+ 119874
1
1198992
119897119900119892 119899 asymp 1
2 119897119900119892 2120587 + 119899 +
1
2 log 119899 minus 119899 +
1
12 119899+ 119874
1
1198992
119897119900119892 119899 asymp 119899 log 119899 minus 119899 (for all terms decreasing faster than linearly with n)
Gaussian Integrals
119868119898 = 2 119909119898 exp(minus1199092)infin
0 119889119909 mgt-1
119868119898 = 2 119910119899 exp(minus119910)infin
0 119889119910 equiv Γ(119899 + 1) 1199092 equiv 119910 2 119889119909 = 1199101 2 119889119910 119899 equiv 1
2 (119898 minus 1)
1198680 = Γ 119899 =1
2 = 120587 m=0 119899 = minus1
2
1198682 119896 = Γ 119896 +1
2 = 119896 minus 1
2 119896 minus 32 3
2 12 120587 even m m=2 kgt0 119899 = 119896 minus 1
2
1198682 119896+1 = Γ 119896 + 1 = 119896 odd m m=2 k+1gt0 119899 = 119896 ge 0
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 22
Lecture 3 Slide 22
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Gaussian Distribution Function
Width of Distribution
(standard deviation)
Center of Distribution
(mean) Distribution
Function
Independent
Variable
Gaussian Distribution Function
Normalization
Constant
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 23
Lecture 3 Slide 23
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Effects of Increasing N on Gaussian Distribution Functions
Consider the limiting distribution function as N infin and dx0
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 24
Lecture 3 Slide 24
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Defining the Gaussian
distribution function
in Mathcad
I suggest you ldquoinvestigaterdquo
these with the Mathcad sheet
on the web site
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 25
Lecture 3 Slide 25
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Using Mathcad to
define other common
distribution functions
Consider the limiting
distribution function
as N ȸ and ∆k0
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 26
Lecture 3 Slide 26
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider the Gaussian distribution function
Use the normalization condition to
evaluate the normalization constant
(see Taylor p 132)
The mean Ẋ is the first moment
of the Gaussian distribution
function (see Taylor p 134)
The standard deviation σx is the standard
deviation of the mean of the Gaussian
distribution function (see Taylor p 143)
Gaussian Distribution Moments
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 27
Lecture 3 Slide 27
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
When is mean x not Xbest
Answer When the
distribution is not
symmetric about X
Example Cauchy
Distribution
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 28
Lecture 3 Slide 28
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
When is mean x not Xbest
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 29
Lecture 3 Slide 29
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
When is mean x not Xbest
Answer When the
distribution is has
more than one peak
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 30
Lecture 3 Slide 30
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
The Gaussian Distribution
Function and Its Relation to
Errors
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 31
Lecture 3 Slide 31
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
A Review of Probabilities in Combination
Probability of a data set of N like measurements (x1x2hellipxN)
P (x1x2hellipxN) = P(x)1P(x2)hellipP(xN)
1 head AND 1 Four
P(H4) = P(H) P(4)
1 head OR 1 Four
P(H4) = P(H) + P(4) - P(H and 4) (true for a ldquonon-mutually exclusiverdquo events)
NOT 1 Six
P(NOT 6) = 1 - P(6)
1 Six OR 1 Four
P(64) = P(6) + P(4) (true for a ldquomutually exclusiverdquo single role)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 32
Lecture 3 Slide 32
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
The Gaussian Distribution Function and Its Relation to Errors
We will use the Gaussian distribution as applied to random
variables to develop the mathematical basis for bull Mean
bull Standard deviation
bull Standard deviation of the mean (SDOM)
bull Moments and expectation values
bull Error propagation formulas
bull Addition of errors in quadrature (for independent and random
measurements)
bull Numerical values for confidence limits (t-test)
bull Principle of maximal likelihood
bull Central limit theorem
bull Weighted distributions and Chi squared
bull Schwartz inequality (ie the uncertainty principle) (next lecture)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 33
Lecture 3 Slide 33
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider the Gaussian distribution function
Use the normalization condition to
evaluate the normalization constant
(see Taylor p 132)
The mean Ẋ is the first moment
of the Gaussian distribution
function (see Taylor p 134)
The standard deviation σx is the standard
deviation of the mean of the Gaussian
distribution function (see Taylor p 143)
Gaussian Distribution Moments
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 34
Lecture 3 Slide 34
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Standard Deviation of Gaussian Distribution
Area under curve
(probability that
ndashσltxlt+σ) is 68
5 or ~2σ ldquoSignificantrdquo
1 or ~3σ ldquoHighly Significantrdquo
1σ ldquoWithin errorsrdquo
5 ppm or ~5σ ldquoValid for HEPrdquo
See Sec 106 Testing of
Hypotheses
More complete Table in App A and B
Ah thatrsquos
highly
significant
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 35
Lecture 3 Slide 35
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Function of Gaussian Distribution
Area under curve
(probability that
ndashtσltxlt+tσ) is given
in Table at right
Probable Error (probability that
ndash067σltxlt+067σ) is 50
Error Function (probability that ndashtσltxlt+tσ )
Error Function Tabulated values-see App A
Complementary Error Function
(probability that ndashxlt-tσ AND xgt+tσ )
Prob(x outside tσ) = 1 - Prob(x within tσ)
More complete Table in App A and B
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 36
Lecture 3 Slide 36
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Useful Points on Gaussian Distribution
Full Width at Half Maximum FWHM
(See Prob 512)
Points of Inflection Occur at plusmnσ
(See Prob 513)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 37
Lecture 3 Slide 37
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Analysis and Gaussian Distribution
Adding a Constant Multiplying by a Constant
XX+A XBX and σB σ
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 38
Lecture 3 Slide 38
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Sum of Two Variables
Consider the derived quantity
Z=X + Y
(with X=0 and Y=0)
Error Propagation Addition
Error in Z
Multiple two probabilities 119875119903119900119887 119909 119910 = 119875119903119900119887 119909 ∙ 119875119903119900119887 119909 prop 119890119909119901 minus
1
2
1199092
1205901199092+
1199102
1205901199102
prop 119890119909119901 minus1
2
(119909+119910)2
1205901199092+120590119910
2 minus1199112
prop 119890119909119901 minus1
2
(119909+119910)2
1205901199092+120590119910
2 119890119909119901 minus
1199112
2
ICBST (Eq 553)
X+YZ and σx2 + σy
2 σz
2
(addition in quadrature for random independent variables)
Integrates to radic2π
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 39
Lecture 3 Slide 39
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Error Propagation
General formula for error propagation see [Taylor Secs 35 and 39]
Uncertainty as a function of one variable [Taylor Sec 35]
1 Consider a graphical method of estimating error
a) Consider an arbitaray function q(x)
b) Plot q(x) vs x
c) On the graph label
(1) qbest = q(xbest)
(2) qhi = q(xbest + δx)
(3) qlow = q(xbest- δx)
d) Making a linear approximation
x
qqxslopeqq
x
qqxslopeqq
bestbestlow
bestbesthi
e) Therefore
xx
Note the absolute value
How do we determine the error of a derived quantity Z(XYhellip) from errors in XYhellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 40
Lecture 3 Slide 40
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Error Propagation
General formula for uncertainty of a function of one variable
xx
[Taylor Eq 323]
Can you now derive for specific rules of error propagation
1 Addition and Subtraction [Taylor p 49]
2 Multiplication and Division [Taylor p 51]
3 Multiplication by a constant (exact number) [Taylor p 54]
4 Exponentiation (powers) [Taylor p 56]
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 41
Lecture 3 Slide 41
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Multiple Variables
Uncertainty of a function of multiple variables [Taylor Sec 311]
1 It can easily (no really) be shown that (see Taylor Sec 311) for a
function of several variables
)(
z
z
qy
y
qx
x
qzyxq
[Taylor Eq 347]
2 More correctly it can be shown that (see Taylor Sec 311) for a
function of several variables
)(
z
z
qy
y
qx
x
qzyxq
[Taylor Eq 347]
where the equals sign represents an upper bound as discussed above
3 For a function of several independent and random variables
)(
222
z
z
qy
y
qx
x
qzyxq
[Taylor Eq 348]
Again the proof is left for Ch 5
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 42
Lecture 3 Slide 42
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Product of Two Variables
Consider the arbitrary derived quantity
q(xy) of two independent random variables x and y
Expand q(xy) in a Taylor series about the expected values of x and y
(ie at points near X and Y)
Error Propagation General Case
119902 119909 119910 = 119902 119883 119884 + 120597119902
120597119909
119883 119909 minus 119883 +
120597119902
120597119910
119884
(119910 minus 119884)
Fixed shifts peak of distribution
Fixed Distribution centered at X with width σX
120575119902 119909 119910 = 120590119902 = 119902 119883 119884 + 120597119902
120597119909
119883120590119909
2
+ 120597119902
120597119910
119884
120590119910
2
0
How do we determine the error of a derived quantity Z(XYhellip) from errors in XYhellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 43
Lecture 3 Slide 43
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
SDOM of Gaussian Distribution
Standard Deviation of the Mean
The SDOM decreases as the square root of the number of measurements
That is the relative width σẊ of the distribution gets narrower as more
measurements are made
Each measurement
has similar σXi= σẊ
and similar partial
derivatives
Thushellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 44
Lecture 3 Slide 44
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Two Key Theorems from Probability
Central Limit Theorem
For random independent measurements (each with a well-define
expectation value and well-defined variance) the arithmetic mean
(average) will be approximately normally distributed
Principle of Maximum Likelihood
Given the N observed measurements x1 x2hellipxN the best estimates
for Ẋ and σ are those values for which the observed x1 x2hellipxN are
most likely
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 45
Lecture 3 Slide 45
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Mean of Gaussian Distribution as ldquoBest Estimaterdquo
Principle of Maximum Likelihood
Best Estimate of X is from maximum probability or minimum summation
Consider a data set
Each randomly distributed with
x1 x2 x3 hellipxN
To find the most likely value of the mean (the best estimate of ẋ)
find X that yields the highest probability for the data set
The combined probability for the full data set is the product
Minimize
Sum
Solve for
derivative wrst
X set to 0
Best
estimate
of X
prop1
120590119890minus(1199091minus119883)2 2120590 times
1
120590119890minus(1199092minus119883)2 2120590 times hellip times
1
120590119890minus(119909119873minus119883)2 2120590 =
1
120590119873119890 minus(119909119894minus119883)2 2120590
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 46
Lecture 3 Slide 46
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Uncertainty of ldquoBest Estimatesrdquo of Gaussian Distribution Principle of Maximum Likelihood
Best Estimate of X is from maximum probability or minimum summation
Consider a data set x1 x2 x3 hellipxN
To find the most likely value of the standard deviation (the best estimate of
the width of the x distribution) find σx that yields the highest probability for
the data set
The combined probability for the full data set is the product
Minimize
Sum
Solve for
derivative
wrst X set to 0
Best
estimate
of X
Best Estimate of σ is from maximum probability or minimum summation
Minimize
Sum
Best
estimate
of σ
Solve for
derivative
wrst σ set to 0
See
Prob 526
prop1
120590119890minus(1199091minus119883)2 2120590 times
1
120590119890minus(1199092minus119883)2 2120590 times hellip times
1
120590119890minus(119909119873minus119883)2 2120590 =
1
120590119873119890 minus(119909119894minus119883)2 2120590
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 47
Lecture 3 Slide 47
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Combining Data Sets
Weighted Averages
References Taylor Ch 7
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 48
Lecture 3 Slide 48
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Weighted Averages
Question How can we properly combine two or more separate
independent measurements of the same randomly distributed
quantity to determine a best combined value with uncertainty
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 49
Lecture 3 Slide 49
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider two measurements of the same quantity described by a random Gaussian distribution
ltx1gt x1 and ltx2gt x2
The probability of measuring two such measurements is
119875119903119900119887119909 1199091 1199092 = 119875119903119900119887119909 1199091 119875119903119900119887119909 1199092
= 1
12059011205902119890minus12059422 119908ℎ119890119903119890 1205942 equiv
1199091 minus 119883
1205901 2 +
1199092 minus 119883
1205902 2
To find the best value for χ find the maximum Prob or minimum χ2
Weighted Averages
Assume negligible systematic errors
Note χ2 or Chi squared is the sum of the squares of the deviations from the mean divided by
the corresponding uncertainty
Such methods are called ldquoMethods of Least Squaresrdquo They follow directly from the Principle of
Maximum Likelihood
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 50
Lecture 3 Slide 50
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
The probability of measuring two such measurements is
119875119903119900119887119909 1199091 1199092 = 119875119903119900119887119909 1199091 119875119903119900119887119909 1199092
= 1
12059011205902119890minus12059422 119908ℎ119890119903119890 1205942 equiv
1199091 minus 119883
1205901 2 +
1199092 minus 119883
1205902 2
To find the best value for χ find the maximum Prob or minimum χ2
Weighted Averages
This leads to 119909119882_119886119907119892 = 11990811199091+11990821199092
1199081+1199082=
119908119894 119909119894
119908119894 119908ℎ119890119903119890 119908119894 = 1
120590119894 2
Best Estimate of χ is from maximum probibility or minimum summation
Minimize Sum Solve for best estimate of χ Solve for derivative wrst χ set to 0
Note If w1=w2 we recover the standard result Xwavg= (12) (x1+x2)
Finally the width of a weighted average distribution is
120590119908119894119890119892 ℎ119905119890119889 119886119907119892 = 1
119908119894119894
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 51
Lecture 3 Slide 51
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Weighted Averages on Steroids
A very powerful method for combining data from different sources
with different methods and uncertainties (or indeed data of
related measured and calculated quantities) is Kalman filtering
The Kalman filter also known as
linear quadratic estimation (LQE)
is an algorithm that uses a series
of measurements observed over
time containing noise (random
variations) and other
inaccuracies and produces
estimates of unknown variables
that tend to be more precise than
those based on a single
measurement alone
The Kalman filter keeps track of the estimated state of the system and the variance or uncertainty of the
estimate The estimate is updated using a state transition model and measurements xk|k-1 denotes the
estimate of the systems state at time step k before the kth measurement yk has been taken into account Pk|k-1
is the corresponding uncertainty --Wikipedia 2013
Ludger Scherliess of USU Physics is a world expert at using Kalman filtering for the assimilation
of satellite and ground-based data and the USU GAMES model to predict space weather
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 52
Lecture 3 Slide 52
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Rejecting Data
Chauvenetrsquos Criterion
References Taylor Ch 6
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 53
Lecture 3 Slide 53
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly
ldquounreasonablerdquo data point from a set of randomly distributed
measurements
bull Never
bull Whenever it makes things look better
bull Chauvenetrsquos criterion provides a (quantitative) compromise
Rejecting Data
What is a good criteria for
rejecting data
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 54
Lecture 3 Slide 54
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly
ldquounreasonablerdquo data point from a set of randomly distributed
measurements
bull Never
Rejecting Data
Zallenrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 55
Lecture 3 Slide 55
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly
ldquounreasonablerdquo data point from a set of randomly distributed
measurements
bull Whenever it makes things look better
Rejecting Data
Disneyrsquos Criterion
Disneyrsquos First Law
Wishing will make it so
Disneyrsquos Second Law
Dreams are more colorful than reality
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 56
Lecture 3 Slide 56
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Data may be rejected if the expected number of measurements at
least as deviant as the suspect measurement is less than 50
Consider a set of N measurements of a single quantity
x1x2 helliphellipxN
Calculate ltxgt and σx and then determine the fractional deviations from the mean of all
the points
For the suspect point(s) xsuspect find the probability of such a point occurring in N
measurements
n = (expected number as deviant as xsuspect)
= N Prob(outside xsuspectσx)
Rejecting Data
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 57
Lecture 3 Slide 57
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Function of Gaussian Distribution
Area under curve
(probability that
ndashtσltxlt+tσ) is given
in Table at right
Probable Error (probability that
ndash067σltxlt+067σ) is 50
Error Function (probability that ndashtσltxlt+tσ )
Error Function Tabulated values-see App A
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 58
Lecture 3 Slide 58
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 59
Lecture 3 Slide 59
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashExample 1
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 60
Lecture 3 Slide 60
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Details (1)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 61
Lecture 3 Slide 61
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashDetails (2)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 62
Lecture 3 Slide 62
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos
Criterionmdash
Details (3)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 63
Lecture 3 Slide 63
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos
Criterionmdash
Example 2
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 64
Lecture 3 Slide 64
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Summary of Probability Theory
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 65
Lecture 3 Slide 65
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of
Probability
Theory-I
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 66
Lecture 3 Slide 66
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of
Probability
Theory-II
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 67
Lecture 3 Slide 67
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-III
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 68
Lecture 3 Slide 68
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-IV
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 18
Lecture 3 Slide 18
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Example of Continuous Distribution Functions and Expectation Values Finite Square Well Example from Quantum Mechanics
Expectation Values The expectation value of a QM operator O(x) is O x equiv Ψlowast x 119874 119909
+infin
minusinfinΨ x 119889119909
Ψlowast x Ψ x +infin
minusinfin119889119909
For a finite square well of width L Ψn x = 2L sin
n π x
L
Ψnlowast x |Ψn x equiv
Ψnlowast x 119874 119909
+infin
minusinfinΨn x 119889119909
Ψnlowast x Ψn x
+infin
minusinfin119889119909
= 1
119909 = Ψnlowast x |x|Ψn x equiv
Ψnlowast x 119909
+infin
minusinfinΨn x 119889119909
Ψnlowast x Ψn x
+infin
minusinfin119889119909
= 1198712
119901 = Ψnlowast x |
ℏ
i
part
partx|Ψn x equiv
Ψnlowast x
ℏi
partpartx
+infin
minusinfinΨn x 119889119909
Ψnlowast x Ψn x
+infin
minusinfin119889119909
= 0
119864119899 = Ψnlowast x |iℏ
part
partt|Ψn x equiv
Ψnlowast x iℏ
partpartt
+infin
minusinfinΨn x 119889119909
Ψnlowast x Ψn x
+infin
minusinfin119889119909
=11989921205872ℏ2
2 119898 1198712
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 19
Lecture 3 Slide 19
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Distribution Functions
Available
on web site
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 20
Lecture 3 Slide 20
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
The Gaussian Distribution
Function
References Taylor Ch 5
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 21
Lecture 3 Slide 21
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Gaussian Integrals
Factorial Approximations
119899 asymp 2120587119899 1 2 119899119899 119890119909119901 minus119899 + 1
12 119899+ 119874
1
1198992
119897119900119892 119899 asymp 1
2 119897119900119892 2120587 + 119899 +
1
2 log 119899 minus 119899 +
1
12 119899+ 119874
1
1198992
119897119900119892 119899 asymp 119899 log 119899 minus 119899 (for all terms decreasing faster than linearly with n)
Gaussian Integrals
119868119898 = 2 119909119898 exp(minus1199092)infin
0 119889119909 mgt-1
119868119898 = 2 119910119899 exp(minus119910)infin
0 119889119910 equiv Γ(119899 + 1) 1199092 equiv 119910 2 119889119909 = 1199101 2 119889119910 119899 equiv 1
2 (119898 minus 1)
1198680 = Γ 119899 =1
2 = 120587 m=0 119899 = minus1
2
1198682 119896 = Γ 119896 +1
2 = 119896 minus 1
2 119896 minus 32 3
2 12 120587 even m m=2 kgt0 119899 = 119896 minus 1
2
1198682 119896+1 = Γ 119896 + 1 = 119896 odd m m=2 k+1gt0 119899 = 119896 ge 0
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 22
Lecture 3 Slide 22
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Gaussian Distribution Function
Width of Distribution
(standard deviation)
Center of Distribution
(mean) Distribution
Function
Independent
Variable
Gaussian Distribution Function
Normalization
Constant
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 23
Lecture 3 Slide 23
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Effects of Increasing N on Gaussian Distribution Functions
Consider the limiting distribution function as N infin and dx0
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 24
Lecture 3 Slide 24
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Defining the Gaussian
distribution function
in Mathcad
I suggest you ldquoinvestigaterdquo
these with the Mathcad sheet
on the web site
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 25
Lecture 3 Slide 25
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Using Mathcad to
define other common
distribution functions
Consider the limiting
distribution function
as N ȸ and ∆k0
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 26
Lecture 3 Slide 26
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider the Gaussian distribution function
Use the normalization condition to
evaluate the normalization constant
(see Taylor p 132)
The mean Ẋ is the first moment
of the Gaussian distribution
function (see Taylor p 134)
The standard deviation σx is the standard
deviation of the mean of the Gaussian
distribution function (see Taylor p 143)
Gaussian Distribution Moments
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 27
Lecture 3 Slide 27
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
When is mean x not Xbest
Answer When the
distribution is not
symmetric about X
Example Cauchy
Distribution
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 28
Lecture 3 Slide 28
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
When is mean x not Xbest
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 29
Lecture 3 Slide 29
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
When is mean x not Xbest
Answer When the
distribution is has
more than one peak
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 30
Lecture 3 Slide 30
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
The Gaussian Distribution
Function and Its Relation to
Errors
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 31
Lecture 3 Slide 31
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
A Review of Probabilities in Combination
Probability of a data set of N like measurements (x1x2hellipxN)
P (x1x2hellipxN) = P(x)1P(x2)hellipP(xN)
1 head AND 1 Four
P(H4) = P(H) P(4)
1 head OR 1 Four
P(H4) = P(H) + P(4) - P(H and 4) (true for a ldquonon-mutually exclusiverdquo events)
NOT 1 Six
P(NOT 6) = 1 - P(6)
1 Six OR 1 Four
P(64) = P(6) + P(4) (true for a ldquomutually exclusiverdquo single role)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 32
Lecture 3 Slide 32
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
The Gaussian Distribution Function and Its Relation to Errors
We will use the Gaussian distribution as applied to random
variables to develop the mathematical basis for bull Mean
bull Standard deviation
bull Standard deviation of the mean (SDOM)
bull Moments and expectation values
bull Error propagation formulas
bull Addition of errors in quadrature (for independent and random
measurements)
bull Numerical values for confidence limits (t-test)
bull Principle of maximal likelihood
bull Central limit theorem
bull Weighted distributions and Chi squared
bull Schwartz inequality (ie the uncertainty principle) (next lecture)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 33
Lecture 3 Slide 33
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider the Gaussian distribution function
Use the normalization condition to
evaluate the normalization constant
(see Taylor p 132)
The mean Ẋ is the first moment
of the Gaussian distribution
function (see Taylor p 134)
The standard deviation σx is the standard
deviation of the mean of the Gaussian
distribution function (see Taylor p 143)
Gaussian Distribution Moments
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 34
Lecture 3 Slide 34
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Standard Deviation of Gaussian Distribution
Area under curve
(probability that
ndashσltxlt+σ) is 68
5 or ~2σ ldquoSignificantrdquo
1 or ~3σ ldquoHighly Significantrdquo
1σ ldquoWithin errorsrdquo
5 ppm or ~5σ ldquoValid for HEPrdquo
See Sec 106 Testing of
Hypotheses
More complete Table in App A and B
Ah thatrsquos
highly
significant
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 35
Lecture 3 Slide 35
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Function of Gaussian Distribution
Area under curve
(probability that
ndashtσltxlt+tσ) is given
in Table at right
Probable Error (probability that
ndash067σltxlt+067σ) is 50
Error Function (probability that ndashtσltxlt+tσ )
Error Function Tabulated values-see App A
Complementary Error Function
(probability that ndashxlt-tσ AND xgt+tσ )
Prob(x outside tσ) = 1 - Prob(x within tσ)
More complete Table in App A and B
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 36
Lecture 3 Slide 36
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Useful Points on Gaussian Distribution
Full Width at Half Maximum FWHM
(See Prob 512)
Points of Inflection Occur at plusmnσ
(See Prob 513)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 37
Lecture 3 Slide 37
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Analysis and Gaussian Distribution
Adding a Constant Multiplying by a Constant
XX+A XBX and σB σ
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 38
Lecture 3 Slide 38
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Sum of Two Variables
Consider the derived quantity
Z=X + Y
(with X=0 and Y=0)
Error Propagation Addition
Error in Z
Multiple two probabilities 119875119903119900119887 119909 119910 = 119875119903119900119887 119909 ∙ 119875119903119900119887 119909 prop 119890119909119901 minus
1
2
1199092
1205901199092+
1199102
1205901199102
prop 119890119909119901 minus1
2
(119909+119910)2
1205901199092+120590119910
2 minus1199112
prop 119890119909119901 minus1
2
(119909+119910)2
1205901199092+120590119910
2 119890119909119901 minus
1199112
2
ICBST (Eq 553)
X+YZ and σx2 + σy
2 σz
2
(addition in quadrature for random independent variables)
Integrates to radic2π
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 39
Lecture 3 Slide 39
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Error Propagation
General formula for error propagation see [Taylor Secs 35 and 39]
Uncertainty as a function of one variable [Taylor Sec 35]
1 Consider a graphical method of estimating error
a) Consider an arbitaray function q(x)
b) Plot q(x) vs x
c) On the graph label
(1) qbest = q(xbest)
(2) qhi = q(xbest + δx)
(3) qlow = q(xbest- δx)
d) Making a linear approximation
x
qqxslopeqq
x
qqxslopeqq
bestbestlow
bestbesthi
e) Therefore
xx
Note the absolute value
How do we determine the error of a derived quantity Z(XYhellip) from errors in XYhellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 40
Lecture 3 Slide 40
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Error Propagation
General formula for uncertainty of a function of one variable
xx
[Taylor Eq 323]
Can you now derive for specific rules of error propagation
1 Addition and Subtraction [Taylor p 49]
2 Multiplication and Division [Taylor p 51]
3 Multiplication by a constant (exact number) [Taylor p 54]
4 Exponentiation (powers) [Taylor p 56]
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 41
Lecture 3 Slide 41
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Multiple Variables
Uncertainty of a function of multiple variables [Taylor Sec 311]
1 It can easily (no really) be shown that (see Taylor Sec 311) for a
function of several variables
)(
z
z
qy
y
qx
x
qzyxq
[Taylor Eq 347]
2 More correctly it can be shown that (see Taylor Sec 311) for a
function of several variables
)(
z
z
qy
y
qx
x
qzyxq
[Taylor Eq 347]
where the equals sign represents an upper bound as discussed above
3 For a function of several independent and random variables
)(
222
z
z
qy
y
qx
x
qzyxq
[Taylor Eq 348]
Again the proof is left for Ch 5
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 42
Lecture 3 Slide 42
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Product of Two Variables
Consider the arbitrary derived quantity
q(xy) of two independent random variables x and y
Expand q(xy) in a Taylor series about the expected values of x and y
(ie at points near X and Y)
Error Propagation General Case
119902 119909 119910 = 119902 119883 119884 + 120597119902
120597119909
119883 119909 minus 119883 +
120597119902
120597119910
119884
(119910 minus 119884)
Fixed shifts peak of distribution
Fixed Distribution centered at X with width σX
120575119902 119909 119910 = 120590119902 = 119902 119883 119884 + 120597119902
120597119909
119883120590119909
2
+ 120597119902
120597119910
119884
120590119910
2
0
How do we determine the error of a derived quantity Z(XYhellip) from errors in XYhellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 43
Lecture 3 Slide 43
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
SDOM of Gaussian Distribution
Standard Deviation of the Mean
The SDOM decreases as the square root of the number of measurements
That is the relative width σẊ of the distribution gets narrower as more
measurements are made
Each measurement
has similar σXi= σẊ
and similar partial
derivatives
Thushellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 44
Lecture 3 Slide 44
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Two Key Theorems from Probability
Central Limit Theorem
For random independent measurements (each with a well-define
expectation value and well-defined variance) the arithmetic mean
(average) will be approximately normally distributed
Principle of Maximum Likelihood
Given the N observed measurements x1 x2hellipxN the best estimates
for Ẋ and σ are those values for which the observed x1 x2hellipxN are
most likely
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 45
Lecture 3 Slide 45
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Mean of Gaussian Distribution as ldquoBest Estimaterdquo
Principle of Maximum Likelihood
Best Estimate of X is from maximum probability or minimum summation
Consider a data set
Each randomly distributed with
x1 x2 x3 hellipxN
To find the most likely value of the mean (the best estimate of ẋ)
find X that yields the highest probability for the data set
The combined probability for the full data set is the product
Minimize
Sum
Solve for
derivative wrst
X set to 0
Best
estimate
of X
prop1
120590119890minus(1199091minus119883)2 2120590 times
1
120590119890minus(1199092minus119883)2 2120590 times hellip times
1
120590119890minus(119909119873minus119883)2 2120590 =
1
120590119873119890 minus(119909119894minus119883)2 2120590
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 46
Lecture 3 Slide 46
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Uncertainty of ldquoBest Estimatesrdquo of Gaussian Distribution Principle of Maximum Likelihood
Best Estimate of X is from maximum probability or minimum summation
Consider a data set x1 x2 x3 hellipxN
To find the most likely value of the standard deviation (the best estimate of
the width of the x distribution) find σx that yields the highest probability for
the data set
The combined probability for the full data set is the product
Minimize
Sum
Solve for
derivative
wrst X set to 0
Best
estimate
of X
Best Estimate of σ is from maximum probability or minimum summation
Minimize
Sum
Best
estimate
of σ
Solve for
derivative
wrst σ set to 0
See
Prob 526
prop1
120590119890minus(1199091minus119883)2 2120590 times
1
120590119890minus(1199092minus119883)2 2120590 times hellip times
1
120590119890minus(119909119873minus119883)2 2120590 =
1
120590119873119890 minus(119909119894minus119883)2 2120590
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 47
Lecture 3 Slide 47
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Combining Data Sets
Weighted Averages
References Taylor Ch 7
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 48
Lecture 3 Slide 48
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Weighted Averages
Question How can we properly combine two or more separate
independent measurements of the same randomly distributed
quantity to determine a best combined value with uncertainty
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 49
Lecture 3 Slide 49
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider two measurements of the same quantity described by a random Gaussian distribution
ltx1gt x1 and ltx2gt x2
The probability of measuring two such measurements is
119875119903119900119887119909 1199091 1199092 = 119875119903119900119887119909 1199091 119875119903119900119887119909 1199092
= 1
12059011205902119890minus12059422 119908ℎ119890119903119890 1205942 equiv
1199091 minus 119883
1205901 2 +
1199092 minus 119883
1205902 2
To find the best value for χ find the maximum Prob or minimum χ2
Weighted Averages
Assume negligible systematic errors
Note χ2 or Chi squared is the sum of the squares of the deviations from the mean divided by
the corresponding uncertainty
Such methods are called ldquoMethods of Least Squaresrdquo They follow directly from the Principle of
Maximum Likelihood
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 50
Lecture 3 Slide 50
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
The probability of measuring two such measurements is
119875119903119900119887119909 1199091 1199092 = 119875119903119900119887119909 1199091 119875119903119900119887119909 1199092
= 1
12059011205902119890minus12059422 119908ℎ119890119903119890 1205942 equiv
1199091 minus 119883
1205901 2 +
1199092 minus 119883
1205902 2
To find the best value for χ find the maximum Prob or minimum χ2
Weighted Averages
This leads to 119909119882_119886119907119892 = 11990811199091+11990821199092
1199081+1199082=
119908119894 119909119894
119908119894 119908ℎ119890119903119890 119908119894 = 1
120590119894 2
Best Estimate of χ is from maximum probibility or minimum summation
Minimize Sum Solve for best estimate of χ Solve for derivative wrst χ set to 0
Note If w1=w2 we recover the standard result Xwavg= (12) (x1+x2)
Finally the width of a weighted average distribution is
120590119908119894119890119892 ℎ119905119890119889 119886119907119892 = 1
119908119894119894
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 51
Lecture 3 Slide 51
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Weighted Averages on Steroids
A very powerful method for combining data from different sources
with different methods and uncertainties (or indeed data of
related measured and calculated quantities) is Kalman filtering
The Kalman filter also known as
linear quadratic estimation (LQE)
is an algorithm that uses a series
of measurements observed over
time containing noise (random
variations) and other
inaccuracies and produces
estimates of unknown variables
that tend to be more precise than
those based on a single
measurement alone
The Kalman filter keeps track of the estimated state of the system and the variance or uncertainty of the
estimate The estimate is updated using a state transition model and measurements xk|k-1 denotes the
estimate of the systems state at time step k before the kth measurement yk has been taken into account Pk|k-1
is the corresponding uncertainty --Wikipedia 2013
Ludger Scherliess of USU Physics is a world expert at using Kalman filtering for the assimilation
of satellite and ground-based data and the USU GAMES model to predict space weather
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 52
Lecture 3 Slide 52
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Rejecting Data
Chauvenetrsquos Criterion
References Taylor Ch 6
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 53
Lecture 3 Slide 53
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly
ldquounreasonablerdquo data point from a set of randomly distributed
measurements
bull Never
bull Whenever it makes things look better
bull Chauvenetrsquos criterion provides a (quantitative) compromise
Rejecting Data
What is a good criteria for
rejecting data
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 54
Lecture 3 Slide 54
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly
ldquounreasonablerdquo data point from a set of randomly distributed
measurements
bull Never
Rejecting Data
Zallenrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 55
Lecture 3 Slide 55
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly
ldquounreasonablerdquo data point from a set of randomly distributed
measurements
bull Whenever it makes things look better
Rejecting Data
Disneyrsquos Criterion
Disneyrsquos First Law
Wishing will make it so
Disneyrsquos Second Law
Dreams are more colorful than reality
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 56
Lecture 3 Slide 56
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Data may be rejected if the expected number of measurements at
least as deviant as the suspect measurement is less than 50
Consider a set of N measurements of a single quantity
x1x2 helliphellipxN
Calculate ltxgt and σx and then determine the fractional deviations from the mean of all
the points
For the suspect point(s) xsuspect find the probability of such a point occurring in N
measurements
n = (expected number as deviant as xsuspect)
= N Prob(outside xsuspectσx)
Rejecting Data
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 57
Lecture 3 Slide 57
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Function of Gaussian Distribution
Area under curve
(probability that
ndashtσltxlt+tσ) is given
in Table at right
Probable Error (probability that
ndash067σltxlt+067σ) is 50
Error Function (probability that ndashtσltxlt+tσ )
Error Function Tabulated values-see App A
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 58
Lecture 3 Slide 58
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 59
Lecture 3 Slide 59
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashExample 1
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 60
Lecture 3 Slide 60
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Details (1)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 61
Lecture 3 Slide 61
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashDetails (2)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 62
Lecture 3 Slide 62
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos
Criterionmdash
Details (3)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 63
Lecture 3 Slide 63
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos
Criterionmdash
Example 2
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 64
Lecture 3 Slide 64
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Summary of Probability Theory
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 65
Lecture 3 Slide 65
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of
Probability
Theory-I
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 66
Lecture 3 Slide 66
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of
Probability
Theory-II
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 67
Lecture 3 Slide 67
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-III
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 68
Lecture 3 Slide 68
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-IV
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 19
Lecture 3 Slide 19
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Distribution Functions
Available
on web site
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 20
Lecture 3 Slide 20
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
The Gaussian Distribution
Function
References Taylor Ch 5
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 21
Lecture 3 Slide 21
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Gaussian Integrals
Factorial Approximations
119899 asymp 2120587119899 1 2 119899119899 119890119909119901 minus119899 + 1
12 119899+ 119874
1
1198992
119897119900119892 119899 asymp 1
2 119897119900119892 2120587 + 119899 +
1
2 log 119899 minus 119899 +
1
12 119899+ 119874
1
1198992
119897119900119892 119899 asymp 119899 log 119899 minus 119899 (for all terms decreasing faster than linearly with n)
Gaussian Integrals
119868119898 = 2 119909119898 exp(minus1199092)infin
0 119889119909 mgt-1
119868119898 = 2 119910119899 exp(minus119910)infin
0 119889119910 equiv Γ(119899 + 1) 1199092 equiv 119910 2 119889119909 = 1199101 2 119889119910 119899 equiv 1
2 (119898 minus 1)
1198680 = Γ 119899 =1
2 = 120587 m=0 119899 = minus1
2
1198682 119896 = Γ 119896 +1
2 = 119896 minus 1
2 119896 minus 32 3
2 12 120587 even m m=2 kgt0 119899 = 119896 minus 1
2
1198682 119896+1 = Γ 119896 + 1 = 119896 odd m m=2 k+1gt0 119899 = 119896 ge 0
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 22
Lecture 3 Slide 22
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Gaussian Distribution Function
Width of Distribution
(standard deviation)
Center of Distribution
(mean) Distribution
Function
Independent
Variable
Gaussian Distribution Function
Normalization
Constant
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 23
Lecture 3 Slide 23
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Effects of Increasing N on Gaussian Distribution Functions
Consider the limiting distribution function as N infin and dx0
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 24
Lecture 3 Slide 24
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Defining the Gaussian
distribution function
in Mathcad
I suggest you ldquoinvestigaterdquo
these with the Mathcad sheet
on the web site
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 25
Lecture 3 Slide 25
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Using Mathcad to
define other common
distribution functions
Consider the limiting
distribution function
as N ȸ and ∆k0
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 26
Lecture 3 Slide 26
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider the Gaussian distribution function
Use the normalization condition to
evaluate the normalization constant
(see Taylor p 132)
The mean Ẋ is the first moment
of the Gaussian distribution
function (see Taylor p 134)
The standard deviation σx is the standard
deviation of the mean of the Gaussian
distribution function (see Taylor p 143)
Gaussian Distribution Moments
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 27
Lecture 3 Slide 27
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
When is mean x not Xbest
Answer When the
distribution is not
symmetric about X
Example Cauchy
Distribution
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 28
Lecture 3 Slide 28
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
When is mean x not Xbest
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 29
Lecture 3 Slide 29
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
When is mean x not Xbest
Answer When the
distribution is has
more than one peak
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 30
Lecture 3 Slide 30
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
The Gaussian Distribution
Function and Its Relation to
Errors
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 31
Lecture 3 Slide 31
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
A Review of Probabilities in Combination
Probability of a data set of N like measurements (x1x2hellipxN)
P (x1x2hellipxN) = P(x)1P(x2)hellipP(xN)
1 head AND 1 Four
P(H4) = P(H) P(4)
1 head OR 1 Four
P(H4) = P(H) + P(4) - P(H and 4) (true for a ldquonon-mutually exclusiverdquo events)
NOT 1 Six
P(NOT 6) = 1 - P(6)
1 Six OR 1 Four
P(64) = P(6) + P(4) (true for a ldquomutually exclusiverdquo single role)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 32
Lecture 3 Slide 32
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
The Gaussian Distribution Function and Its Relation to Errors
We will use the Gaussian distribution as applied to random
variables to develop the mathematical basis for bull Mean
bull Standard deviation
bull Standard deviation of the mean (SDOM)
bull Moments and expectation values
bull Error propagation formulas
bull Addition of errors in quadrature (for independent and random
measurements)
bull Numerical values for confidence limits (t-test)
bull Principle of maximal likelihood
bull Central limit theorem
bull Weighted distributions and Chi squared
bull Schwartz inequality (ie the uncertainty principle) (next lecture)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 33
Lecture 3 Slide 33
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider the Gaussian distribution function
Use the normalization condition to
evaluate the normalization constant
(see Taylor p 132)
The mean Ẋ is the first moment
of the Gaussian distribution
function (see Taylor p 134)
The standard deviation σx is the standard
deviation of the mean of the Gaussian
distribution function (see Taylor p 143)
Gaussian Distribution Moments
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 34
Lecture 3 Slide 34
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Standard Deviation of Gaussian Distribution
Area under curve
(probability that
ndashσltxlt+σ) is 68
5 or ~2σ ldquoSignificantrdquo
1 or ~3σ ldquoHighly Significantrdquo
1σ ldquoWithin errorsrdquo
5 ppm or ~5σ ldquoValid for HEPrdquo
See Sec 106 Testing of
Hypotheses
More complete Table in App A and B
Ah thatrsquos
highly
significant
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 35
Lecture 3 Slide 35
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Function of Gaussian Distribution
Area under curve
(probability that
ndashtσltxlt+tσ) is given
in Table at right
Probable Error (probability that
ndash067σltxlt+067σ) is 50
Error Function (probability that ndashtσltxlt+tσ )
Error Function Tabulated values-see App A
Complementary Error Function
(probability that ndashxlt-tσ AND xgt+tσ )
Prob(x outside tσ) = 1 - Prob(x within tσ)
More complete Table in App A and B
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 36
Lecture 3 Slide 36
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Useful Points on Gaussian Distribution
Full Width at Half Maximum FWHM
(See Prob 512)
Points of Inflection Occur at plusmnσ
(See Prob 513)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 37
Lecture 3 Slide 37
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Analysis and Gaussian Distribution
Adding a Constant Multiplying by a Constant
XX+A XBX and σB σ
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 38
Lecture 3 Slide 38
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Sum of Two Variables
Consider the derived quantity
Z=X + Y
(with X=0 and Y=0)
Error Propagation Addition
Error in Z
Multiple two probabilities 119875119903119900119887 119909 119910 = 119875119903119900119887 119909 ∙ 119875119903119900119887 119909 prop 119890119909119901 minus
1
2
1199092
1205901199092+
1199102
1205901199102
prop 119890119909119901 minus1
2
(119909+119910)2
1205901199092+120590119910
2 minus1199112
prop 119890119909119901 minus1
2
(119909+119910)2
1205901199092+120590119910
2 119890119909119901 minus
1199112
2
ICBST (Eq 553)
X+YZ and σx2 + σy
2 σz
2
(addition in quadrature for random independent variables)
Integrates to radic2π
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 39
Lecture 3 Slide 39
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Error Propagation
General formula for error propagation see [Taylor Secs 35 and 39]
Uncertainty as a function of one variable [Taylor Sec 35]
1 Consider a graphical method of estimating error
a) Consider an arbitaray function q(x)
b) Plot q(x) vs x
c) On the graph label
(1) qbest = q(xbest)
(2) qhi = q(xbest + δx)
(3) qlow = q(xbest- δx)
d) Making a linear approximation
x
qqxslopeqq
x
qqxslopeqq
bestbestlow
bestbesthi
e) Therefore
xx
Note the absolute value
How do we determine the error of a derived quantity Z(XYhellip) from errors in XYhellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 40
Lecture 3 Slide 40
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Error Propagation
General formula for uncertainty of a function of one variable
xx
[Taylor Eq 323]
Can you now derive for specific rules of error propagation
1 Addition and Subtraction [Taylor p 49]
2 Multiplication and Division [Taylor p 51]
3 Multiplication by a constant (exact number) [Taylor p 54]
4 Exponentiation (powers) [Taylor p 56]
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 41
Lecture 3 Slide 41
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Multiple Variables
Uncertainty of a function of multiple variables [Taylor Sec 311]
1 It can easily (no really) be shown that (see Taylor Sec 311) for a
function of several variables
)(
z
z
qy
y
qx
x
qzyxq
[Taylor Eq 347]
2 More correctly it can be shown that (see Taylor Sec 311) for a
function of several variables
)(
z
z
qy
y
qx
x
qzyxq
[Taylor Eq 347]
where the equals sign represents an upper bound as discussed above
3 For a function of several independent and random variables
)(
222
z
z
qy
y
qx
x
qzyxq
[Taylor Eq 348]
Again the proof is left for Ch 5
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 42
Lecture 3 Slide 42
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Product of Two Variables
Consider the arbitrary derived quantity
q(xy) of two independent random variables x and y
Expand q(xy) in a Taylor series about the expected values of x and y
(ie at points near X and Y)
Error Propagation General Case
119902 119909 119910 = 119902 119883 119884 + 120597119902
120597119909
119883 119909 minus 119883 +
120597119902
120597119910
119884
(119910 minus 119884)
Fixed shifts peak of distribution
Fixed Distribution centered at X with width σX
120575119902 119909 119910 = 120590119902 = 119902 119883 119884 + 120597119902
120597119909
119883120590119909
2
+ 120597119902
120597119910
119884
120590119910
2
0
How do we determine the error of a derived quantity Z(XYhellip) from errors in XYhellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 43
Lecture 3 Slide 43
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
SDOM of Gaussian Distribution
Standard Deviation of the Mean
The SDOM decreases as the square root of the number of measurements
That is the relative width σẊ of the distribution gets narrower as more
measurements are made
Each measurement
has similar σXi= σẊ
and similar partial
derivatives
Thushellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 44
Lecture 3 Slide 44
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Two Key Theorems from Probability
Central Limit Theorem
For random independent measurements (each with a well-define
expectation value and well-defined variance) the arithmetic mean
(average) will be approximately normally distributed
Principle of Maximum Likelihood
Given the N observed measurements x1 x2hellipxN the best estimates
for Ẋ and σ are those values for which the observed x1 x2hellipxN are
most likely
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 45
Lecture 3 Slide 45
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Mean of Gaussian Distribution as ldquoBest Estimaterdquo
Principle of Maximum Likelihood
Best Estimate of X is from maximum probability or minimum summation
Consider a data set
Each randomly distributed with
x1 x2 x3 hellipxN
To find the most likely value of the mean (the best estimate of ẋ)
find X that yields the highest probability for the data set
The combined probability for the full data set is the product
Minimize
Sum
Solve for
derivative wrst
X set to 0
Best
estimate
of X
prop1
120590119890minus(1199091minus119883)2 2120590 times
1
120590119890minus(1199092minus119883)2 2120590 times hellip times
1
120590119890minus(119909119873minus119883)2 2120590 =
1
120590119873119890 minus(119909119894minus119883)2 2120590
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 46
Lecture 3 Slide 46
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Uncertainty of ldquoBest Estimatesrdquo of Gaussian Distribution Principle of Maximum Likelihood
Best Estimate of X is from maximum probability or minimum summation
Consider a data set x1 x2 x3 hellipxN
To find the most likely value of the standard deviation (the best estimate of
the width of the x distribution) find σx that yields the highest probability for
the data set
The combined probability for the full data set is the product
Minimize
Sum
Solve for
derivative
wrst X set to 0
Best
estimate
of X
Best Estimate of σ is from maximum probability or minimum summation
Minimize
Sum
Best
estimate
of σ
Solve for
derivative
wrst σ set to 0
See
Prob 526
prop1
120590119890minus(1199091minus119883)2 2120590 times
1
120590119890minus(1199092minus119883)2 2120590 times hellip times
1
120590119890minus(119909119873minus119883)2 2120590 =
1
120590119873119890 minus(119909119894minus119883)2 2120590
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 47
Lecture 3 Slide 47
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Combining Data Sets
Weighted Averages
References Taylor Ch 7
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 48
Lecture 3 Slide 48
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Weighted Averages
Question How can we properly combine two or more separate
independent measurements of the same randomly distributed
quantity to determine a best combined value with uncertainty
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 49
Lecture 3 Slide 49
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider two measurements of the same quantity described by a random Gaussian distribution
ltx1gt x1 and ltx2gt x2
The probability of measuring two such measurements is
119875119903119900119887119909 1199091 1199092 = 119875119903119900119887119909 1199091 119875119903119900119887119909 1199092
= 1
12059011205902119890minus12059422 119908ℎ119890119903119890 1205942 equiv
1199091 minus 119883
1205901 2 +
1199092 minus 119883
1205902 2
To find the best value for χ find the maximum Prob or minimum χ2
Weighted Averages
Assume negligible systematic errors
Note χ2 or Chi squared is the sum of the squares of the deviations from the mean divided by
the corresponding uncertainty
Such methods are called ldquoMethods of Least Squaresrdquo They follow directly from the Principle of
Maximum Likelihood
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 50
Lecture 3 Slide 50
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
The probability of measuring two such measurements is
119875119903119900119887119909 1199091 1199092 = 119875119903119900119887119909 1199091 119875119903119900119887119909 1199092
= 1
12059011205902119890minus12059422 119908ℎ119890119903119890 1205942 equiv
1199091 minus 119883
1205901 2 +
1199092 minus 119883
1205902 2
To find the best value for χ find the maximum Prob or minimum χ2
Weighted Averages
This leads to 119909119882_119886119907119892 = 11990811199091+11990821199092
1199081+1199082=
119908119894 119909119894
119908119894 119908ℎ119890119903119890 119908119894 = 1
120590119894 2
Best Estimate of χ is from maximum probibility or minimum summation
Minimize Sum Solve for best estimate of χ Solve for derivative wrst χ set to 0
Note If w1=w2 we recover the standard result Xwavg= (12) (x1+x2)
Finally the width of a weighted average distribution is
120590119908119894119890119892 ℎ119905119890119889 119886119907119892 = 1
119908119894119894
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 51
Lecture 3 Slide 51
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Weighted Averages on Steroids
A very powerful method for combining data from different sources
with different methods and uncertainties (or indeed data of
related measured and calculated quantities) is Kalman filtering
The Kalman filter also known as
linear quadratic estimation (LQE)
is an algorithm that uses a series
of measurements observed over
time containing noise (random
variations) and other
inaccuracies and produces
estimates of unknown variables
that tend to be more precise than
those based on a single
measurement alone
The Kalman filter keeps track of the estimated state of the system and the variance or uncertainty of the
estimate The estimate is updated using a state transition model and measurements xk|k-1 denotes the
estimate of the systems state at time step k before the kth measurement yk has been taken into account Pk|k-1
is the corresponding uncertainty --Wikipedia 2013
Ludger Scherliess of USU Physics is a world expert at using Kalman filtering for the assimilation
of satellite and ground-based data and the USU GAMES model to predict space weather
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 52
Lecture 3 Slide 52
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Rejecting Data
Chauvenetrsquos Criterion
References Taylor Ch 6
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 53
Lecture 3 Slide 53
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly
ldquounreasonablerdquo data point from a set of randomly distributed
measurements
bull Never
bull Whenever it makes things look better
bull Chauvenetrsquos criterion provides a (quantitative) compromise
Rejecting Data
What is a good criteria for
rejecting data
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 54
Lecture 3 Slide 54
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly
ldquounreasonablerdquo data point from a set of randomly distributed
measurements
bull Never
Rejecting Data
Zallenrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 55
Lecture 3 Slide 55
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly
ldquounreasonablerdquo data point from a set of randomly distributed
measurements
bull Whenever it makes things look better
Rejecting Data
Disneyrsquos Criterion
Disneyrsquos First Law
Wishing will make it so
Disneyrsquos Second Law
Dreams are more colorful than reality
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 56
Lecture 3 Slide 56
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Data may be rejected if the expected number of measurements at
least as deviant as the suspect measurement is less than 50
Consider a set of N measurements of a single quantity
x1x2 helliphellipxN
Calculate ltxgt and σx and then determine the fractional deviations from the mean of all
the points
For the suspect point(s) xsuspect find the probability of such a point occurring in N
measurements
n = (expected number as deviant as xsuspect)
= N Prob(outside xsuspectσx)
Rejecting Data
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 57
Lecture 3 Slide 57
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Function of Gaussian Distribution
Area under curve
(probability that
ndashtσltxlt+tσ) is given
in Table at right
Probable Error (probability that
ndash067σltxlt+067σ) is 50
Error Function (probability that ndashtσltxlt+tσ )
Error Function Tabulated values-see App A
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 58
Lecture 3 Slide 58
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 59
Lecture 3 Slide 59
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashExample 1
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 60
Lecture 3 Slide 60
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Details (1)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 61
Lecture 3 Slide 61
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashDetails (2)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 62
Lecture 3 Slide 62
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos
Criterionmdash
Details (3)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 63
Lecture 3 Slide 63
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos
Criterionmdash
Example 2
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 64
Lecture 3 Slide 64
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Summary of Probability Theory
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 65
Lecture 3 Slide 65
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of
Probability
Theory-I
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 66
Lecture 3 Slide 66
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of
Probability
Theory-II
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 67
Lecture 3 Slide 67
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-III
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 68
Lecture 3 Slide 68
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-IV
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 20
Lecture 3 Slide 20
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
The Gaussian Distribution
Function
References Taylor Ch 5
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 21
Lecture 3 Slide 21
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Gaussian Integrals
Factorial Approximations
119899 asymp 2120587119899 1 2 119899119899 119890119909119901 minus119899 + 1
12 119899+ 119874
1
1198992
119897119900119892 119899 asymp 1
2 119897119900119892 2120587 + 119899 +
1
2 log 119899 minus 119899 +
1
12 119899+ 119874
1
1198992
119897119900119892 119899 asymp 119899 log 119899 minus 119899 (for all terms decreasing faster than linearly with n)
Gaussian Integrals
119868119898 = 2 119909119898 exp(minus1199092)infin
0 119889119909 mgt-1
119868119898 = 2 119910119899 exp(minus119910)infin
0 119889119910 equiv Γ(119899 + 1) 1199092 equiv 119910 2 119889119909 = 1199101 2 119889119910 119899 equiv 1
2 (119898 minus 1)
1198680 = Γ 119899 =1
2 = 120587 m=0 119899 = minus1
2
1198682 119896 = Γ 119896 +1
2 = 119896 minus 1
2 119896 minus 32 3
2 12 120587 even m m=2 kgt0 119899 = 119896 minus 1
2
1198682 119896+1 = Γ 119896 + 1 = 119896 odd m m=2 k+1gt0 119899 = 119896 ge 0
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 22
Lecture 3 Slide 22
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Gaussian Distribution Function
Width of Distribution
(standard deviation)
Center of Distribution
(mean) Distribution
Function
Independent
Variable
Gaussian Distribution Function
Normalization
Constant
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 23
Lecture 3 Slide 23
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Effects of Increasing N on Gaussian Distribution Functions
Consider the limiting distribution function as N infin and dx0
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 24
Lecture 3 Slide 24
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Defining the Gaussian
distribution function
in Mathcad
I suggest you ldquoinvestigaterdquo
these with the Mathcad sheet
on the web site
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 25
Lecture 3 Slide 25
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Using Mathcad to
define other common
distribution functions
Consider the limiting
distribution function
as N ȸ and ∆k0
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 26
Lecture 3 Slide 26
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider the Gaussian distribution function
Use the normalization condition to
evaluate the normalization constant
(see Taylor p 132)
The mean Ẋ is the first moment
of the Gaussian distribution
function (see Taylor p 134)
The standard deviation σx is the standard
deviation of the mean of the Gaussian
distribution function (see Taylor p 143)
Gaussian Distribution Moments
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 27
Lecture 3 Slide 27
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
When is mean x not Xbest
Answer When the
distribution is not
symmetric about X
Example Cauchy
Distribution
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 28
Lecture 3 Slide 28
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
When is mean x not Xbest
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 29
Lecture 3 Slide 29
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
When is mean x not Xbest
Answer When the
distribution is has
more than one peak
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 30
Lecture 3 Slide 30
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
The Gaussian Distribution
Function and Its Relation to
Errors
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 31
Lecture 3 Slide 31
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
A Review of Probabilities in Combination
Probability of a data set of N like measurements (x1x2hellipxN)
P (x1x2hellipxN) = P(x)1P(x2)hellipP(xN)
1 head AND 1 Four
P(H4) = P(H) P(4)
1 head OR 1 Four
P(H4) = P(H) + P(4) - P(H and 4) (true for a ldquonon-mutually exclusiverdquo events)
NOT 1 Six
P(NOT 6) = 1 - P(6)
1 Six OR 1 Four
P(64) = P(6) + P(4) (true for a ldquomutually exclusiverdquo single role)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 32
Lecture 3 Slide 32
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
The Gaussian Distribution Function and Its Relation to Errors
We will use the Gaussian distribution as applied to random
variables to develop the mathematical basis for bull Mean
bull Standard deviation
bull Standard deviation of the mean (SDOM)
bull Moments and expectation values
bull Error propagation formulas
bull Addition of errors in quadrature (for independent and random
measurements)
bull Numerical values for confidence limits (t-test)
bull Principle of maximal likelihood
bull Central limit theorem
bull Weighted distributions and Chi squared
bull Schwartz inequality (ie the uncertainty principle) (next lecture)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 33
Lecture 3 Slide 33
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider the Gaussian distribution function
Use the normalization condition to
evaluate the normalization constant
(see Taylor p 132)
The mean Ẋ is the first moment
of the Gaussian distribution
function (see Taylor p 134)
The standard deviation σx is the standard
deviation of the mean of the Gaussian
distribution function (see Taylor p 143)
Gaussian Distribution Moments
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 34
Lecture 3 Slide 34
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Standard Deviation of Gaussian Distribution
Area under curve
(probability that
ndashσltxlt+σ) is 68
5 or ~2σ ldquoSignificantrdquo
1 or ~3σ ldquoHighly Significantrdquo
1σ ldquoWithin errorsrdquo
5 ppm or ~5σ ldquoValid for HEPrdquo
See Sec 106 Testing of
Hypotheses
More complete Table in App A and B
Ah thatrsquos
highly
significant
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 35
Lecture 3 Slide 35
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Function of Gaussian Distribution
Area under curve
(probability that
ndashtσltxlt+tσ) is given
in Table at right
Probable Error (probability that
ndash067σltxlt+067σ) is 50
Error Function (probability that ndashtσltxlt+tσ )
Error Function Tabulated values-see App A
Complementary Error Function
(probability that ndashxlt-tσ AND xgt+tσ )
Prob(x outside tσ) = 1 - Prob(x within tσ)
More complete Table in App A and B
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 36
Lecture 3 Slide 36
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Useful Points on Gaussian Distribution
Full Width at Half Maximum FWHM
(See Prob 512)
Points of Inflection Occur at plusmnσ
(See Prob 513)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 37
Lecture 3 Slide 37
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Analysis and Gaussian Distribution
Adding a Constant Multiplying by a Constant
XX+A XBX and σB σ
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 38
Lecture 3 Slide 38
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Sum of Two Variables
Consider the derived quantity
Z=X + Y
(with X=0 and Y=0)
Error Propagation Addition
Error in Z
Multiple two probabilities 119875119903119900119887 119909 119910 = 119875119903119900119887 119909 ∙ 119875119903119900119887 119909 prop 119890119909119901 minus
1
2
1199092
1205901199092+
1199102
1205901199102
prop 119890119909119901 minus1
2
(119909+119910)2
1205901199092+120590119910
2 minus1199112
prop 119890119909119901 minus1
2
(119909+119910)2
1205901199092+120590119910
2 119890119909119901 minus
1199112
2
ICBST (Eq 553)
X+YZ and σx2 + σy
2 σz
2
(addition in quadrature for random independent variables)
Integrates to radic2π
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 39
Lecture 3 Slide 39
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Error Propagation
General formula for error propagation see [Taylor Secs 35 and 39]
Uncertainty as a function of one variable [Taylor Sec 35]
1 Consider a graphical method of estimating error
a) Consider an arbitaray function q(x)
b) Plot q(x) vs x
c) On the graph label
(1) qbest = q(xbest)
(2) qhi = q(xbest + δx)
(3) qlow = q(xbest- δx)
d) Making a linear approximation
x
qqxslopeqq
x
qqxslopeqq
bestbestlow
bestbesthi
e) Therefore
xx
Note the absolute value
How do we determine the error of a derived quantity Z(XYhellip) from errors in XYhellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 40
Lecture 3 Slide 40
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Error Propagation
General formula for uncertainty of a function of one variable
xx
[Taylor Eq 323]
Can you now derive for specific rules of error propagation
1 Addition and Subtraction [Taylor p 49]
2 Multiplication and Division [Taylor p 51]
3 Multiplication by a constant (exact number) [Taylor p 54]
4 Exponentiation (powers) [Taylor p 56]
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 41
Lecture 3 Slide 41
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Multiple Variables
Uncertainty of a function of multiple variables [Taylor Sec 311]
1 It can easily (no really) be shown that (see Taylor Sec 311) for a
function of several variables
)(
z
z
qy
y
qx
x
qzyxq
[Taylor Eq 347]
2 More correctly it can be shown that (see Taylor Sec 311) for a
function of several variables
)(
z
z
qy
y
qx
x
qzyxq
[Taylor Eq 347]
where the equals sign represents an upper bound as discussed above
3 For a function of several independent and random variables
)(
222
z
z
qy
y
qx
x
qzyxq
[Taylor Eq 348]
Again the proof is left for Ch 5
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 42
Lecture 3 Slide 42
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Product of Two Variables
Consider the arbitrary derived quantity
q(xy) of two independent random variables x and y
Expand q(xy) in a Taylor series about the expected values of x and y
(ie at points near X and Y)
Error Propagation General Case
119902 119909 119910 = 119902 119883 119884 + 120597119902
120597119909
119883 119909 minus 119883 +
120597119902
120597119910
119884
(119910 minus 119884)
Fixed shifts peak of distribution
Fixed Distribution centered at X with width σX
120575119902 119909 119910 = 120590119902 = 119902 119883 119884 + 120597119902
120597119909
119883120590119909
2
+ 120597119902
120597119910
119884
120590119910
2
0
How do we determine the error of a derived quantity Z(XYhellip) from errors in XYhellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 43
Lecture 3 Slide 43
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
SDOM of Gaussian Distribution
Standard Deviation of the Mean
The SDOM decreases as the square root of the number of measurements
That is the relative width σẊ of the distribution gets narrower as more
measurements are made
Each measurement
has similar σXi= σẊ
and similar partial
derivatives
Thushellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 44
Lecture 3 Slide 44
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Two Key Theorems from Probability
Central Limit Theorem
For random independent measurements (each with a well-define
expectation value and well-defined variance) the arithmetic mean
(average) will be approximately normally distributed
Principle of Maximum Likelihood
Given the N observed measurements x1 x2hellipxN the best estimates
for Ẋ and σ are those values for which the observed x1 x2hellipxN are
most likely
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 45
Lecture 3 Slide 45
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Mean of Gaussian Distribution as ldquoBest Estimaterdquo
Principle of Maximum Likelihood
Best Estimate of X is from maximum probability or minimum summation
Consider a data set
Each randomly distributed with
x1 x2 x3 hellipxN
To find the most likely value of the mean (the best estimate of ẋ)
find X that yields the highest probability for the data set
The combined probability for the full data set is the product
Minimize
Sum
Solve for
derivative wrst
X set to 0
Best
estimate
of X
prop1
120590119890minus(1199091minus119883)2 2120590 times
1
120590119890minus(1199092minus119883)2 2120590 times hellip times
1
120590119890minus(119909119873minus119883)2 2120590 =
1
120590119873119890 minus(119909119894minus119883)2 2120590
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 46
Lecture 3 Slide 46
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Uncertainty of ldquoBest Estimatesrdquo of Gaussian Distribution Principle of Maximum Likelihood
Best Estimate of X is from maximum probability or minimum summation
Consider a data set x1 x2 x3 hellipxN
To find the most likely value of the standard deviation (the best estimate of
the width of the x distribution) find σx that yields the highest probability for
the data set
The combined probability for the full data set is the product
Minimize
Sum
Solve for
derivative
wrst X set to 0
Best
estimate
of X
Best Estimate of σ is from maximum probability or minimum summation
Minimize
Sum
Best
estimate
of σ
Solve for
derivative
wrst σ set to 0
See
Prob 526
prop1
120590119890minus(1199091minus119883)2 2120590 times
1
120590119890minus(1199092minus119883)2 2120590 times hellip times
1
120590119890minus(119909119873minus119883)2 2120590 =
1
120590119873119890 minus(119909119894minus119883)2 2120590
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 47
Lecture 3 Slide 47
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Combining Data Sets
Weighted Averages
References Taylor Ch 7
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 48
Lecture 3 Slide 48
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Weighted Averages
Question How can we properly combine two or more separate
independent measurements of the same randomly distributed
quantity to determine a best combined value with uncertainty
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 49
Lecture 3 Slide 49
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider two measurements of the same quantity described by a random Gaussian distribution
ltx1gt x1 and ltx2gt x2
The probability of measuring two such measurements is
119875119903119900119887119909 1199091 1199092 = 119875119903119900119887119909 1199091 119875119903119900119887119909 1199092
= 1
12059011205902119890minus12059422 119908ℎ119890119903119890 1205942 equiv
1199091 minus 119883
1205901 2 +
1199092 minus 119883
1205902 2
To find the best value for χ find the maximum Prob or minimum χ2
Weighted Averages
Assume negligible systematic errors
Note χ2 or Chi squared is the sum of the squares of the deviations from the mean divided by
the corresponding uncertainty
Such methods are called ldquoMethods of Least Squaresrdquo They follow directly from the Principle of
Maximum Likelihood
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 50
Lecture 3 Slide 50
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
The probability of measuring two such measurements is
119875119903119900119887119909 1199091 1199092 = 119875119903119900119887119909 1199091 119875119903119900119887119909 1199092
= 1
12059011205902119890minus12059422 119908ℎ119890119903119890 1205942 equiv
1199091 minus 119883
1205901 2 +
1199092 minus 119883
1205902 2
To find the best value for χ find the maximum Prob or minimum χ2
Weighted Averages
This leads to 119909119882_119886119907119892 = 11990811199091+11990821199092
1199081+1199082=
119908119894 119909119894
119908119894 119908ℎ119890119903119890 119908119894 = 1
120590119894 2
Best Estimate of χ is from maximum probibility or minimum summation
Minimize Sum Solve for best estimate of χ Solve for derivative wrst χ set to 0
Note If w1=w2 we recover the standard result Xwavg= (12) (x1+x2)
Finally the width of a weighted average distribution is
120590119908119894119890119892 ℎ119905119890119889 119886119907119892 = 1
119908119894119894
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 51
Lecture 3 Slide 51
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Weighted Averages on Steroids
A very powerful method for combining data from different sources
with different methods and uncertainties (or indeed data of
related measured and calculated quantities) is Kalman filtering
The Kalman filter also known as
linear quadratic estimation (LQE)
is an algorithm that uses a series
of measurements observed over
time containing noise (random
variations) and other
inaccuracies and produces
estimates of unknown variables
that tend to be more precise than
those based on a single
measurement alone
The Kalman filter keeps track of the estimated state of the system and the variance or uncertainty of the
estimate The estimate is updated using a state transition model and measurements xk|k-1 denotes the
estimate of the systems state at time step k before the kth measurement yk has been taken into account Pk|k-1
is the corresponding uncertainty --Wikipedia 2013
Ludger Scherliess of USU Physics is a world expert at using Kalman filtering for the assimilation
of satellite and ground-based data and the USU GAMES model to predict space weather
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 52
Lecture 3 Slide 52
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Rejecting Data
Chauvenetrsquos Criterion
References Taylor Ch 6
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 53
Lecture 3 Slide 53
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly
ldquounreasonablerdquo data point from a set of randomly distributed
measurements
bull Never
bull Whenever it makes things look better
bull Chauvenetrsquos criterion provides a (quantitative) compromise
Rejecting Data
What is a good criteria for
rejecting data
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 54
Lecture 3 Slide 54
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly
ldquounreasonablerdquo data point from a set of randomly distributed
measurements
bull Never
Rejecting Data
Zallenrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 55
Lecture 3 Slide 55
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly
ldquounreasonablerdquo data point from a set of randomly distributed
measurements
bull Whenever it makes things look better
Rejecting Data
Disneyrsquos Criterion
Disneyrsquos First Law
Wishing will make it so
Disneyrsquos Second Law
Dreams are more colorful than reality
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 56
Lecture 3 Slide 56
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Data may be rejected if the expected number of measurements at
least as deviant as the suspect measurement is less than 50
Consider a set of N measurements of a single quantity
x1x2 helliphellipxN
Calculate ltxgt and σx and then determine the fractional deviations from the mean of all
the points
For the suspect point(s) xsuspect find the probability of such a point occurring in N
measurements
n = (expected number as deviant as xsuspect)
= N Prob(outside xsuspectσx)
Rejecting Data
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 57
Lecture 3 Slide 57
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Function of Gaussian Distribution
Area under curve
(probability that
ndashtσltxlt+tσ) is given
in Table at right
Probable Error (probability that
ndash067σltxlt+067σ) is 50
Error Function (probability that ndashtσltxlt+tσ )
Error Function Tabulated values-see App A
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 58
Lecture 3 Slide 58
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 59
Lecture 3 Slide 59
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashExample 1
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 60
Lecture 3 Slide 60
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Details (1)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 61
Lecture 3 Slide 61
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashDetails (2)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 62
Lecture 3 Slide 62
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos
Criterionmdash
Details (3)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 63
Lecture 3 Slide 63
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos
Criterionmdash
Example 2
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 64
Lecture 3 Slide 64
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Summary of Probability Theory
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 65
Lecture 3 Slide 65
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of
Probability
Theory-I
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 66
Lecture 3 Slide 66
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of
Probability
Theory-II
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 67
Lecture 3 Slide 67
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-III
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 68
Lecture 3 Slide 68
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-IV
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 21
Lecture 3 Slide 21
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Gaussian Integrals
Factorial Approximations
119899 asymp 2120587119899 1 2 119899119899 119890119909119901 minus119899 + 1
12 119899+ 119874
1
1198992
119897119900119892 119899 asymp 1
2 119897119900119892 2120587 + 119899 +
1
2 log 119899 minus 119899 +
1
12 119899+ 119874
1
1198992
119897119900119892 119899 asymp 119899 log 119899 minus 119899 (for all terms decreasing faster than linearly with n)
Gaussian Integrals
119868119898 = 2 119909119898 exp(minus1199092)infin
0 119889119909 mgt-1
119868119898 = 2 119910119899 exp(minus119910)infin
0 119889119910 equiv Γ(119899 + 1) 1199092 equiv 119910 2 119889119909 = 1199101 2 119889119910 119899 equiv 1
2 (119898 minus 1)
1198680 = Γ 119899 =1
2 = 120587 m=0 119899 = minus1
2
1198682 119896 = Γ 119896 +1
2 = 119896 minus 1
2 119896 minus 32 3
2 12 120587 even m m=2 kgt0 119899 = 119896 minus 1
2
1198682 119896+1 = Γ 119896 + 1 = 119896 odd m m=2 k+1gt0 119899 = 119896 ge 0
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 22
Lecture 3 Slide 22
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Gaussian Distribution Function
Width of Distribution
(standard deviation)
Center of Distribution
(mean) Distribution
Function
Independent
Variable
Gaussian Distribution Function
Normalization
Constant
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 23
Lecture 3 Slide 23
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Effects of Increasing N on Gaussian Distribution Functions
Consider the limiting distribution function as N infin and dx0
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 24
Lecture 3 Slide 24
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Defining the Gaussian
distribution function
in Mathcad
I suggest you ldquoinvestigaterdquo
these with the Mathcad sheet
on the web site
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 25
Lecture 3 Slide 25
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Using Mathcad to
define other common
distribution functions
Consider the limiting
distribution function
as N ȸ and ∆k0
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 26
Lecture 3 Slide 26
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider the Gaussian distribution function
Use the normalization condition to
evaluate the normalization constant
(see Taylor p 132)
The mean Ẋ is the first moment
of the Gaussian distribution
function (see Taylor p 134)
The standard deviation σx is the standard
deviation of the mean of the Gaussian
distribution function (see Taylor p 143)
Gaussian Distribution Moments
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 27
Lecture 3 Slide 27
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
When is mean x not Xbest
Answer When the
distribution is not
symmetric about X
Example Cauchy
Distribution
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 28
Lecture 3 Slide 28
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
When is mean x not Xbest
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 29
Lecture 3 Slide 29
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
When is mean x not Xbest
Answer When the
distribution is has
more than one peak
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 30
Lecture 3 Slide 30
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
The Gaussian Distribution
Function and Its Relation to
Errors
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 31
Lecture 3 Slide 31
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
A Review of Probabilities in Combination
Probability of a data set of N like measurements (x1x2hellipxN)
P (x1x2hellipxN) = P(x)1P(x2)hellipP(xN)
1 head AND 1 Four
P(H4) = P(H) P(4)
1 head OR 1 Four
P(H4) = P(H) + P(4) - P(H and 4) (true for a ldquonon-mutually exclusiverdquo events)
NOT 1 Six
P(NOT 6) = 1 - P(6)
1 Six OR 1 Four
P(64) = P(6) + P(4) (true for a ldquomutually exclusiverdquo single role)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 32
Lecture 3 Slide 32
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
The Gaussian Distribution Function and Its Relation to Errors
We will use the Gaussian distribution as applied to random
variables to develop the mathematical basis for bull Mean
bull Standard deviation
bull Standard deviation of the mean (SDOM)
bull Moments and expectation values
bull Error propagation formulas
bull Addition of errors in quadrature (for independent and random
measurements)
bull Numerical values for confidence limits (t-test)
bull Principle of maximal likelihood
bull Central limit theorem
bull Weighted distributions and Chi squared
bull Schwartz inequality (ie the uncertainty principle) (next lecture)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 33
Lecture 3 Slide 33
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider the Gaussian distribution function
Use the normalization condition to
evaluate the normalization constant
(see Taylor p 132)
The mean Ẋ is the first moment
of the Gaussian distribution
function (see Taylor p 134)
The standard deviation σx is the standard
deviation of the mean of the Gaussian
distribution function (see Taylor p 143)
Gaussian Distribution Moments
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 34
Lecture 3 Slide 34
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Standard Deviation of Gaussian Distribution
Area under curve
(probability that
ndashσltxlt+σ) is 68
5 or ~2σ ldquoSignificantrdquo
1 or ~3σ ldquoHighly Significantrdquo
1σ ldquoWithin errorsrdquo
5 ppm or ~5σ ldquoValid for HEPrdquo
See Sec 106 Testing of
Hypotheses
More complete Table in App A and B
Ah thatrsquos
highly
significant
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 35
Lecture 3 Slide 35
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Function of Gaussian Distribution
Area under curve
(probability that
ndashtσltxlt+tσ) is given
in Table at right
Probable Error (probability that
ndash067σltxlt+067σ) is 50
Error Function (probability that ndashtσltxlt+tσ )
Error Function Tabulated values-see App A
Complementary Error Function
(probability that ndashxlt-tσ AND xgt+tσ )
Prob(x outside tσ) = 1 - Prob(x within tσ)
More complete Table in App A and B
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 36
Lecture 3 Slide 36
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Useful Points on Gaussian Distribution
Full Width at Half Maximum FWHM
(See Prob 512)
Points of Inflection Occur at plusmnσ
(See Prob 513)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 37
Lecture 3 Slide 37
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Analysis and Gaussian Distribution
Adding a Constant Multiplying by a Constant
XX+A XBX and σB σ
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 38
Lecture 3 Slide 38
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Sum of Two Variables
Consider the derived quantity
Z=X + Y
(with X=0 and Y=0)
Error Propagation Addition
Error in Z
Multiple two probabilities 119875119903119900119887 119909 119910 = 119875119903119900119887 119909 ∙ 119875119903119900119887 119909 prop 119890119909119901 minus
1
2
1199092
1205901199092+
1199102
1205901199102
prop 119890119909119901 minus1
2
(119909+119910)2
1205901199092+120590119910
2 minus1199112
prop 119890119909119901 minus1
2
(119909+119910)2
1205901199092+120590119910
2 119890119909119901 minus
1199112
2
ICBST (Eq 553)
X+YZ and σx2 + σy
2 σz
2
(addition in quadrature for random independent variables)
Integrates to radic2π
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 39
Lecture 3 Slide 39
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Error Propagation
General formula for error propagation see [Taylor Secs 35 and 39]
Uncertainty as a function of one variable [Taylor Sec 35]
1 Consider a graphical method of estimating error
a) Consider an arbitaray function q(x)
b) Plot q(x) vs x
c) On the graph label
(1) qbest = q(xbest)
(2) qhi = q(xbest + δx)
(3) qlow = q(xbest- δx)
d) Making a linear approximation
x
qqxslopeqq
x
qqxslopeqq
bestbestlow
bestbesthi
e) Therefore
xx
Note the absolute value
How do we determine the error of a derived quantity Z(XYhellip) from errors in XYhellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 40
Lecture 3 Slide 40
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Error Propagation
General formula for uncertainty of a function of one variable
xx
[Taylor Eq 323]
Can you now derive for specific rules of error propagation
1 Addition and Subtraction [Taylor p 49]
2 Multiplication and Division [Taylor p 51]
3 Multiplication by a constant (exact number) [Taylor p 54]
4 Exponentiation (powers) [Taylor p 56]
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 41
Lecture 3 Slide 41
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Multiple Variables
Uncertainty of a function of multiple variables [Taylor Sec 311]
1 It can easily (no really) be shown that (see Taylor Sec 311) for a
function of several variables
)(
z
z
qy
y
qx
x
qzyxq
[Taylor Eq 347]
2 More correctly it can be shown that (see Taylor Sec 311) for a
function of several variables
)(
z
z
qy
y
qx
x
qzyxq
[Taylor Eq 347]
where the equals sign represents an upper bound as discussed above
3 For a function of several independent and random variables
)(
222
z
z
qy
y
qx
x
qzyxq
[Taylor Eq 348]
Again the proof is left for Ch 5
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 42
Lecture 3 Slide 42
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Product of Two Variables
Consider the arbitrary derived quantity
q(xy) of two independent random variables x and y
Expand q(xy) in a Taylor series about the expected values of x and y
(ie at points near X and Y)
Error Propagation General Case
119902 119909 119910 = 119902 119883 119884 + 120597119902
120597119909
119883 119909 minus 119883 +
120597119902
120597119910
119884
(119910 minus 119884)
Fixed shifts peak of distribution
Fixed Distribution centered at X with width σX
120575119902 119909 119910 = 120590119902 = 119902 119883 119884 + 120597119902
120597119909
119883120590119909
2
+ 120597119902
120597119910
119884
120590119910
2
0
How do we determine the error of a derived quantity Z(XYhellip) from errors in XYhellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 43
Lecture 3 Slide 43
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
SDOM of Gaussian Distribution
Standard Deviation of the Mean
The SDOM decreases as the square root of the number of measurements
That is the relative width σẊ of the distribution gets narrower as more
measurements are made
Each measurement
has similar σXi= σẊ
and similar partial
derivatives
Thushellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 44
Lecture 3 Slide 44
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Two Key Theorems from Probability
Central Limit Theorem
For random independent measurements (each with a well-define
expectation value and well-defined variance) the arithmetic mean
(average) will be approximately normally distributed
Principle of Maximum Likelihood
Given the N observed measurements x1 x2hellipxN the best estimates
for Ẋ and σ are those values for which the observed x1 x2hellipxN are
most likely
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 45
Lecture 3 Slide 45
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Mean of Gaussian Distribution as ldquoBest Estimaterdquo
Principle of Maximum Likelihood
Best Estimate of X is from maximum probability or minimum summation
Consider a data set
Each randomly distributed with
x1 x2 x3 hellipxN
To find the most likely value of the mean (the best estimate of ẋ)
find X that yields the highest probability for the data set
The combined probability for the full data set is the product
Minimize
Sum
Solve for
derivative wrst
X set to 0
Best
estimate
of X
prop1
120590119890minus(1199091minus119883)2 2120590 times
1
120590119890minus(1199092minus119883)2 2120590 times hellip times
1
120590119890minus(119909119873minus119883)2 2120590 =
1
120590119873119890 minus(119909119894minus119883)2 2120590
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 46
Lecture 3 Slide 46
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Uncertainty of ldquoBest Estimatesrdquo of Gaussian Distribution Principle of Maximum Likelihood
Best Estimate of X is from maximum probability or minimum summation
Consider a data set x1 x2 x3 hellipxN
To find the most likely value of the standard deviation (the best estimate of
the width of the x distribution) find σx that yields the highest probability for
the data set
The combined probability for the full data set is the product
Minimize
Sum
Solve for
derivative
wrst X set to 0
Best
estimate
of X
Best Estimate of σ is from maximum probability or minimum summation
Minimize
Sum
Best
estimate
of σ
Solve for
derivative
wrst σ set to 0
See
Prob 526
prop1
120590119890minus(1199091minus119883)2 2120590 times
1
120590119890minus(1199092minus119883)2 2120590 times hellip times
1
120590119890minus(119909119873minus119883)2 2120590 =
1
120590119873119890 minus(119909119894minus119883)2 2120590
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 47
Lecture 3 Slide 47
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Combining Data Sets
Weighted Averages
References Taylor Ch 7
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 48
Lecture 3 Slide 48
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Weighted Averages
Question How can we properly combine two or more separate
independent measurements of the same randomly distributed
quantity to determine a best combined value with uncertainty
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 49
Lecture 3 Slide 49
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider two measurements of the same quantity described by a random Gaussian distribution
ltx1gt x1 and ltx2gt x2
The probability of measuring two such measurements is
119875119903119900119887119909 1199091 1199092 = 119875119903119900119887119909 1199091 119875119903119900119887119909 1199092
= 1
12059011205902119890minus12059422 119908ℎ119890119903119890 1205942 equiv
1199091 minus 119883
1205901 2 +
1199092 minus 119883
1205902 2
To find the best value for χ find the maximum Prob or minimum χ2
Weighted Averages
Assume negligible systematic errors
Note χ2 or Chi squared is the sum of the squares of the deviations from the mean divided by
the corresponding uncertainty
Such methods are called ldquoMethods of Least Squaresrdquo They follow directly from the Principle of
Maximum Likelihood
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 50
Lecture 3 Slide 50
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
The probability of measuring two such measurements is
119875119903119900119887119909 1199091 1199092 = 119875119903119900119887119909 1199091 119875119903119900119887119909 1199092
= 1
12059011205902119890minus12059422 119908ℎ119890119903119890 1205942 equiv
1199091 minus 119883
1205901 2 +
1199092 minus 119883
1205902 2
To find the best value for χ find the maximum Prob or minimum χ2
Weighted Averages
This leads to 119909119882_119886119907119892 = 11990811199091+11990821199092
1199081+1199082=
119908119894 119909119894
119908119894 119908ℎ119890119903119890 119908119894 = 1
120590119894 2
Best Estimate of χ is from maximum probibility or minimum summation
Minimize Sum Solve for best estimate of χ Solve for derivative wrst χ set to 0
Note If w1=w2 we recover the standard result Xwavg= (12) (x1+x2)
Finally the width of a weighted average distribution is
120590119908119894119890119892 ℎ119905119890119889 119886119907119892 = 1
119908119894119894
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 51
Lecture 3 Slide 51
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Weighted Averages on Steroids
A very powerful method for combining data from different sources
with different methods and uncertainties (or indeed data of
related measured and calculated quantities) is Kalman filtering
The Kalman filter also known as
linear quadratic estimation (LQE)
is an algorithm that uses a series
of measurements observed over
time containing noise (random
variations) and other
inaccuracies and produces
estimates of unknown variables
that tend to be more precise than
those based on a single
measurement alone
The Kalman filter keeps track of the estimated state of the system and the variance or uncertainty of the
estimate The estimate is updated using a state transition model and measurements xk|k-1 denotes the
estimate of the systems state at time step k before the kth measurement yk has been taken into account Pk|k-1
is the corresponding uncertainty --Wikipedia 2013
Ludger Scherliess of USU Physics is a world expert at using Kalman filtering for the assimilation
of satellite and ground-based data and the USU GAMES model to predict space weather
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 52
Lecture 3 Slide 52
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Rejecting Data
Chauvenetrsquos Criterion
References Taylor Ch 6
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 53
Lecture 3 Slide 53
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly
ldquounreasonablerdquo data point from a set of randomly distributed
measurements
bull Never
bull Whenever it makes things look better
bull Chauvenetrsquos criterion provides a (quantitative) compromise
Rejecting Data
What is a good criteria for
rejecting data
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 54
Lecture 3 Slide 54
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly
ldquounreasonablerdquo data point from a set of randomly distributed
measurements
bull Never
Rejecting Data
Zallenrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 55
Lecture 3 Slide 55
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly
ldquounreasonablerdquo data point from a set of randomly distributed
measurements
bull Whenever it makes things look better
Rejecting Data
Disneyrsquos Criterion
Disneyrsquos First Law
Wishing will make it so
Disneyrsquos Second Law
Dreams are more colorful than reality
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 56
Lecture 3 Slide 56
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Data may be rejected if the expected number of measurements at
least as deviant as the suspect measurement is less than 50
Consider a set of N measurements of a single quantity
x1x2 helliphellipxN
Calculate ltxgt and σx and then determine the fractional deviations from the mean of all
the points
For the suspect point(s) xsuspect find the probability of such a point occurring in N
measurements
n = (expected number as deviant as xsuspect)
= N Prob(outside xsuspectσx)
Rejecting Data
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 57
Lecture 3 Slide 57
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Function of Gaussian Distribution
Area under curve
(probability that
ndashtσltxlt+tσ) is given
in Table at right
Probable Error (probability that
ndash067σltxlt+067σ) is 50
Error Function (probability that ndashtσltxlt+tσ )
Error Function Tabulated values-see App A
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 58
Lecture 3 Slide 58
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 59
Lecture 3 Slide 59
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashExample 1
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 60
Lecture 3 Slide 60
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Details (1)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 61
Lecture 3 Slide 61
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashDetails (2)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 62
Lecture 3 Slide 62
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos
Criterionmdash
Details (3)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 63
Lecture 3 Slide 63
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos
Criterionmdash
Example 2
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 64
Lecture 3 Slide 64
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Summary of Probability Theory
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 65
Lecture 3 Slide 65
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of
Probability
Theory-I
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 66
Lecture 3 Slide 66
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of
Probability
Theory-II
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 67
Lecture 3 Slide 67
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-III
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 68
Lecture 3 Slide 68
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-IV
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 22
Lecture 3 Slide 22
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Gaussian Distribution Function
Width of Distribution
(standard deviation)
Center of Distribution
(mean) Distribution
Function
Independent
Variable
Gaussian Distribution Function
Normalization
Constant
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 23
Lecture 3 Slide 23
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Effects of Increasing N on Gaussian Distribution Functions
Consider the limiting distribution function as N infin and dx0
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 24
Lecture 3 Slide 24
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Defining the Gaussian
distribution function
in Mathcad
I suggest you ldquoinvestigaterdquo
these with the Mathcad sheet
on the web site
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 25
Lecture 3 Slide 25
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Using Mathcad to
define other common
distribution functions
Consider the limiting
distribution function
as N ȸ and ∆k0
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 26
Lecture 3 Slide 26
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider the Gaussian distribution function
Use the normalization condition to
evaluate the normalization constant
(see Taylor p 132)
The mean Ẋ is the first moment
of the Gaussian distribution
function (see Taylor p 134)
The standard deviation σx is the standard
deviation of the mean of the Gaussian
distribution function (see Taylor p 143)
Gaussian Distribution Moments
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 27
Lecture 3 Slide 27
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
When is mean x not Xbest
Answer When the
distribution is not
symmetric about X
Example Cauchy
Distribution
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 28
Lecture 3 Slide 28
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
When is mean x not Xbest
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 29
Lecture 3 Slide 29
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
When is mean x not Xbest
Answer When the
distribution is has
more than one peak
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 30
Lecture 3 Slide 30
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
The Gaussian Distribution
Function and Its Relation to
Errors
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 31
Lecture 3 Slide 31
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
A Review of Probabilities in Combination
Probability of a data set of N like measurements (x1x2hellipxN)
P (x1x2hellipxN) = P(x)1P(x2)hellipP(xN)
1 head AND 1 Four
P(H4) = P(H) P(4)
1 head OR 1 Four
P(H4) = P(H) + P(4) - P(H and 4) (true for a ldquonon-mutually exclusiverdquo events)
NOT 1 Six
P(NOT 6) = 1 - P(6)
1 Six OR 1 Four
P(64) = P(6) + P(4) (true for a ldquomutually exclusiverdquo single role)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 32
Lecture 3 Slide 32
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
The Gaussian Distribution Function and Its Relation to Errors
We will use the Gaussian distribution as applied to random
variables to develop the mathematical basis for bull Mean
bull Standard deviation
bull Standard deviation of the mean (SDOM)
bull Moments and expectation values
bull Error propagation formulas
bull Addition of errors in quadrature (for independent and random
measurements)
bull Numerical values for confidence limits (t-test)
bull Principle of maximal likelihood
bull Central limit theorem
bull Weighted distributions and Chi squared
bull Schwartz inequality (ie the uncertainty principle) (next lecture)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 33
Lecture 3 Slide 33
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider the Gaussian distribution function
Use the normalization condition to
evaluate the normalization constant
(see Taylor p 132)
The mean Ẋ is the first moment
of the Gaussian distribution
function (see Taylor p 134)
The standard deviation σx is the standard
deviation of the mean of the Gaussian
distribution function (see Taylor p 143)
Gaussian Distribution Moments
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 34
Lecture 3 Slide 34
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Standard Deviation of Gaussian Distribution
Area under curve
(probability that
ndashσltxlt+σ) is 68
5 or ~2σ ldquoSignificantrdquo
1 or ~3σ ldquoHighly Significantrdquo
1σ ldquoWithin errorsrdquo
5 ppm or ~5σ ldquoValid for HEPrdquo
See Sec 106 Testing of
Hypotheses
More complete Table in App A and B
Ah thatrsquos
highly
significant
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 35
Lecture 3 Slide 35
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Function of Gaussian Distribution
Area under curve
(probability that
ndashtσltxlt+tσ) is given
in Table at right
Probable Error (probability that
ndash067σltxlt+067σ) is 50
Error Function (probability that ndashtσltxlt+tσ )
Error Function Tabulated values-see App A
Complementary Error Function
(probability that ndashxlt-tσ AND xgt+tσ )
Prob(x outside tσ) = 1 - Prob(x within tσ)
More complete Table in App A and B
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 36
Lecture 3 Slide 36
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Useful Points on Gaussian Distribution
Full Width at Half Maximum FWHM
(See Prob 512)
Points of Inflection Occur at plusmnσ
(See Prob 513)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 37
Lecture 3 Slide 37
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Analysis and Gaussian Distribution
Adding a Constant Multiplying by a Constant
XX+A XBX and σB σ
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 38
Lecture 3 Slide 38
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Sum of Two Variables
Consider the derived quantity
Z=X + Y
(with X=0 and Y=0)
Error Propagation Addition
Error in Z
Multiple two probabilities 119875119903119900119887 119909 119910 = 119875119903119900119887 119909 ∙ 119875119903119900119887 119909 prop 119890119909119901 minus
1
2
1199092
1205901199092+
1199102
1205901199102
prop 119890119909119901 minus1
2
(119909+119910)2
1205901199092+120590119910
2 minus1199112
prop 119890119909119901 minus1
2
(119909+119910)2
1205901199092+120590119910
2 119890119909119901 minus
1199112
2
ICBST (Eq 553)
X+YZ and σx2 + σy
2 σz
2
(addition in quadrature for random independent variables)
Integrates to radic2π
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 39
Lecture 3 Slide 39
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Error Propagation
General formula for error propagation see [Taylor Secs 35 and 39]
Uncertainty as a function of one variable [Taylor Sec 35]
1 Consider a graphical method of estimating error
a) Consider an arbitaray function q(x)
b) Plot q(x) vs x
c) On the graph label
(1) qbest = q(xbest)
(2) qhi = q(xbest + δx)
(3) qlow = q(xbest- δx)
d) Making a linear approximation
x
qqxslopeqq
x
qqxslopeqq
bestbestlow
bestbesthi
e) Therefore
xx
Note the absolute value
How do we determine the error of a derived quantity Z(XYhellip) from errors in XYhellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 40
Lecture 3 Slide 40
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Error Propagation
General formula for uncertainty of a function of one variable
xx
[Taylor Eq 323]
Can you now derive for specific rules of error propagation
1 Addition and Subtraction [Taylor p 49]
2 Multiplication and Division [Taylor p 51]
3 Multiplication by a constant (exact number) [Taylor p 54]
4 Exponentiation (powers) [Taylor p 56]
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 41
Lecture 3 Slide 41
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Multiple Variables
Uncertainty of a function of multiple variables [Taylor Sec 311]
1 It can easily (no really) be shown that (see Taylor Sec 311) for a
function of several variables
)(
z
z
qy
y
qx
x
qzyxq
[Taylor Eq 347]
2 More correctly it can be shown that (see Taylor Sec 311) for a
function of several variables
)(
z
z
qy
y
qx
x
qzyxq
[Taylor Eq 347]
where the equals sign represents an upper bound as discussed above
3 For a function of several independent and random variables
)(
222
z
z
qy
y
qx
x
qzyxq
[Taylor Eq 348]
Again the proof is left for Ch 5
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 42
Lecture 3 Slide 42
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Product of Two Variables
Consider the arbitrary derived quantity
q(xy) of two independent random variables x and y
Expand q(xy) in a Taylor series about the expected values of x and y
(ie at points near X and Y)
Error Propagation General Case
119902 119909 119910 = 119902 119883 119884 + 120597119902
120597119909
119883 119909 minus 119883 +
120597119902
120597119910
119884
(119910 minus 119884)
Fixed shifts peak of distribution
Fixed Distribution centered at X with width σX
120575119902 119909 119910 = 120590119902 = 119902 119883 119884 + 120597119902
120597119909
119883120590119909
2
+ 120597119902
120597119910
119884
120590119910
2
0
How do we determine the error of a derived quantity Z(XYhellip) from errors in XYhellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 43
Lecture 3 Slide 43
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
SDOM of Gaussian Distribution
Standard Deviation of the Mean
The SDOM decreases as the square root of the number of measurements
That is the relative width σẊ of the distribution gets narrower as more
measurements are made
Each measurement
has similar σXi= σẊ
and similar partial
derivatives
Thushellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 44
Lecture 3 Slide 44
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Two Key Theorems from Probability
Central Limit Theorem
For random independent measurements (each with a well-define
expectation value and well-defined variance) the arithmetic mean
(average) will be approximately normally distributed
Principle of Maximum Likelihood
Given the N observed measurements x1 x2hellipxN the best estimates
for Ẋ and σ are those values for which the observed x1 x2hellipxN are
most likely
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 45
Lecture 3 Slide 45
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Mean of Gaussian Distribution as ldquoBest Estimaterdquo
Principle of Maximum Likelihood
Best Estimate of X is from maximum probability or minimum summation
Consider a data set
Each randomly distributed with
x1 x2 x3 hellipxN
To find the most likely value of the mean (the best estimate of ẋ)
find X that yields the highest probability for the data set
The combined probability for the full data set is the product
Minimize
Sum
Solve for
derivative wrst
X set to 0
Best
estimate
of X
prop1
120590119890minus(1199091minus119883)2 2120590 times
1
120590119890minus(1199092minus119883)2 2120590 times hellip times
1
120590119890minus(119909119873minus119883)2 2120590 =
1
120590119873119890 minus(119909119894minus119883)2 2120590
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 46
Lecture 3 Slide 46
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Uncertainty of ldquoBest Estimatesrdquo of Gaussian Distribution Principle of Maximum Likelihood
Best Estimate of X is from maximum probability or minimum summation
Consider a data set x1 x2 x3 hellipxN
To find the most likely value of the standard deviation (the best estimate of
the width of the x distribution) find σx that yields the highest probability for
the data set
The combined probability for the full data set is the product
Minimize
Sum
Solve for
derivative
wrst X set to 0
Best
estimate
of X
Best Estimate of σ is from maximum probability or minimum summation
Minimize
Sum
Best
estimate
of σ
Solve for
derivative
wrst σ set to 0
See
Prob 526
prop1
120590119890minus(1199091minus119883)2 2120590 times
1
120590119890minus(1199092minus119883)2 2120590 times hellip times
1
120590119890minus(119909119873minus119883)2 2120590 =
1
120590119873119890 minus(119909119894minus119883)2 2120590
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 47
Lecture 3 Slide 47
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Combining Data Sets
Weighted Averages
References Taylor Ch 7
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 48
Lecture 3 Slide 48
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Weighted Averages
Question How can we properly combine two or more separate
independent measurements of the same randomly distributed
quantity to determine a best combined value with uncertainty
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 49
Lecture 3 Slide 49
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider two measurements of the same quantity described by a random Gaussian distribution
ltx1gt x1 and ltx2gt x2
The probability of measuring two such measurements is
119875119903119900119887119909 1199091 1199092 = 119875119903119900119887119909 1199091 119875119903119900119887119909 1199092
= 1
12059011205902119890minus12059422 119908ℎ119890119903119890 1205942 equiv
1199091 minus 119883
1205901 2 +
1199092 minus 119883
1205902 2
To find the best value for χ find the maximum Prob or minimum χ2
Weighted Averages
Assume negligible systematic errors
Note χ2 or Chi squared is the sum of the squares of the deviations from the mean divided by
the corresponding uncertainty
Such methods are called ldquoMethods of Least Squaresrdquo They follow directly from the Principle of
Maximum Likelihood
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 50
Lecture 3 Slide 50
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
The probability of measuring two such measurements is
119875119903119900119887119909 1199091 1199092 = 119875119903119900119887119909 1199091 119875119903119900119887119909 1199092
= 1
12059011205902119890minus12059422 119908ℎ119890119903119890 1205942 equiv
1199091 minus 119883
1205901 2 +
1199092 minus 119883
1205902 2
To find the best value for χ find the maximum Prob or minimum χ2
Weighted Averages
This leads to 119909119882_119886119907119892 = 11990811199091+11990821199092
1199081+1199082=
119908119894 119909119894
119908119894 119908ℎ119890119903119890 119908119894 = 1
120590119894 2
Best Estimate of χ is from maximum probibility or minimum summation
Minimize Sum Solve for best estimate of χ Solve for derivative wrst χ set to 0
Note If w1=w2 we recover the standard result Xwavg= (12) (x1+x2)
Finally the width of a weighted average distribution is
120590119908119894119890119892 ℎ119905119890119889 119886119907119892 = 1
119908119894119894
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 51
Lecture 3 Slide 51
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Weighted Averages on Steroids
A very powerful method for combining data from different sources
with different methods and uncertainties (or indeed data of
related measured and calculated quantities) is Kalman filtering
The Kalman filter also known as
linear quadratic estimation (LQE)
is an algorithm that uses a series
of measurements observed over
time containing noise (random
variations) and other
inaccuracies and produces
estimates of unknown variables
that tend to be more precise than
those based on a single
measurement alone
The Kalman filter keeps track of the estimated state of the system and the variance or uncertainty of the
estimate The estimate is updated using a state transition model and measurements xk|k-1 denotes the
estimate of the systems state at time step k before the kth measurement yk has been taken into account Pk|k-1
is the corresponding uncertainty --Wikipedia 2013
Ludger Scherliess of USU Physics is a world expert at using Kalman filtering for the assimilation
of satellite and ground-based data and the USU GAMES model to predict space weather
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 52
Lecture 3 Slide 52
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Rejecting Data
Chauvenetrsquos Criterion
References Taylor Ch 6
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 53
Lecture 3 Slide 53
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly
ldquounreasonablerdquo data point from a set of randomly distributed
measurements
bull Never
bull Whenever it makes things look better
bull Chauvenetrsquos criterion provides a (quantitative) compromise
Rejecting Data
What is a good criteria for
rejecting data
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 54
Lecture 3 Slide 54
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly
ldquounreasonablerdquo data point from a set of randomly distributed
measurements
bull Never
Rejecting Data
Zallenrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 55
Lecture 3 Slide 55
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly
ldquounreasonablerdquo data point from a set of randomly distributed
measurements
bull Whenever it makes things look better
Rejecting Data
Disneyrsquos Criterion
Disneyrsquos First Law
Wishing will make it so
Disneyrsquos Second Law
Dreams are more colorful than reality
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 56
Lecture 3 Slide 56
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Data may be rejected if the expected number of measurements at
least as deviant as the suspect measurement is less than 50
Consider a set of N measurements of a single quantity
x1x2 helliphellipxN
Calculate ltxgt and σx and then determine the fractional deviations from the mean of all
the points
For the suspect point(s) xsuspect find the probability of such a point occurring in N
measurements
n = (expected number as deviant as xsuspect)
= N Prob(outside xsuspectσx)
Rejecting Data
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 57
Lecture 3 Slide 57
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Function of Gaussian Distribution
Area under curve
(probability that
ndashtσltxlt+tσ) is given
in Table at right
Probable Error (probability that
ndash067σltxlt+067σ) is 50
Error Function (probability that ndashtσltxlt+tσ )
Error Function Tabulated values-see App A
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 58
Lecture 3 Slide 58
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 59
Lecture 3 Slide 59
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashExample 1
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 60
Lecture 3 Slide 60
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Details (1)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 61
Lecture 3 Slide 61
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashDetails (2)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 62
Lecture 3 Slide 62
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos
Criterionmdash
Details (3)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 63
Lecture 3 Slide 63
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos
Criterionmdash
Example 2
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 64
Lecture 3 Slide 64
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Summary of Probability Theory
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 65
Lecture 3 Slide 65
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of
Probability
Theory-I
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 66
Lecture 3 Slide 66
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of
Probability
Theory-II
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 67
Lecture 3 Slide 67
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-III
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 68
Lecture 3 Slide 68
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-IV
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 23
Lecture 3 Slide 23
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Effects of Increasing N on Gaussian Distribution Functions
Consider the limiting distribution function as N infin and dx0
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 24
Lecture 3 Slide 24
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Defining the Gaussian
distribution function
in Mathcad
I suggest you ldquoinvestigaterdquo
these with the Mathcad sheet
on the web site
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 25
Lecture 3 Slide 25
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Using Mathcad to
define other common
distribution functions
Consider the limiting
distribution function
as N ȸ and ∆k0
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 26
Lecture 3 Slide 26
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider the Gaussian distribution function
Use the normalization condition to
evaluate the normalization constant
(see Taylor p 132)
The mean Ẋ is the first moment
of the Gaussian distribution
function (see Taylor p 134)
The standard deviation σx is the standard
deviation of the mean of the Gaussian
distribution function (see Taylor p 143)
Gaussian Distribution Moments
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 27
Lecture 3 Slide 27
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
When is mean x not Xbest
Answer When the
distribution is not
symmetric about X
Example Cauchy
Distribution
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 28
Lecture 3 Slide 28
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
When is mean x not Xbest
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 29
Lecture 3 Slide 29
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
When is mean x not Xbest
Answer When the
distribution is has
more than one peak
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 30
Lecture 3 Slide 30
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
The Gaussian Distribution
Function and Its Relation to
Errors
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 31
Lecture 3 Slide 31
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
A Review of Probabilities in Combination
Probability of a data set of N like measurements (x1x2hellipxN)
P (x1x2hellipxN) = P(x)1P(x2)hellipP(xN)
1 head AND 1 Four
P(H4) = P(H) P(4)
1 head OR 1 Four
P(H4) = P(H) + P(4) - P(H and 4) (true for a ldquonon-mutually exclusiverdquo events)
NOT 1 Six
P(NOT 6) = 1 - P(6)
1 Six OR 1 Four
P(64) = P(6) + P(4) (true for a ldquomutually exclusiverdquo single role)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 32
Lecture 3 Slide 32
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
The Gaussian Distribution Function and Its Relation to Errors
We will use the Gaussian distribution as applied to random
variables to develop the mathematical basis for bull Mean
bull Standard deviation
bull Standard deviation of the mean (SDOM)
bull Moments and expectation values
bull Error propagation formulas
bull Addition of errors in quadrature (for independent and random
measurements)
bull Numerical values for confidence limits (t-test)
bull Principle of maximal likelihood
bull Central limit theorem
bull Weighted distributions and Chi squared
bull Schwartz inequality (ie the uncertainty principle) (next lecture)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 33
Lecture 3 Slide 33
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider the Gaussian distribution function
Use the normalization condition to
evaluate the normalization constant
(see Taylor p 132)
The mean Ẋ is the first moment
of the Gaussian distribution
function (see Taylor p 134)
The standard deviation σx is the standard
deviation of the mean of the Gaussian
distribution function (see Taylor p 143)
Gaussian Distribution Moments
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 34
Lecture 3 Slide 34
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Standard Deviation of Gaussian Distribution
Area under curve
(probability that
ndashσltxlt+σ) is 68
5 or ~2σ ldquoSignificantrdquo
1 or ~3σ ldquoHighly Significantrdquo
1σ ldquoWithin errorsrdquo
5 ppm or ~5σ ldquoValid for HEPrdquo
See Sec 106 Testing of
Hypotheses
More complete Table in App A and B
Ah thatrsquos
highly
significant
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 35
Lecture 3 Slide 35
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Function of Gaussian Distribution
Area under curve
(probability that
ndashtσltxlt+tσ) is given
in Table at right
Probable Error (probability that
ndash067σltxlt+067σ) is 50
Error Function (probability that ndashtσltxlt+tσ )
Error Function Tabulated values-see App A
Complementary Error Function
(probability that ndashxlt-tσ AND xgt+tσ )
Prob(x outside tσ) = 1 - Prob(x within tσ)
More complete Table in App A and B
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 36
Lecture 3 Slide 36
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Useful Points on Gaussian Distribution
Full Width at Half Maximum FWHM
(See Prob 512)
Points of Inflection Occur at plusmnσ
(See Prob 513)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 37
Lecture 3 Slide 37
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Analysis and Gaussian Distribution
Adding a Constant Multiplying by a Constant
XX+A XBX and σB σ
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 38
Lecture 3 Slide 38
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Sum of Two Variables
Consider the derived quantity
Z=X + Y
(with X=0 and Y=0)
Error Propagation Addition
Error in Z
Multiple two probabilities 119875119903119900119887 119909 119910 = 119875119903119900119887 119909 ∙ 119875119903119900119887 119909 prop 119890119909119901 minus
1
2
1199092
1205901199092+
1199102
1205901199102
prop 119890119909119901 minus1
2
(119909+119910)2
1205901199092+120590119910
2 minus1199112
prop 119890119909119901 minus1
2
(119909+119910)2
1205901199092+120590119910
2 119890119909119901 minus
1199112
2
ICBST (Eq 553)
X+YZ and σx2 + σy
2 σz
2
(addition in quadrature for random independent variables)
Integrates to radic2π
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 39
Lecture 3 Slide 39
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Error Propagation
General formula for error propagation see [Taylor Secs 35 and 39]
Uncertainty as a function of one variable [Taylor Sec 35]
1 Consider a graphical method of estimating error
a) Consider an arbitaray function q(x)
b) Plot q(x) vs x
c) On the graph label
(1) qbest = q(xbest)
(2) qhi = q(xbest + δx)
(3) qlow = q(xbest- δx)
d) Making a linear approximation
x
qqxslopeqq
x
qqxslopeqq
bestbestlow
bestbesthi
e) Therefore
xx
Note the absolute value
How do we determine the error of a derived quantity Z(XYhellip) from errors in XYhellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 40
Lecture 3 Slide 40
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Error Propagation
General formula for uncertainty of a function of one variable
xx
[Taylor Eq 323]
Can you now derive for specific rules of error propagation
1 Addition and Subtraction [Taylor p 49]
2 Multiplication and Division [Taylor p 51]
3 Multiplication by a constant (exact number) [Taylor p 54]
4 Exponentiation (powers) [Taylor p 56]
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 41
Lecture 3 Slide 41
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Multiple Variables
Uncertainty of a function of multiple variables [Taylor Sec 311]
1 It can easily (no really) be shown that (see Taylor Sec 311) for a
function of several variables
)(
z
z
qy
y
qx
x
qzyxq
[Taylor Eq 347]
2 More correctly it can be shown that (see Taylor Sec 311) for a
function of several variables
)(
z
z
qy
y
qx
x
qzyxq
[Taylor Eq 347]
where the equals sign represents an upper bound as discussed above
3 For a function of several independent and random variables
)(
222
z
z
qy
y
qx
x
qzyxq
[Taylor Eq 348]
Again the proof is left for Ch 5
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 42
Lecture 3 Slide 42
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Product of Two Variables
Consider the arbitrary derived quantity
q(xy) of two independent random variables x and y
Expand q(xy) in a Taylor series about the expected values of x and y
(ie at points near X and Y)
Error Propagation General Case
119902 119909 119910 = 119902 119883 119884 + 120597119902
120597119909
119883 119909 minus 119883 +
120597119902
120597119910
119884
(119910 minus 119884)
Fixed shifts peak of distribution
Fixed Distribution centered at X with width σX
120575119902 119909 119910 = 120590119902 = 119902 119883 119884 + 120597119902
120597119909
119883120590119909
2
+ 120597119902
120597119910
119884
120590119910
2
0
How do we determine the error of a derived quantity Z(XYhellip) from errors in XYhellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 43
Lecture 3 Slide 43
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
SDOM of Gaussian Distribution
Standard Deviation of the Mean
The SDOM decreases as the square root of the number of measurements
That is the relative width σẊ of the distribution gets narrower as more
measurements are made
Each measurement
has similar σXi= σẊ
and similar partial
derivatives
Thushellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 44
Lecture 3 Slide 44
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Two Key Theorems from Probability
Central Limit Theorem
For random independent measurements (each with a well-define
expectation value and well-defined variance) the arithmetic mean
(average) will be approximately normally distributed
Principle of Maximum Likelihood
Given the N observed measurements x1 x2hellipxN the best estimates
for Ẋ and σ are those values for which the observed x1 x2hellipxN are
most likely
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 45
Lecture 3 Slide 45
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Mean of Gaussian Distribution as ldquoBest Estimaterdquo
Principle of Maximum Likelihood
Best Estimate of X is from maximum probability or minimum summation
Consider a data set
Each randomly distributed with
x1 x2 x3 hellipxN
To find the most likely value of the mean (the best estimate of ẋ)
find X that yields the highest probability for the data set
The combined probability for the full data set is the product
Minimize
Sum
Solve for
derivative wrst
X set to 0
Best
estimate
of X
prop1
120590119890minus(1199091minus119883)2 2120590 times
1
120590119890minus(1199092minus119883)2 2120590 times hellip times
1
120590119890minus(119909119873minus119883)2 2120590 =
1
120590119873119890 minus(119909119894minus119883)2 2120590
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 46
Lecture 3 Slide 46
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Uncertainty of ldquoBest Estimatesrdquo of Gaussian Distribution Principle of Maximum Likelihood
Best Estimate of X is from maximum probability or minimum summation
Consider a data set x1 x2 x3 hellipxN
To find the most likely value of the standard deviation (the best estimate of
the width of the x distribution) find σx that yields the highest probability for
the data set
The combined probability for the full data set is the product
Minimize
Sum
Solve for
derivative
wrst X set to 0
Best
estimate
of X
Best Estimate of σ is from maximum probability or minimum summation
Minimize
Sum
Best
estimate
of σ
Solve for
derivative
wrst σ set to 0
See
Prob 526
prop1
120590119890minus(1199091minus119883)2 2120590 times
1
120590119890minus(1199092minus119883)2 2120590 times hellip times
1
120590119890minus(119909119873minus119883)2 2120590 =
1
120590119873119890 minus(119909119894minus119883)2 2120590
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 47
Lecture 3 Slide 47
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Combining Data Sets
Weighted Averages
References Taylor Ch 7
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 48
Lecture 3 Slide 48
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Weighted Averages
Question How can we properly combine two or more separate
independent measurements of the same randomly distributed
quantity to determine a best combined value with uncertainty
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 49
Lecture 3 Slide 49
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider two measurements of the same quantity described by a random Gaussian distribution
ltx1gt x1 and ltx2gt x2
The probability of measuring two such measurements is
119875119903119900119887119909 1199091 1199092 = 119875119903119900119887119909 1199091 119875119903119900119887119909 1199092
= 1
12059011205902119890minus12059422 119908ℎ119890119903119890 1205942 equiv
1199091 minus 119883
1205901 2 +
1199092 minus 119883
1205902 2
To find the best value for χ find the maximum Prob or minimum χ2
Weighted Averages
Assume negligible systematic errors
Note χ2 or Chi squared is the sum of the squares of the deviations from the mean divided by
the corresponding uncertainty
Such methods are called ldquoMethods of Least Squaresrdquo They follow directly from the Principle of
Maximum Likelihood
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 50
Lecture 3 Slide 50
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
The probability of measuring two such measurements is
119875119903119900119887119909 1199091 1199092 = 119875119903119900119887119909 1199091 119875119903119900119887119909 1199092
= 1
12059011205902119890minus12059422 119908ℎ119890119903119890 1205942 equiv
1199091 minus 119883
1205901 2 +
1199092 minus 119883
1205902 2
To find the best value for χ find the maximum Prob or minimum χ2
Weighted Averages
This leads to 119909119882_119886119907119892 = 11990811199091+11990821199092
1199081+1199082=
119908119894 119909119894
119908119894 119908ℎ119890119903119890 119908119894 = 1
120590119894 2
Best Estimate of χ is from maximum probibility or minimum summation
Minimize Sum Solve for best estimate of χ Solve for derivative wrst χ set to 0
Note If w1=w2 we recover the standard result Xwavg= (12) (x1+x2)
Finally the width of a weighted average distribution is
120590119908119894119890119892 ℎ119905119890119889 119886119907119892 = 1
119908119894119894
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 51
Lecture 3 Slide 51
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Weighted Averages on Steroids
A very powerful method for combining data from different sources
with different methods and uncertainties (or indeed data of
related measured and calculated quantities) is Kalman filtering
The Kalman filter also known as
linear quadratic estimation (LQE)
is an algorithm that uses a series
of measurements observed over
time containing noise (random
variations) and other
inaccuracies and produces
estimates of unknown variables
that tend to be more precise than
those based on a single
measurement alone
The Kalman filter keeps track of the estimated state of the system and the variance or uncertainty of the
estimate The estimate is updated using a state transition model and measurements xk|k-1 denotes the
estimate of the systems state at time step k before the kth measurement yk has been taken into account Pk|k-1
is the corresponding uncertainty --Wikipedia 2013
Ludger Scherliess of USU Physics is a world expert at using Kalman filtering for the assimilation
of satellite and ground-based data and the USU GAMES model to predict space weather
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 52
Lecture 3 Slide 52
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Rejecting Data
Chauvenetrsquos Criterion
References Taylor Ch 6
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 53
Lecture 3 Slide 53
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly
ldquounreasonablerdquo data point from a set of randomly distributed
measurements
bull Never
bull Whenever it makes things look better
bull Chauvenetrsquos criterion provides a (quantitative) compromise
Rejecting Data
What is a good criteria for
rejecting data
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 54
Lecture 3 Slide 54
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly
ldquounreasonablerdquo data point from a set of randomly distributed
measurements
bull Never
Rejecting Data
Zallenrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 55
Lecture 3 Slide 55
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly
ldquounreasonablerdquo data point from a set of randomly distributed
measurements
bull Whenever it makes things look better
Rejecting Data
Disneyrsquos Criterion
Disneyrsquos First Law
Wishing will make it so
Disneyrsquos Second Law
Dreams are more colorful than reality
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 56
Lecture 3 Slide 56
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Data may be rejected if the expected number of measurements at
least as deviant as the suspect measurement is less than 50
Consider a set of N measurements of a single quantity
x1x2 helliphellipxN
Calculate ltxgt and σx and then determine the fractional deviations from the mean of all
the points
For the suspect point(s) xsuspect find the probability of such a point occurring in N
measurements
n = (expected number as deviant as xsuspect)
= N Prob(outside xsuspectσx)
Rejecting Data
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 57
Lecture 3 Slide 57
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Function of Gaussian Distribution
Area under curve
(probability that
ndashtσltxlt+tσ) is given
in Table at right
Probable Error (probability that
ndash067σltxlt+067σ) is 50
Error Function (probability that ndashtσltxlt+tσ )
Error Function Tabulated values-see App A
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 58
Lecture 3 Slide 58
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 59
Lecture 3 Slide 59
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashExample 1
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 60
Lecture 3 Slide 60
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Details (1)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 61
Lecture 3 Slide 61
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashDetails (2)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 62
Lecture 3 Slide 62
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos
Criterionmdash
Details (3)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 63
Lecture 3 Slide 63
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos
Criterionmdash
Example 2
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 64
Lecture 3 Slide 64
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Summary of Probability Theory
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 65
Lecture 3 Slide 65
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of
Probability
Theory-I
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 66
Lecture 3 Slide 66
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of
Probability
Theory-II
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 67
Lecture 3 Slide 67
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-III
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 68
Lecture 3 Slide 68
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-IV
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 24
Lecture 3 Slide 24
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Defining the Gaussian
distribution function
in Mathcad
I suggest you ldquoinvestigaterdquo
these with the Mathcad sheet
on the web site
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 25
Lecture 3 Slide 25
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Using Mathcad to
define other common
distribution functions
Consider the limiting
distribution function
as N ȸ and ∆k0
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 26
Lecture 3 Slide 26
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider the Gaussian distribution function
Use the normalization condition to
evaluate the normalization constant
(see Taylor p 132)
The mean Ẋ is the first moment
of the Gaussian distribution
function (see Taylor p 134)
The standard deviation σx is the standard
deviation of the mean of the Gaussian
distribution function (see Taylor p 143)
Gaussian Distribution Moments
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 27
Lecture 3 Slide 27
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
When is mean x not Xbest
Answer When the
distribution is not
symmetric about X
Example Cauchy
Distribution
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 28
Lecture 3 Slide 28
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
When is mean x not Xbest
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 29
Lecture 3 Slide 29
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
When is mean x not Xbest
Answer When the
distribution is has
more than one peak
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 30
Lecture 3 Slide 30
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
The Gaussian Distribution
Function and Its Relation to
Errors
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 31
Lecture 3 Slide 31
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
A Review of Probabilities in Combination
Probability of a data set of N like measurements (x1x2hellipxN)
P (x1x2hellipxN) = P(x)1P(x2)hellipP(xN)
1 head AND 1 Four
P(H4) = P(H) P(4)
1 head OR 1 Four
P(H4) = P(H) + P(4) - P(H and 4) (true for a ldquonon-mutually exclusiverdquo events)
NOT 1 Six
P(NOT 6) = 1 - P(6)
1 Six OR 1 Four
P(64) = P(6) + P(4) (true for a ldquomutually exclusiverdquo single role)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 32
Lecture 3 Slide 32
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
The Gaussian Distribution Function and Its Relation to Errors
We will use the Gaussian distribution as applied to random
variables to develop the mathematical basis for bull Mean
bull Standard deviation
bull Standard deviation of the mean (SDOM)
bull Moments and expectation values
bull Error propagation formulas
bull Addition of errors in quadrature (for independent and random
measurements)
bull Numerical values for confidence limits (t-test)
bull Principle of maximal likelihood
bull Central limit theorem
bull Weighted distributions and Chi squared
bull Schwartz inequality (ie the uncertainty principle) (next lecture)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 33
Lecture 3 Slide 33
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider the Gaussian distribution function
Use the normalization condition to
evaluate the normalization constant
(see Taylor p 132)
The mean Ẋ is the first moment
of the Gaussian distribution
function (see Taylor p 134)
The standard deviation σx is the standard
deviation of the mean of the Gaussian
distribution function (see Taylor p 143)
Gaussian Distribution Moments
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 34
Lecture 3 Slide 34
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Standard Deviation of Gaussian Distribution
Area under curve
(probability that
ndashσltxlt+σ) is 68
5 or ~2σ ldquoSignificantrdquo
1 or ~3σ ldquoHighly Significantrdquo
1σ ldquoWithin errorsrdquo
5 ppm or ~5σ ldquoValid for HEPrdquo
See Sec 106 Testing of
Hypotheses
More complete Table in App A and B
Ah thatrsquos
highly
significant
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 35
Lecture 3 Slide 35
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Function of Gaussian Distribution
Area under curve
(probability that
ndashtσltxlt+tσ) is given
in Table at right
Probable Error (probability that
ndash067σltxlt+067σ) is 50
Error Function (probability that ndashtσltxlt+tσ )
Error Function Tabulated values-see App A
Complementary Error Function
(probability that ndashxlt-tσ AND xgt+tσ )
Prob(x outside tσ) = 1 - Prob(x within tσ)
More complete Table in App A and B
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 36
Lecture 3 Slide 36
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Useful Points on Gaussian Distribution
Full Width at Half Maximum FWHM
(See Prob 512)
Points of Inflection Occur at plusmnσ
(See Prob 513)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 37
Lecture 3 Slide 37
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Analysis and Gaussian Distribution
Adding a Constant Multiplying by a Constant
XX+A XBX and σB σ
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 38
Lecture 3 Slide 38
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Sum of Two Variables
Consider the derived quantity
Z=X + Y
(with X=0 and Y=0)
Error Propagation Addition
Error in Z
Multiple two probabilities 119875119903119900119887 119909 119910 = 119875119903119900119887 119909 ∙ 119875119903119900119887 119909 prop 119890119909119901 minus
1
2
1199092
1205901199092+
1199102
1205901199102
prop 119890119909119901 minus1
2
(119909+119910)2
1205901199092+120590119910
2 minus1199112
prop 119890119909119901 minus1
2
(119909+119910)2
1205901199092+120590119910
2 119890119909119901 minus
1199112
2
ICBST (Eq 553)
X+YZ and σx2 + σy
2 σz
2
(addition in quadrature for random independent variables)
Integrates to radic2π
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 39
Lecture 3 Slide 39
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Error Propagation
General formula for error propagation see [Taylor Secs 35 and 39]
Uncertainty as a function of one variable [Taylor Sec 35]
1 Consider a graphical method of estimating error
a) Consider an arbitaray function q(x)
b) Plot q(x) vs x
c) On the graph label
(1) qbest = q(xbest)
(2) qhi = q(xbest + δx)
(3) qlow = q(xbest- δx)
d) Making a linear approximation
x
qqxslopeqq
x
qqxslopeqq
bestbestlow
bestbesthi
e) Therefore
xx
Note the absolute value
How do we determine the error of a derived quantity Z(XYhellip) from errors in XYhellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 40
Lecture 3 Slide 40
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Error Propagation
General formula for uncertainty of a function of one variable
xx
[Taylor Eq 323]
Can you now derive for specific rules of error propagation
1 Addition and Subtraction [Taylor p 49]
2 Multiplication and Division [Taylor p 51]
3 Multiplication by a constant (exact number) [Taylor p 54]
4 Exponentiation (powers) [Taylor p 56]
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 41
Lecture 3 Slide 41
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Multiple Variables
Uncertainty of a function of multiple variables [Taylor Sec 311]
1 It can easily (no really) be shown that (see Taylor Sec 311) for a
function of several variables
)(
z
z
qy
y
qx
x
qzyxq
[Taylor Eq 347]
2 More correctly it can be shown that (see Taylor Sec 311) for a
function of several variables
)(
z
z
qy
y
qx
x
qzyxq
[Taylor Eq 347]
where the equals sign represents an upper bound as discussed above
3 For a function of several independent and random variables
)(
222
z
z
qy
y
qx
x
qzyxq
[Taylor Eq 348]
Again the proof is left for Ch 5
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 42
Lecture 3 Slide 42
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Product of Two Variables
Consider the arbitrary derived quantity
q(xy) of two independent random variables x and y
Expand q(xy) in a Taylor series about the expected values of x and y
(ie at points near X and Y)
Error Propagation General Case
119902 119909 119910 = 119902 119883 119884 + 120597119902
120597119909
119883 119909 minus 119883 +
120597119902
120597119910
119884
(119910 minus 119884)
Fixed shifts peak of distribution
Fixed Distribution centered at X with width σX
120575119902 119909 119910 = 120590119902 = 119902 119883 119884 + 120597119902
120597119909
119883120590119909
2
+ 120597119902
120597119910
119884
120590119910
2
0
How do we determine the error of a derived quantity Z(XYhellip) from errors in XYhellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 43
Lecture 3 Slide 43
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
SDOM of Gaussian Distribution
Standard Deviation of the Mean
The SDOM decreases as the square root of the number of measurements
That is the relative width σẊ of the distribution gets narrower as more
measurements are made
Each measurement
has similar σXi= σẊ
and similar partial
derivatives
Thushellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 44
Lecture 3 Slide 44
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Two Key Theorems from Probability
Central Limit Theorem
For random independent measurements (each with a well-define
expectation value and well-defined variance) the arithmetic mean
(average) will be approximately normally distributed
Principle of Maximum Likelihood
Given the N observed measurements x1 x2hellipxN the best estimates
for Ẋ and σ are those values for which the observed x1 x2hellipxN are
most likely
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 45
Lecture 3 Slide 45
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Mean of Gaussian Distribution as ldquoBest Estimaterdquo
Principle of Maximum Likelihood
Best Estimate of X is from maximum probability or minimum summation
Consider a data set
Each randomly distributed with
x1 x2 x3 hellipxN
To find the most likely value of the mean (the best estimate of ẋ)
find X that yields the highest probability for the data set
The combined probability for the full data set is the product
Minimize
Sum
Solve for
derivative wrst
X set to 0
Best
estimate
of X
prop1
120590119890minus(1199091minus119883)2 2120590 times
1
120590119890minus(1199092minus119883)2 2120590 times hellip times
1
120590119890minus(119909119873minus119883)2 2120590 =
1
120590119873119890 minus(119909119894minus119883)2 2120590
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 46
Lecture 3 Slide 46
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Uncertainty of ldquoBest Estimatesrdquo of Gaussian Distribution Principle of Maximum Likelihood
Best Estimate of X is from maximum probability or minimum summation
Consider a data set x1 x2 x3 hellipxN
To find the most likely value of the standard deviation (the best estimate of
the width of the x distribution) find σx that yields the highest probability for
the data set
The combined probability for the full data set is the product
Minimize
Sum
Solve for
derivative
wrst X set to 0
Best
estimate
of X
Best Estimate of σ is from maximum probability or minimum summation
Minimize
Sum
Best
estimate
of σ
Solve for
derivative
wrst σ set to 0
See
Prob 526
prop1
120590119890minus(1199091minus119883)2 2120590 times
1
120590119890minus(1199092minus119883)2 2120590 times hellip times
1
120590119890minus(119909119873minus119883)2 2120590 =
1
120590119873119890 minus(119909119894minus119883)2 2120590
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 47
Lecture 3 Slide 47
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Combining Data Sets
Weighted Averages
References Taylor Ch 7
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 48
Lecture 3 Slide 48
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Weighted Averages
Question How can we properly combine two or more separate
independent measurements of the same randomly distributed
quantity to determine a best combined value with uncertainty
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 49
Lecture 3 Slide 49
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider two measurements of the same quantity described by a random Gaussian distribution
ltx1gt x1 and ltx2gt x2
The probability of measuring two such measurements is
119875119903119900119887119909 1199091 1199092 = 119875119903119900119887119909 1199091 119875119903119900119887119909 1199092
= 1
12059011205902119890minus12059422 119908ℎ119890119903119890 1205942 equiv
1199091 minus 119883
1205901 2 +
1199092 minus 119883
1205902 2
To find the best value for χ find the maximum Prob or minimum χ2
Weighted Averages
Assume negligible systematic errors
Note χ2 or Chi squared is the sum of the squares of the deviations from the mean divided by
the corresponding uncertainty
Such methods are called ldquoMethods of Least Squaresrdquo They follow directly from the Principle of
Maximum Likelihood
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 50
Lecture 3 Slide 50
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
The probability of measuring two such measurements is
119875119903119900119887119909 1199091 1199092 = 119875119903119900119887119909 1199091 119875119903119900119887119909 1199092
= 1
12059011205902119890minus12059422 119908ℎ119890119903119890 1205942 equiv
1199091 minus 119883
1205901 2 +
1199092 minus 119883
1205902 2
To find the best value for χ find the maximum Prob or minimum χ2
Weighted Averages
This leads to 119909119882_119886119907119892 = 11990811199091+11990821199092
1199081+1199082=
119908119894 119909119894
119908119894 119908ℎ119890119903119890 119908119894 = 1
120590119894 2
Best Estimate of χ is from maximum probibility or minimum summation
Minimize Sum Solve for best estimate of χ Solve for derivative wrst χ set to 0
Note If w1=w2 we recover the standard result Xwavg= (12) (x1+x2)
Finally the width of a weighted average distribution is
120590119908119894119890119892 ℎ119905119890119889 119886119907119892 = 1
119908119894119894
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 51
Lecture 3 Slide 51
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Weighted Averages on Steroids
A very powerful method for combining data from different sources
with different methods and uncertainties (or indeed data of
related measured and calculated quantities) is Kalman filtering
The Kalman filter also known as
linear quadratic estimation (LQE)
is an algorithm that uses a series
of measurements observed over
time containing noise (random
variations) and other
inaccuracies and produces
estimates of unknown variables
that tend to be more precise than
those based on a single
measurement alone
The Kalman filter keeps track of the estimated state of the system and the variance or uncertainty of the
estimate The estimate is updated using a state transition model and measurements xk|k-1 denotes the
estimate of the systems state at time step k before the kth measurement yk has been taken into account Pk|k-1
is the corresponding uncertainty --Wikipedia 2013
Ludger Scherliess of USU Physics is a world expert at using Kalman filtering for the assimilation
of satellite and ground-based data and the USU GAMES model to predict space weather
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 52
Lecture 3 Slide 52
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Rejecting Data
Chauvenetrsquos Criterion
References Taylor Ch 6
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 53
Lecture 3 Slide 53
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly
ldquounreasonablerdquo data point from a set of randomly distributed
measurements
bull Never
bull Whenever it makes things look better
bull Chauvenetrsquos criterion provides a (quantitative) compromise
Rejecting Data
What is a good criteria for
rejecting data
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 54
Lecture 3 Slide 54
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly
ldquounreasonablerdquo data point from a set of randomly distributed
measurements
bull Never
Rejecting Data
Zallenrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 55
Lecture 3 Slide 55
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly
ldquounreasonablerdquo data point from a set of randomly distributed
measurements
bull Whenever it makes things look better
Rejecting Data
Disneyrsquos Criterion
Disneyrsquos First Law
Wishing will make it so
Disneyrsquos Second Law
Dreams are more colorful than reality
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 56
Lecture 3 Slide 56
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Data may be rejected if the expected number of measurements at
least as deviant as the suspect measurement is less than 50
Consider a set of N measurements of a single quantity
x1x2 helliphellipxN
Calculate ltxgt and σx and then determine the fractional deviations from the mean of all
the points
For the suspect point(s) xsuspect find the probability of such a point occurring in N
measurements
n = (expected number as deviant as xsuspect)
= N Prob(outside xsuspectσx)
Rejecting Data
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 57
Lecture 3 Slide 57
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Function of Gaussian Distribution
Area under curve
(probability that
ndashtσltxlt+tσ) is given
in Table at right
Probable Error (probability that
ndash067σltxlt+067σ) is 50
Error Function (probability that ndashtσltxlt+tσ )
Error Function Tabulated values-see App A
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 58
Lecture 3 Slide 58
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 59
Lecture 3 Slide 59
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashExample 1
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 60
Lecture 3 Slide 60
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Details (1)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 61
Lecture 3 Slide 61
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashDetails (2)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 62
Lecture 3 Slide 62
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos
Criterionmdash
Details (3)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 63
Lecture 3 Slide 63
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos
Criterionmdash
Example 2
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 64
Lecture 3 Slide 64
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Summary of Probability Theory
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 65
Lecture 3 Slide 65
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of
Probability
Theory-I
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 66
Lecture 3 Slide 66
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of
Probability
Theory-II
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 67
Lecture 3 Slide 67
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-III
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 68
Lecture 3 Slide 68
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-IV
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 25
Lecture 3 Slide 25
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Using Mathcad to
define other common
distribution functions
Consider the limiting
distribution function
as N ȸ and ∆k0
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 26
Lecture 3 Slide 26
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider the Gaussian distribution function
Use the normalization condition to
evaluate the normalization constant
(see Taylor p 132)
The mean Ẋ is the first moment
of the Gaussian distribution
function (see Taylor p 134)
The standard deviation σx is the standard
deviation of the mean of the Gaussian
distribution function (see Taylor p 143)
Gaussian Distribution Moments
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 27
Lecture 3 Slide 27
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
When is mean x not Xbest
Answer When the
distribution is not
symmetric about X
Example Cauchy
Distribution
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 28
Lecture 3 Slide 28
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
When is mean x not Xbest
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 29
Lecture 3 Slide 29
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
When is mean x not Xbest
Answer When the
distribution is has
more than one peak
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 30
Lecture 3 Slide 30
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
The Gaussian Distribution
Function and Its Relation to
Errors
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 31
Lecture 3 Slide 31
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
A Review of Probabilities in Combination
Probability of a data set of N like measurements (x1x2hellipxN)
P (x1x2hellipxN) = P(x)1P(x2)hellipP(xN)
1 head AND 1 Four
P(H4) = P(H) P(4)
1 head OR 1 Four
P(H4) = P(H) + P(4) - P(H and 4) (true for a ldquonon-mutually exclusiverdquo events)
NOT 1 Six
P(NOT 6) = 1 - P(6)
1 Six OR 1 Four
P(64) = P(6) + P(4) (true for a ldquomutually exclusiverdquo single role)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 32
Lecture 3 Slide 32
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
The Gaussian Distribution Function and Its Relation to Errors
We will use the Gaussian distribution as applied to random
variables to develop the mathematical basis for bull Mean
bull Standard deviation
bull Standard deviation of the mean (SDOM)
bull Moments and expectation values
bull Error propagation formulas
bull Addition of errors in quadrature (for independent and random
measurements)
bull Numerical values for confidence limits (t-test)
bull Principle of maximal likelihood
bull Central limit theorem
bull Weighted distributions and Chi squared
bull Schwartz inequality (ie the uncertainty principle) (next lecture)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 33
Lecture 3 Slide 33
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider the Gaussian distribution function
Use the normalization condition to
evaluate the normalization constant
(see Taylor p 132)
The mean Ẋ is the first moment
of the Gaussian distribution
function (see Taylor p 134)
The standard deviation σx is the standard
deviation of the mean of the Gaussian
distribution function (see Taylor p 143)
Gaussian Distribution Moments
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 34
Lecture 3 Slide 34
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Standard Deviation of Gaussian Distribution
Area under curve
(probability that
ndashσltxlt+σ) is 68
5 or ~2σ ldquoSignificantrdquo
1 or ~3σ ldquoHighly Significantrdquo
1σ ldquoWithin errorsrdquo
5 ppm or ~5σ ldquoValid for HEPrdquo
See Sec 106 Testing of
Hypotheses
More complete Table in App A and B
Ah thatrsquos
highly
significant
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 35
Lecture 3 Slide 35
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Function of Gaussian Distribution
Area under curve
(probability that
ndashtσltxlt+tσ) is given
in Table at right
Probable Error (probability that
ndash067σltxlt+067σ) is 50
Error Function (probability that ndashtσltxlt+tσ )
Error Function Tabulated values-see App A
Complementary Error Function
(probability that ndashxlt-tσ AND xgt+tσ )
Prob(x outside tσ) = 1 - Prob(x within tσ)
More complete Table in App A and B
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 36
Lecture 3 Slide 36
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Useful Points on Gaussian Distribution
Full Width at Half Maximum FWHM
(See Prob 512)
Points of Inflection Occur at plusmnσ
(See Prob 513)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 37
Lecture 3 Slide 37
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Analysis and Gaussian Distribution
Adding a Constant Multiplying by a Constant
XX+A XBX and σB σ
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 38
Lecture 3 Slide 38
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Sum of Two Variables
Consider the derived quantity
Z=X + Y
(with X=0 and Y=0)
Error Propagation Addition
Error in Z
Multiple two probabilities 119875119903119900119887 119909 119910 = 119875119903119900119887 119909 ∙ 119875119903119900119887 119909 prop 119890119909119901 minus
1
2
1199092
1205901199092+
1199102
1205901199102
prop 119890119909119901 minus1
2
(119909+119910)2
1205901199092+120590119910
2 minus1199112
prop 119890119909119901 minus1
2
(119909+119910)2
1205901199092+120590119910
2 119890119909119901 minus
1199112
2
ICBST (Eq 553)
X+YZ and σx2 + σy
2 σz
2
(addition in quadrature for random independent variables)
Integrates to radic2π
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 39
Lecture 3 Slide 39
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Error Propagation
General formula for error propagation see [Taylor Secs 35 and 39]
Uncertainty as a function of one variable [Taylor Sec 35]
1 Consider a graphical method of estimating error
a) Consider an arbitaray function q(x)
b) Plot q(x) vs x
c) On the graph label
(1) qbest = q(xbest)
(2) qhi = q(xbest + δx)
(3) qlow = q(xbest- δx)
d) Making a linear approximation
x
qqxslopeqq
x
qqxslopeqq
bestbestlow
bestbesthi
e) Therefore
xx
Note the absolute value
How do we determine the error of a derived quantity Z(XYhellip) from errors in XYhellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 40
Lecture 3 Slide 40
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Error Propagation
General formula for uncertainty of a function of one variable
xx
[Taylor Eq 323]
Can you now derive for specific rules of error propagation
1 Addition and Subtraction [Taylor p 49]
2 Multiplication and Division [Taylor p 51]
3 Multiplication by a constant (exact number) [Taylor p 54]
4 Exponentiation (powers) [Taylor p 56]
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 41
Lecture 3 Slide 41
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Multiple Variables
Uncertainty of a function of multiple variables [Taylor Sec 311]
1 It can easily (no really) be shown that (see Taylor Sec 311) for a
function of several variables
)(
z
z
qy
y
qx
x
qzyxq
[Taylor Eq 347]
2 More correctly it can be shown that (see Taylor Sec 311) for a
function of several variables
)(
z
z
qy
y
qx
x
qzyxq
[Taylor Eq 347]
where the equals sign represents an upper bound as discussed above
3 For a function of several independent and random variables
)(
222
z
z
qy
y
qx
x
qzyxq
[Taylor Eq 348]
Again the proof is left for Ch 5
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 42
Lecture 3 Slide 42
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Product of Two Variables
Consider the arbitrary derived quantity
q(xy) of two independent random variables x and y
Expand q(xy) in a Taylor series about the expected values of x and y
(ie at points near X and Y)
Error Propagation General Case
119902 119909 119910 = 119902 119883 119884 + 120597119902
120597119909
119883 119909 minus 119883 +
120597119902
120597119910
119884
(119910 minus 119884)
Fixed shifts peak of distribution
Fixed Distribution centered at X with width σX
120575119902 119909 119910 = 120590119902 = 119902 119883 119884 + 120597119902
120597119909
119883120590119909
2
+ 120597119902
120597119910
119884
120590119910
2
0
How do we determine the error of a derived quantity Z(XYhellip) from errors in XYhellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 43
Lecture 3 Slide 43
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
SDOM of Gaussian Distribution
Standard Deviation of the Mean
The SDOM decreases as the square root of the number of measurements
That is the relative width σẊ of the distribution gets narrower as more
measurements are made
Each measurement
has similar σXi= σẊ
and similar partial
derivatives
Thushellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 44
Lecture 3 Slide 44
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Two Key Theorems from Probability
Central Limit Theorem
For random independent measurements (each with a well-define
expectation value and well-defined variance) the arithmetic mean
(average) will be approximately normally distributed
Principle of Maximum Likelihood
Given the N observed measurements x1 x2hellipxN the best estimates
for Ẋ and σ are those values for which the observed x1 x2hellipxN are
most likely
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 45
Lecture 3 Slide 45
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Mean of Gaussian Distribution as ldquoBest Estimaterdquo
Principle of Maximum Likelihood
Best Estimate of X is from maximum probability or minimum summation
Consider a data set
Each randomly distributed with
x1 x2 x3 hellipxN
To find the most likely value of the mean (the best estimate of ẋ)
find X that yields the highest probability for the data set
The combined probability for the full data set is the product
Minimize
Sum
Solve for
derivative wrst
X set to 0
Best
estimate
of X
prop1
120590119890minus(1199091minus119883)2 2120590 times
1
120590119890minus(1199092minus119883)2 2120590 times hellip times
1
120590119890minus(119909119873minus119883)2 2120590 =
1
120590119873119890 minus(119909119894minus119883)2 2120590
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 46
Lecture 3 Slide 46
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Uncertainty of ldquoBest Estimatesrdquo of Gaussian Distribution Principle of Maximum Likelihood
Best Estimate of X is from maximum probability or minimum summation
Consider a data set x1 x2 x3 hellipxN
To find the most likely value of the standard deviation (the best estimate of
the width of the x distribution) find σx that yields the highest probability for
the data set
The combined probability for the full data set is the product
Minimize
Sum
Solve for
derivative
wrst X set to 0
Best
estimate
of X
Best Estimate of σ is from maximum probability or minimum summation
Minimize
Sum
Best
estimate
of σ
Solve for
derivative
wrst σ set to 0
See
Prob 526
prop1
120590119890minus(1199091minus119883)2 2120590 times
1
120590119890minus(1199092minus119883)2 2120590 times hellip times
1
120590119890minus(119909119873minus119883)2 2120590 =
1
120590119873119890 minus(119909119894minus119883)2 2120590
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 47
Lecture 3 Slide 47
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Combining Data Sets
Weighted Averages
References Taylor Ch 7
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 48
Lecture 3 Slide 48
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Weighted Averages
Question How can we properly combine two or more separate
independent measurements of the same randomly distributed
quantity to determine a best combined value with uncertainty
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 49
Lecture 3 Slide 49
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider two measurements of the same quantity described by a random Gaussian distribution
ltx1gt x1 and ltx2gt x2
The probability of measuring two such measurements is
119875119903119900119887119909 1199091 1199092 = 119875119903119900119887119909 1199091 119875119903119900119887119909 1199092
= 1
12059011205902119890minus12059422 119908ℎ119890119903119890 1205942 equiv
1199091 minus 119883
1205901 2 +
1199092 minus 119883
1205902 2
To find the best value for χ find the maximum Prob or minimum χ2
Weighted Averages
Assume negligible systematic errors
Note χ2 or Chi squared is the sum of the squares of the deviations from the mean divided by
the corresponding uncertainty
Such methods are called ldquoMethods of Least Squaresrdquo They follow directly from the Principle of
Maximum Likelihood
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 50
Lecture 3 Slide 50
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
The probability of measuring two such measurements is
119875119903119900119887119909 1199091 1199092 = 119875119903119900119887119909 1199091 119875119903119900119887119909 1199092
= 1
12059011205902119890minus12059422 119908ℎ119890119903119890 1205942 equiv
1199091 minus 119883
1205901 2 +
1199092 minus 119883
1205902 2
To find the best value for χ find the maximum Prob or minimum χ2
Weighted Averages
This leads to 119909119882_119886119907119892 = 11990811199091+11990821199092
1199081+1199082=
119908119894 119909119894
119908119894 119908ℎ119890119903119890 119908119894 = 1
120590119894 2
Best Estimate of χ is from maximum probibility or minimum summation
Minimize Sum Solve for best estimate of χ Solve for derivative wrst χ set to 0
Note If w1=w2 we recover the standard result Xwavg= (12) (x1+x2)
Finally the width of a weighted average distribution is
120590119908119894119890119892 ℎ119905119890119889 119886119907119892 = 1
119908119894119894
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 51
Lecture 3 Slide 51
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Weighted Averages on Steroids
A very powerful method for combining data from different sources
with different methods and uncertainties (or indeed data of
related measured and calculated quantities) is Kalman filtering
The Kalman filter also known as
linear quadratic estimation (LQE)
is an algorithm that uses a series
of measurements observed over
time containing noise (random
variations) and other
inaccuracies and produces
estimates of unknown variables
that tend to be more precise than
those based on a single
measurement alone
The Kalman filter keeps track of the estimated state of the system and the variance or uncertainty of the
estimate The estimate is updated using a state transition model and measurements xk|k-1 denotes the
estimate of the systems state at time step k before the kth measurement yk has been taken into account Pk|k-1
is the corresponding uncertainty --Wikipedia 2013
Ludger Scherliess of USU Physics is a world expert at using Kalman filtering for the assimilation
of satellite and ground-based data and the USU GAMES model to predict space weather
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 52
Lecture 3 Slide 52
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Rejecting Data
Chauvenetrsquos Criterion
References Taylor Ch 6
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 53
Lecture 3 Slide 53
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly
ldquounreasonablerdquo data point from a set of randomly distributed
measurements
bull Never
bull Whenever it makes things look better
bull Chauvenetrsquos criterion provides a (quantitative) compromise
Rejecting Data
What is a good criteria for
rejecting data
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 54
Lecture 3 Slide 54
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly
ldquounreasonablerdquo data point from a set of randomly distributed
measurements
bull Never
Rejecting Data
Zallenrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 55
Lecture 3 Slide 55
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly
ldquounreasonablerdquo data point from a set of randomly distributed
measurements
bull Whenever it makes things look better
Rejecting Data
Disneyrsquos Criterion
Disneyrsquos First Law
Wishing will make it so
Disneyrsquos Second Law
Dreams are more colorful than reality
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 56
Lecture 3 Slide 56
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Data may be rejected if the expected number of measurements at
least as deviant as the suspect measurement is less than 50
Consider a set of N measurements of a single quantity
x1x2 helliphellipxN
Calculate ltxgt and σx and then determine the fractional deviations from the mean of all
the points
For the suspect point(s) xsuspect find the probability of such a point occurring in N
measurements
n = (expected number as deviant as xsuspect)
= N Prob(outside xsuspectσx)
Rejecting Data
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 57
Lecture 3 Slide 57
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Function of Gaussian Distribution
Area under curve
(probability that
ndashtσltxlt+tσ) is given
in Table at right
Probable Error (probability that
ndash067σltxlt+067σ) is 50
Error Function (probability that ndashtσltxlt+tσ )
Error Function Tabulated values-see App A
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 58
Lecture 3 Slide 58
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 59
Lecture 3 Slide 59
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashExample 1
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 60
Lecture 3 Slide 60
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Details (1)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 61
Lecture 3 Slide 61
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashDetails (2)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 62
Lecture 3 Slide 62
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos
Criterionmdash
Details (3)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 63
Lecture 3 Slide 63
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos
Criterionmdash
Example 2
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 64
Lecture 3 Slide 64
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Summary of Probability Theory
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 65
Lecture 3 Slide 65
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of
Probability
Theory-I
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 66
Lecture 3 Slide 66
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of
Probability
Theory-II
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 67
Lecture 3 Slide 67
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-III
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 68
Lecture 3 Slide 68
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-IV
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 26
Lecture 3 Slide 26
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider the Gaussian distribution function
Use the normalization condition to
evaluate the normalization constant
(see Taylor p 132)
The mean Ẋ is the first moment
of the Gaussian distribution
function (see Taylor p 134)
The standard deviation σx is the standard
deviation of the mean of the Gaussian
distribution function (see Taylor p 143)
Gaussian Distribution Moments
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 27
Lecture 3 Slide 27
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
When is mean x not Xbest
Answer When the
distribution is not
symmetric about X
Example Cauchy
Distribution
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 28
Lecture 3 Slide 28
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
When is mean x not Xbest
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 29
Lecture 3 Slide 29
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
When is mean x not Xbest
Answer When the
distribution is has
more than one peak
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 30
Lecture 3 Slide 30
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
The Gaussian Distribution
Function and Its Relation to
Errors
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 31
Lecture 3 Slide 31
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
A Review of Probabilities in Combination
Probability of a data set of N like measurements (x1x2hellipxN)
P (x1x2hellipxN) = P(x)1P(x2)hellipP(xN)
1 head AND 1 Four
P(H4) = P(H) P(4)
1 head OR 1 Four
P(H4) = P(H) + P(4) - P(H and 4) (true for a ldquonon-mutually exclusiverdquo events)
NOT 1 Six
P(NOT 6) = 1 - P(6)
1 Six OR 1 Four
P(64) = P(6) + P(4) (true for a ldquomutually exclusiverdquo single role)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 32
Lecture 3 Slide 32
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
The Gaussian Distribution Function and Its Relation to Errors
We will use the Gaussian distribution as applied to random
variables to develop the mathematical basis for bull Mean
bull Standard deviation
bull Standard deviation of the mean (SDOM)
bull Moments and expectation values
bull Error propagation formulas
bull Addition of errors in quadrature (for independent and random
measurements)
bull Numerical values for confidence limits (t-test)
bull Principle of maximal likelihood
bull Central limit theorem
bull Weighted distributions and Chi squared
bull Schwartz inequality (ie the uncertainty principle) (next lecture)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 33
Lecture 3 Slide 33
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider the Gaussian distribution function
Use the normalization condition to
evaluate the normalization constant
(see Taylor p 132)
The mean Ẋ is the first moment
of the Gaussian distribution
function (see Taylor p 134)
The standard deviation σx is the standard
deviation of the mean of the Gaussian
distribution function (see Taylor p 143)
Gaussian Distribution Moments
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 34
Lecture 3 Slide 34
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Standard Deviation of Gaussian Distribution
Area under curve
(probability that
ndashσltxlt+σ) is 68
5 or ~2σ ldquoSignificantrdquo
1 or ~3σ ldquoHighly Significantrdquo
1σ ldquoWithin errorsrdquo
5 ppm or ~5σ ldquoValid for HEPrdquo
See Sec 106 Testing of
Hypotheses
More complete Table in App A and B
Ah thatrsquos
highly
significant
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 35
Lecture 3 Slide 35
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Function of Gaussian Distribution
Area under curve
(probability that
ndashtσltxlt+tσ) is given
in Table at right
Probable Error (probability that
ndash067σltxlt+067σ) is 50
Error Function (probability that ndashtσltxlt+tσ )
Error Function Tabulated values-see App A
Complementary Error Function
(probability that ndashxlt-tσ AND xgt+tσ )
Prob(x outside tσ) = 1 - Prob(x within tσ)
More complete Table in App A and B
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 36
Lecture 3 Slide 36
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Useful Points on Gaussian Distribution
Full Width at Half Maximum FWHM
(See Prob 512)
Points of Inflection Occur at plusmnσ
(See Prob 513)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 37
Lecture 3 Slide 37
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Analysis and Gaussian Distribution
Adding a Constant Multiplying by a Constant
XX+A XBX and σB σ
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 38
Lecture 3 Slide 38
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Sum of Two Variables
Consider the derived quantity
Z=X + Y
(with X=0 and Y=0)
Error Propagation Addition
Error in Z
Multiple two probabilities 119875119903119900119887 119909 119910 = 119875119903119900119887 119909 ∙ 119875119903119900119887 119909 prop 119890119909119901 minus
1
2
1199092
1205901199092+
1199102
1205901199102
prop 119890119909119901 minus1
2
(119909+119910)2
1205901199092+120590119910
2 minus1199112
prop 119890119909119901 minus1
2
(119909+119910)2
1205901199092+120590119910
2 119890119909119901 minus
1199112
2
ICBST (Eq 553)
X+YZ and σx2 + σy
2 σz
2
(addition in quadrature for random independent variables)
Integrates to radic2π
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 39
Lecture 3 Slide 39
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Error Propagation
General formula for error propagation see [Taylor Secs 35 and 39]
Uncertainty as a function of one variable [Taylor Sec 35]
1 Consider a graphical method of estimating error
a) Consider an arbitaray function q(x)
b) Plot q(x) vs x
c) On the graph label
(1) qbest = q(xbest)
(2) qhi = q(xbest + δx)
(3) qlow = q(xbest- δx)
d) Making a linear approximation
x
qqxslopeqq
x
qqxslopeqq
bestbestlow
bestbesthi
e) Therefore
xx
Note the absolute value
How do we determine the error of a derived quantity Z(XYhellip) from errors in XYhellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 40
Lecture 3 Slide 40
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Error Propagation
General formula for uncertainty of a function of one variable
xx
[Taylor Eq 323]
Can you now derive for specific rules of error propagation
1 Addition and Subtraction [Taylor p 49]
2 Multiplication and Division [Taylor p 51]
3 Multiplication by a constant (exact number) [Taylor p 54]
4 Exponentiation (powers) [Taylor p 56]
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 41
Lecture 3 Slide 41
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Multiple Variables
Uncertainty of a function of multiple variables [Taylor Sec 311]
1 It can easily (no really) be shown that (see Taylor Sec 311) for a
function of several variables
)(
z
z
qy
y
qx
x
qzyxq
[Taylor Eq 347]
2 More correctly it can be shown that (see Taylor Sec 311) for a
function of several variables
)(
z
z
qy
y
qx
x
qzyxq
[Taylor Eq 347]
where the equals sign represents an upper bound as discussed above
3 For a function of several independent and random variables
)(
222
z
z
qy
y
qx
x
qzyxq
[Taylor Eq 348]
Again the proof is left for Ch 5
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 42
Lecture 3 Slide 42
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Product of Two Variables
Consider the arbitrary derived quantity
q(xy) of two independent random variables x and y
Expand q(xy) in a Taylor series about the expected values of x and y
(ie at points near X and Y)
Error Propagation General Case
119902 119909 119910 = 119902 119883 119884 + 120597119902
120597119909
119883 119909 minus 119883 +
120597119902
120597119910
119884
(119910 minus 119884)
Fixed shifts peak of distribution
Fixed Distribution centered at X with width σX
120575119902 119909 119910 = 120590119902 = 119902 119883 119884 + 120597119902
120597119909
119883120590119909
2
+ 120597119902
120597119910
119884
120590119910
2
0
How do we determine the error of a derived quantity Z(XYhellip) from errors in XYhellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 43
Lecture 3 Slide 43
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
SDOM of Gaussian Distribution
Standard Deviation of the Mean
The SDOM decreases as the square root of the number of measurements
That is the relative width σẊ of the distribution gets narrower as more
measurements are made
Each measurement
has similar σXi= σẊ
and similar partial
derivatives
Thushellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 44
Lecture 3 Slide 44
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Two Key Theorems from Probability
Central Limit Theorem
For random independent measurements (each with a well-define
expectation value and well-defined variance) the arithmetic mean
(average) will be approximately normally distributed
Principle of Maximum Likelihood
Given the N observed measurements x1 x2hellipxN the best estimates
for Ẋ and σ are those values for which the observed x1 x2hellipxN are
most likely
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 45
Lecture 3 Slide 45
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Mean of Gaussian Distribution as ldquoBest Estimaterdquo
Principle of Maximum Likelihood
Best Estimate of X is from maximum probability or minimum summation
Consider a data set
Each randomly distributed with
x1 x2 x3 hellipxN
To find the most likely value of the mean (the best estimate of ẋ)
find X that yields the highest probability for the data set
The combined probability for the full data set is the product
Minimize
Sum
Solve for
derivative wrst
X set to 0
Best
estimate
of X
prop1
120590119890minus(1199091minus119883)2 2120590 times
1
120590119890minus(1199092minus119883)2 2120590 times hellip times
1
120590119890minus(119909119873minus119883)2 2120590 =
1
120590119873119890 minus(119909119894minus119883)2 2120590
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 46
Lecture 3 Slide 46
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Uncertainty of ldquoBest Estimatesrdquo of Gaussian Distribution Principle of Maximum Likelihood
Best Estimate of X is from maximum probability or minimum summation
Consider a data set x1 x2 x3 hellipxN
To find the most likely value of the standard deviation (the best estimate of
the width of the x distribution) find σx that yields the highest probability for
the data set
The combined probability for the full data set is the product
Minimize
Sum
Solve for
derivative
wrst X set to 0
Best
estimate
of X
Best Estimate of σ is from maximum probability or minimum summation
Minimize
Sum
Best
estimate
of σ
Solve for
derivative
wrst σ set to 0
See
Prob 526
prop1
120590119890minus(1199091minus119883)2 2120590 times
1
120590119890minus(1199092minus119883)2 2120590 times hellip times
1
120590119890minus(119909119873minus119883)2 2120590 =
1
120590119873119890 minus(119909119894minus119883)2 2120590
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 47
Lecture 3 Slide 47
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Combining Data Sets
Weighted Averages
References Taylor Ch 7
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 48
Lecture 3 Slide 48
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Weighted Averages
Question How can we properly combine two or more separate
independent measurements of the same randomly distributed
quantity to determine a best combined value with uncertainty
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 49
Lecture 3 Slide 49
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider two measurements of the same quantity described by a random Gaussian distribution
ltx1gt x1 and ltx2gt x2
The probability of measuring two such measurements is
119875119903119900119887119909 1199091 1199092 = 119875119903119900119887119909 1199091 119875119903119900119887119909 1199092
= 1
12059011205902119890minus12059422 119908ℎ119890119903119890 1205942 equiv
1199091 minus 119883
1205901 2 +
1199092 minus 119883
1205902 2
To find the best value for χ find the maximum Prob or minimum χ2
Weighted Averages
Assume negligible systematic errors
Note χ2 or Chi squared is the sum of the squares of the deviations from the mean divided by
the corresponding uncertainty
Such methods are called ldquoMethods of Least Squaresrdquo They follow directly from the Principle of
Maximum Likelihood
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 50
Lecture 3 Slide 50
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
The probability of measuring two such measurements is
119875119903119900119887119909 1199091 1199092 = 119875119903119900119887119909 1199091 119875119903119900119887119909 1199092
= 1
12059011205902119890minus12059422 119908ℎ119890119903119890 1205942 equiv
1199091 minus 119883
1205901 2 +
1199092 minus 119883
1205902 2
To find the best value for χ find the maximum Prob or minimum χ2
Weighted Averages
This leads to 119909119882_119886119907119892 = 11990811199091+11990821199092
1199081+1199082=
119908119894 119909119894
119908119894 119908ℎ119890119903119890 119908119894 = 1
120590119894 2
Best Estimate of χ is from maximum probibility or minimum summation
Minimize Sum Solve for best estimate of χ Solve for derivative wrst χ set to 0
Note If w1=w2 we recover the standard result Xwavg= (12) (x1+x2)
Finally the width of a weighted average distribution is
120590119908119894119890119892 ℎ119905119890119889 119886119907119892 = 1
119908119894119894
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 51
Lecture 3 Slide 51
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Weighted Averages on Steroids
A very powerful method for combining data from different sources
with different methods and uncertainties (or indeed data of
related measured and calculated quantities) is Kalman filtering
The Kalman filter also known as
linear quadratic estimation (LQE)
is an algorithm that uses a series
of measurements observed over
time containing noise (random
variations) and other
inaccuracies and produces
estimates of unknown variables
that tend to be more precise than
those based on a single
measurement alone
The Kalman filter keeps track of the estimated state of the system and the variance or uncertainty of the
estimate The estimate is updated using a state transition model and measurements xk|k-1 denotes the
estimate of the systems state at time step k before the kth measurement yk has been taken into account Pk|k-1
is the corresponding uncertainty --Wikipedia 2013
Ludger Scherliess of USU Physics is a world expert at using Kalman filtering for the assimilation
of satellite and ground-based data and the USU GAMES model to predict space weather
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 52
Lecture 3 Slide 52
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Rejecting Data
Chauvenetrsquos Criterion
References Taylor Ch 6
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 53
Lecture 3 Slide 53
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly
ldquounreasonablerdquo data point from a set of randomly distributed
measurements
bull Never
bull Whenever it makes things look better
bull Chauvenetrsquos criterion provides a (quantitative) compromise
Rejecting Data
What is a good criteria for
rejecting data
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 54
Lecture 3 Slide 54
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly
ldquounreasonablerdquo data point from a set of randomly distributed
measurements
bull Never
Rejecting Data
Zallenrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 55
Lecture 3 Slide 55
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly
ldquounreasonablerdquo data point from a set of randomly distributed
measurements
bull Whenever it makes things look better
Rejecting Data
Disneyrsquos Criterion
Disneyrsquos First Law
Wishing will make it so
Disneyrsquos Second Law
Dreams are more colorful than reality
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 56
Lecture 3 Slide 56
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Data may be rejected if the expected number of measurements at
least as deviant as the suspect measurement is less than 50
Consider a set of N measurements of a single quantity
x1x2 helliphellipxN
Calculate ltxgt and σx and then determine the fractional deviations from the mean of all
the points
For the suspect point(s) xsuspect find the probability of such a point occurring in N
measurements
n = (expected number as deviant as xsuspect)
= N Prob(outside xsuspectσx)
Rejecting Data
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 57
Lecture 3 Slide 57
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Function of Gaussian Distribution
Area under curve
(probability that
ndashtσltxlt+tσ) is given
in Table at right
Probable Error (probability that
ndash067σltxlt+067σ) is 50
Error Function (probability that ndashtσltxlt+tσ )
Error Function Tabulated values-see App A
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 58
Lecture 3 Slide 58
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 59
Lecture 3 Slide 59
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashExample 1
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 60
Lecture 3 Slide 60
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Details (1)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 61
Lecture 3 Slide 61
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashDetails (2)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 62
Lecture 3 Slide 62
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos
Criterionmdash
Details (3)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 63
Lecture 3 Slide 63
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos
Criterionmdash
Example 2
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 64
Lecture 3 Slide 64
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Summary of Probability Theory
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 65
Lecture 3 Slide 65
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of
Probability
Theory-I
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 66
Lecture 3 Slide 66
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of
Probability
Theory-II
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 67
Lecture 3 Slide 67
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-III
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 68
Lecture 3 Slide 68
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-IV
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 27
Lecture 3 Slide 27
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
When is mean x not Xbest
Answer When the
distribution is not
symmetric about X
Example Cauchy
Distribution
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 28
Lecture 3 Slide 28
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
When is mean x not Xbest
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 29
Lecture 3 Slide 29
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
When is mean x not Xbest
Answer When the
distribution is has
more than one peak
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 30
Lecture 3 Slide 30
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
The Gaussian Distribution
Function and Its Relation to
Errors
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 31
Lecture 3 Slide 31
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
A Review of Probabilities in Combination
Probability of a data set of N like measurements (x1x2hellipxN)
P (x1x2hellipxN) = P(x)1P(x2)hellipP(xN)
1 head AND 1 Four
P(H4) = P(H) P(4)
1 head OR 1 Four
P(H4) = P(H) + P(4) - P(H and 4) (true for a ldquonon-mutually exclusiverdquo events)
NOT 1 Six
P(NOT 6) = 1 - P(6)
1 Six OR 1 Four
P(64) = P(6) + P(4) (true for a ldquomutually exclusiverdquo single role)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 32
Lecture 3 Slide 32
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
The Gaussian Distribution Function and Its Relation to Errors
We will use the Gaussian distribution as applied to random
variables to develop the mathematical basis for bull Mean
bull Standard deviation
bull Standard deviation of the mean (SDOM)
bull Moments and expectation values
bull Error propagation formulas
bull Addition of errors in quadrature (for independent and random
measurements)
bull Numerical values for confidence limits (t-test)
bull Principle of maximal likelihood
bull Central limit theorem
bull Weighted distributions and Chi squared
bull Schwartz inequality (ie the uncertainty principle) (next lecture)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 33
Lecture 3 Slide 33
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider the Gaussian distribution function
Use the normalization condition to
evaluate the normalization constant
(see Taylor p 132)
The mean Ẋ is the first moment
of the Gaussian distribution
function (see Taylor p 134)
The standard deviation σx is the standard
deviation of the mean of the Gaussian
distribution function (see Taylor p 143)
Gaussian Distribution Moments
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 34
Lecture 3 Slide 34
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Standard Deviation of Gaussian Distribution
Area under curve
(probability that
ndashσltxlt+σ) is 68
5 or ~2σ ldquoSignificantrdquo
1 or ~3σ ldquoHighly Significantrdquo
1σ ldquoWithin errorsrdquo
5 ppm or ~5σ ldquoValid for HEPrdquo
See Sec 106 Testing of
Hypotheses
More complete Table in App A and B
Ah thatrsquos
highly
significant
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 35
Lecture 3 Slide 35
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Function of Gaussian Distribution
Area under curve
(probability that
ndashtσltxlt+tσ) is given
in Table at right
Probable Error (probability that
ndash067σltxlt+067σ) is 50
Error Function (probability that ndashtσltxlt+tσ )
Error Function Tabulated values-see App A
Complementary Error Function
(probability that ndashxlt-tσ AND xgt+tσ )
Prob(x outside tσ) = 1 - Prob(x within tσ)
More complete Table in App A and B
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 36
Lecture 3 Slide 36
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Useful Points on Gaussian Distribution
Full Width at Half Maximum FWHM
(See Prob 512)
Points of Inflection Occur at plusmnσ
(See Prob 513)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 37
Lecture 3 Slide 37
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Analysis and Gaussian Distribution
Adding a Constant Multiplying by a Constant
XX+A XBX and σB σ
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 38
Lecture 3 Slide 38
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Sum of Two Variables
Consider the derived quantity
Z=X + Y
(with X=0 and Y=0)
Error Propagation Addition
Error in Z
Multiple two probabilities 119875119903119900119887 119909 119910 = 119875119903119900119887 119909 ∙ 119875119903119900119887 119909 prop 119890119909119901 minus
1
2
1199092
1205901199092+
1199102
1205901199102
prop 119890119909119901 minus1
2
(119909+119910)2
1205901199092+120590119910
2 minus1199112
prop 119890119909119901 minus1
2
(119909+119910)2
1205901199092+120590119910
2 119890119909119901 minus
1199112
2
ICBST (Eq 553)
X+YZ and σx2 + σy
2 σz
2
(addition in quadrature for random independent variables)
Integrates to radic2π
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 39
Lecture 3 Slide 39
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Error Propagation
General formula for error propagation see [Taylor Secs 35 and 39]
Uncertainty as a function of one variable [Taylor Sec 35]
1 Consider a graphical method of estimating error
a) Consider an arbitaray function q(x)
b) Plot q(x) vs x
c) On the graph label
(1) qbest = q(xbest)
(2) qhi = q(xbest + δx)
(3) qlow = q(xbest- δx)
d) Making a linear approximation
x
qqxslopeqq
x
qqxslopeqq
bestbestlow
bestbesthi
e) Therefore
xx
Note the absolute value
How do we determine the error of a derived quantity Z(XYhellip) from errors in XYhellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 40
Lecture 3 Slide 40
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Error Propagation
General formula for uncertainty of a function of one variable
xx
[Taylor Eq 323]
Can you now derive for specific rules of error propagation
1 Addition and Subtraction [Taylor p 49]
2 Multiplication and Division [Taylor p 51]
3 Multiplication by a constant (exact number) [Taylor p 54]
4 Exponentiation (powers) [Taylor p 56]
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 41
Lecture 3 Slide 41
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Multiple Variables
Uncertainty of a function of multiple variables [Taylor Sec 311]
1 It can easily (no really) be shown that (see Taylor Sec 311) for a
function of several variables
)(
z
z
qy
y
qx
x
qzyxq
[Taylor Eq 347]
2 More correctly it can be shown that (see Taylor Sec 311) for a
function of several variables
)(
z
z
qy
y
qx
x
qzyxq
[Taylor Eq 347]
where the equals sign represents an upper bound as discussed above
3 For a function of several independent and random variables
)(
222
z
z
qy
y
qx
x
qzyxq
[Taylor Eq 348]
Again the proof is left for Ch 5
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 42
Lecture 3 Slide 42
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Product of Two Variables
Consider the arbitrary derived quantity
q(xy) of two independent random variables x and y
Expand q(xy) in a Taylor series about the expected values of x and y
(ie at points near X and Y)
Error Propagation General Case
119902 119909 119910 = 119902 119883 119884 + 120597119902
120597119909
119883 119909 minus 119883 +
120597119902
120597119910
119884
(119910 minus 119884)
Fixed shifts peak of distribution
Fixed Distribution centered at X with width σX
120575119902 119909 119910 = 120590119902 = 119902 119883 119884 + 120597119902
120597119909
119883120590119909
2
+ 120597119902
120597119910
119884
120590119910
2
0
How do we determine the error of a derived quantity Z(XYhellip) from errors in XYhellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 43
Lecture 3 Slide 43
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
SDOM of Gaussian Distribution
Standard Deviation of the Mean
The SDOM decreases as the square root of the number of measurements
That is the relative width σẊ of the distribution gets narrower as more
measurements are made
Each measurement
has similar σXi= σẊ
and similar partial
derivatives
Thushellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 44
Lecture 3 Slide 44
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Two Key Theorems from Probability
Central Limit Theorem
For random independent measurements (each with a well-define
expectation value and well-defined variance) the arithmetic mean
(average) will be approximately normally distributed
Principle of Maximum Likelihood
Given the N observed measurements x1 x2hellipxN the best estimates
for Ẋ and σ are those values for which the observed x1 x2hellipxN are
most likely
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 45
Lecture 3 Slide 45
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Mean of Gaussian Distribution as ldquoBest Estimaterdquo
Principle of Maximum Likelihood
Best Estimate of X is from maximum probability or minimum summation
Consider a data set
Each randomly distributed with
x1 x2 x3 hellipxN
To find the most likely value of the mean (the best estimate of ẋ)
find X that yields the highest probability for the data set
The combined probability for the full data set is the product
Minimize
Sum
Solve for
derivative wrst
X set to 0
Best
estimate
of X
prop1
120590119890minus(1199091minus119883)2 2120590 times
1
120590119890minus(1199092minus119883)2 2120590 times hellip times
1
120590119890minus(119909119873minus119883)2 2120590 =
1
120590119873119890 minus(119909119894minus119883)2 2120590
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 46
Lecture 3 Slide 46
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Uncertainty of ldquoBest Estimatesrdquo of Gaussian Distribution Principle of Maximum Likelihood
Best Estimate of X is from maximum probability or minimum summation
Consider a data set x1 x2 x3 hellipxN
To find the most likely value of the standard deviation (the best estimate of
the width of the x distribution) find σx that yields the highest probability for
the data set
The combined probability for the full data set is the product
Minimize
Sum
Solve for
derivative
wrst X set to 0
Best
estimate
of X
Best Estimate of σ is from maximum probability or minimum summation
Minimize
Sum
Best
estimate
of σ
Solve for
derivative
wrst σ set to 0
See
Prob 526
prop1
120590119890minus(1199091minus119883)2 2120590 times
1
120590119890minus(1199092minus119883)2 2120590 times hellip times
1
120590119890minus(119909119873minus119883)2 2120590 =
1
120590119873119890 minus(119909119894minus119883)2 2120590
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 47
Lecture 3 Slide 47
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Combining Data Sets
Weighted Averages
References Taylor Ch 7
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 48
Lecture 3 Slide 48
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Weighted Averages
Question How can we properly combine two or more separate
independent measurements of the same randomly distributed
quantity to determine a best combined value with uncertainty
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 49
Lecture 3 Slide 49
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider two measurements of the same quantity described by a random Gaussian distribution
ltx1gt x1 and ltx2gt x2
The probability of measuring two such measurements is
119875119903119900119887119909 1199091 1199092 = 119875119903119900119887119909 1199091 119875119903119900119887119909 1199092
= 1
12059011205902119890minus12059422 119908ℎ119890119903119890 1205942 equiv
1199091 minus 119883
1205901 2 +
1199092 minus 119883
1205902 2
To find the best value for χ find the maximum Prob or minimum χ2
Weighted Averages
Assume negligible systematic errors
Note χ2 or Chi squared is the sum of the squares of the deviations from the mean divided by
the corresponding uncertainty
Such methods are called ldquoMethods of Least Squaresrdquo They follow directly from the Principle of
Maximum Likelihood
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 50
Lecture 3 Slide 50
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
The probability of measuring two such measurements is
119875119903119900119887119909 1199091 1199092 = 119875119903119900119887119909 1199091 119875119903119900119887119909 1199092
= 1
12059011205902119890minus12059422 119908ℎ119890119903119890 1205942 equiv
1199091 minus 119883
1205901 2 +
1199092 minus 119883
1205902 2
To find the best value for χ find the maximum Prob or minimum χ2
Weighted Averages
This leads to 119909119882_119886119907119892 = 11990811199091+11990821199092
1199081+1199082=
119908119894 119909119894
119908119894 119908ℎ119890119903119890 119908119894 = 1
120590119894 2
Best Estimate of χ is from maximum probibility or minimum summation
Minimize Sum Solve for best estimate of χ Solve for derivative wrst χ set to 0
Note If w1=w2 we recover the standard result Xwavg= (12) (x1+x2)
Finally the width of a weighted average distribution is
120590119908119894119890119892 ℎ119905119890119889 119886119907119892 = 1
119908119894119894
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 51
Lecture 3 Slide 51
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Weighted Averages on Steroids
A very powerful method for combining data from different sources
with different methods and uncertainties (or indeed data of
related measured and calculated quantities) is Kalman filtering
The Kalman filter also known as
linear quadratic estimation (LQE)
is an algorithm that uses a series
of measurements observed over
time containing noise (random
variations) and other
inaccuracies and produces
estimates of unknown variables
that tend to be more precise than
those based on a single
measurement alone
The Kalman filter keeps track of the estimated state of the system and the variance or uncertainty of the
estimate The estimate is updated using a state transition model and measurements xk|k-1 denotes the
estimate of the systems state at time step k before the kth measurement yk has been taken into account Pk|k-1
is the corresponding uncertainty --Wikipedia 2013
Ludger Scherliess of USU Physics is a world expert at using Kalman filtering for the assimilation
of satellite and ground-based data and the USU GAMES model to predict space weather
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 52
Lecture 3 Slide 52
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Rejecting Data
Chauvenetrsquos Criterion
References Taylor Ch 6
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 53
Lecture 3 Slide 53
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly
ldquounreasonablerdquo data point from a set of randomly distributed
measurements
bull Never
bull Whenever it makes things look better
bull Chauvenetrsquos criterion provides a (quantitative) compromise
Rejecting Data
What is a good criteria for
rejecting data
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 54
Lecture 3 Slide 54
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly
ldquounreasonablerdquo data point from a set of randomly distributed
measurements
bull Never
Rejecting Data
Zallenrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 55
Lecture 3 Slide 55
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly
ldquounreasonablerdquo data point from a set of randomly distributed
measurements
bull Whenever it makes things look better
Rejecting Data
Disneyrsquos Criterion
Disneyrsquos First Law
Wishing will make it so
Disneyrsquos Second Law
Dreams are more colorful than reality
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 56
Lecture 3 Slide 56
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Data may be rejected if the expected number of measurements at
least as deviant as the suspect measurement is less than 50
Consider a set of N measurements of a single quantity
x1x2 helliphellipxN
Calculate ltxgt and σx and then determine the fractional deviations from the mean of all
the points
For the suspect point(s) xsuspect find the probability of such a point occurring in N
measurements
n = (expected number as deviant as xsuspect)
= N Prob(outside xsuspectσx)
Rejecting Data
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 57
Lecture 3 Slide 57
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Function of Gaussian Distribution
Area under curve
(probability that
ndashtσltxlt+tσ) is given
in Table at right
Probable Error (probability that
ndash067σltxlt+067σ) is 50
Error Function (probability that ndashtσltxlt+tσ )
Error Function Tabulated values-see App A
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 58
Lecture 3 Slide 58
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 59
Lecture 3 Slide 59
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashExample 1
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 60
Lecture 3 Slide 60
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Details (1)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 61
Lecture 3 Slide 61
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashDetails (2)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 62
Lecture 3 Slide 62
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos
Criterionmdash
Details (3)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 63
Lecture 3 Slide 63
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos
Criterionmdash
Example 2
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 64
Lecture 3 Slide 64
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Summary of Probability Theory
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 65
Lecture 3 Slide 65
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of
Probability
Theory-I
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 66
Lecture 3 Slide 66
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of
Probability
Theory-II
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 67
Lecture 3 Slide 67
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-III
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 68
Lecture 3 Slide 68
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-IV
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 28
Lecture 3 Slide 28
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
When is mean x not Xbest
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 29
Lecture 3 Slide 29
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
When is mean x not Xbest
Answer When the
distribution is has
more than one peak
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 30
Lecture 3 Slide 30
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
The Gaussian Distribution
Function and Its Relation to
Errors
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 31
Lecture 3 Slide 31
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
A Review of Probabilities in Combination
Probability of a data set of N like measurements (x1x2hellipxN)
P (x1x2hellipxN) = P(x)1P(x2)hellipP(xN)
1 head AND 1 Four
P(H4) = P(H) P(4)
1 head OR 1 Four
P(H4) = P(H) + P(4) - P(H and 4) (true for a ldquonon-mutually exclusiverdquo events)
NOT 1 Six
P(NOT 6) = 1 - P(6)
1 Six OR 1 Four
P(64) = P(6) + P(4) (true for a ldquomutually exclusiverdquo single role)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 32
Lecture 3 Slide 32
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
The Gaussian Distribution Function and Its Relation to Errors
We will use the Gaussian distribution as applied to random
variables to develop the mathematical basis for bull Mean
bull Standard deviation
bull Standard deviation of the mean (SDOM)
bull Moments and expectation values
bull Error propagation formulas
bull Addition of errors in quadrature (for independent and random
measurements)
bull Numerical values for confidence limits (t-test)
bull Principle of maximal likelihood
bull Central limit theorem
bull Weighted distributions and Chi squared
bull Schwartz inequality (ie the uncertainty principle) (next lecture)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 33
Lecture 3 Slide 33
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider the Gaussian distribution function
Use the normalization condition to
evaluate the normalization constant
(see Taylor p 132)
The mean Ẋ is the first moment
of the Gaussian distribution
function (see Taylor p 134)
The standard deviation σx is the standard
deviation of the mean of the Gaussian
distribution function (see Taylor p 143)
Gaussian Distribution Moments
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 34
Lecture 3 Slide 34
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Standard Deviation of Gaussian Distribution
Area under curve
(probability that
ndashσltxlt+σ) is 68
5 or ~2σ ldquoSignificantrdquo
1 or ~3σ ldquoHighly Significantrdquo
1σ ldquoWithin errorsrdquo
5 ppm or ~5σ ldquoValid for HEPrdquo
See Sec 106 Testing of
Hypotheses
More complete Table in App A and B
Ah thatrsquos
highly
significant
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 35
Lecture 3 Slide 35
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Function of Gaussian Distribution
Area under curve
(probability that
ndashtσltxlt+tσ) is given
in Table at right
Probable Error (probability that
ndash067σltxlt+067σ) is 50
Error Function (probability that ndashtσltxlt+tσ )
Error Function Tabulated values-see App A
Complementary Error Function
(probability that ndashxlt-tσ AND xgt+tσ )
Prob(x outside tσ) = 1 - Prob(x within tσ)
More complete Table in App A and B
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 36
Lecture 3 Slide 36
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Useful Points on Gaussian Distribution
Full Width at Half Maximum FWHM
(See Prob 512)
Points of Inflection Occur at plusmnσ
(See Prob 513)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 37
Lecture 3 Slide 37
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Analysis and Gaussian Distribution
Adding a Constant Multiplying by a Constant
XX+A XBX and σB σ
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 38
Lecture 3 Slide 38
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Sum of Two Variables
Consider the derived quantity
Z=X + Y
(with X=0 and Y=0)
Error Propagation Addition
Error in Z
Multiple two probabilities 119875119903119900119887 119909 119910 = 119875119903119900119887 119909 ∙ 119875119903119900119887 119909 prop 119890119909119901 minus
1
2
1199092
1205901199092+
1199102
1205901199102
prop 119890119909119901 minus1
2
(119909+119910)2
1205901199092+120590119910
2 minus1199112
prop 119890119909119901 minus1
2
(119909+119910)2
1205901199092+120590119910
2 119890119909119901 minus
1199112
2
ICBST (Eq 553)
X+YZ and σx2 + σy
2 σz
2
(addition in quadrature for random independent variables)
Integrates to radic2π
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 39
Lecture 3 Slide 39
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Error Propagation
General formula for error propagation see [Taylor Secs 35 and 39]
Uncertainty as a function of one variable [Taylor Sec 35]
1 Consider a graphical method of estimating error
a) Consider an arbitaray function q(x)
b) Plot q(x) vs x
c) On the graph label
(1) qbest = q(xbest)
(2) qhi = q(xbest + δx)
(3) qlow = q(xbest- δx)
d) Making a linear approximation
x
qqxslopeqq
x
qqxslopeqq
bestbestlow
bestbesthi
e) Therefore
xx
Note the absolute value
How do we determine the error of a derived quantity Z(XYhellip) from errors in XYhellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 40
Lecture 3 Slide 40
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Error Propagation
General formula for uncertainty of a function of one variable
xx
[Taylor Eq 323]
Can you now derive for specific rules of error propagation
1 Addition and Subtraction [Taylor p 49]
2 Multiplication and Division [Taylor p 51]
3 Multiplication by a constant (exact number) [Taylor p 54]
4 Exponentiation (powers) [Taylor p 56]
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 41
Lecture 3 Slide 41
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Multiple Variables
Uncertainty of a function of multiple variables [Taylor Sec 311]
1 It can easily (no really) be shown that (see Taylor Sec 311) for a
function of several variables
)(
z
z
qy
y
qx
x
qzyxq
[Taylor Eq 347]
2 More correctly it can be shown that (see Taylor Sec 311) for a
function of several variables
)(
z
z
qy
y
qx
x
qzyxq
[Taylor Eq 347]
where the equals sign represents an upper bound as discussed above
3 For a function of several independent and random variables
)(
222
z
z
qy
y
qx
x
qzyxq
[Taylor Eq 348]
Again the proof is left for Ch 5
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 42
Lecture 3 Slide 42
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Product of Two Variables
Consider the arbitrary derived quantity
q(xy) of two independent random variables x and y
Expand q(xy) in a Taylor series about the expected values of x and y
(ie at points near X and Y)
Error Propagation General Case
119902 119909 119910 = 119902 119883 119884 + 120597119902
120597119909
119883 119909 minus 119883 +
120597119902
120597119910
119884
(119910 minus 119884)
Fixed shifts peak of distribution
Fixed Distribution centered at X with width σX
120575119902 119909 119910 = 120590119902 = 119902 119883 119884 + 120597119902
120597119909
119883120590119909
2
+ 120597119902
120597119910
119884
120590119910
2
0
How do we determine the error of a derived quantity Z(XYhellip) from errors in XYhellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 43
Lecture 3 Slide 43
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
SDOM of Gaussian Distribution
Standard Deviation of the Mean
The SDOM decreases as the square root of the number of measurements
That is the relative width σẊ of the distribution gets narrower as more
measurements are made
Each measurement
has similar σXi= σẊ
and similar partial
derivatives
Thushellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 44
Lecture 3 Slide 44
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Two Key Theorems from Probability
Central Limit Theorem
For random independent measurements (each with a well-define
expectation value and well-defined variance) the arithmetic mean
(average) will be approximately normally distributed
Principle of Maximum Likelihood
Given the N observed measurements x1 x2hellipxN the best estimates
for Ẋ and σ are those values for which the observed x1 x2hellipxN are
most likely
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 45
Lecture 3 Slide 45
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Mean of Gaussian Distribution as ldquoBest Estimaterdquo
Principle of Maximum Likelihood
Best Estimate of X is from maximum probability or minimum summation
Consider a data set
Each randomly distributed with
x1 x2 x3 hellipxN
To find the most likely value of the mean (the best estimate of ẋ)
find X that yields the highest probability for the data set
The combined probability for the full data set is the product
Minimize
Sum
Solve for
derivative wrst
X set to 0
Best
estimate
of X
prop1
120590119890minus(1199091minus119883)2 2120590 times
1
120590119890minus(1199092minus119883)2 2120590 times hellip times
1
120590119890minus(119909119873minus119883)2 2120590 =
1
120590119873119890 minus(119909119894minus119883)2 2120590
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 46
Lecture 3 Slide 46
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Uncertainty of ldquoBest Estimatesrdquo of Gaussian Distribution Principle of Maximum Likelihood
Best Estimate of X is from maximum probability or minimum summation
Consider a data set x1 x2 x3 hellipxN
To find the most likely value of the standard deviation (the best estimate of
the width of the x distribution) find σx that yields the highest probability for
the data set
The combined probability for the full data set is the product
Minimize
Sum
Solve for
derivative
wrst X set to 0
Best
estimate
of X
Best Estimate of σ is from maximum probability or minimum summation
Minimize
Sum
Best
estimate
of σ
Solve for
derivative
wrst σ set to 0
See
Prob 526
prop1
120590119890minus(1199091minus119883)2 2120590 times
1
120590119890minus(1199092minus119883)2 2120590 times hellip times
1
120590119890minus(119909119873minus119883)2 2120590 =
1
120590119873119890 minus(119909119894minus119883)2 2120590
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 47
Lecture 3 Slide 47
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Combining Data Sets
Weighted Averages
References Taylor Ch 7
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 48
Lecture 3 Slide 48
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Weighted Averages
Question How can we properly combine two or more separate
independent measurements of the same randomly distributed
quantity to determine a best combined value with uncertainty
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 49
Lecture 3 Slide 49
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider two measurements of the same quantity described by a random Gaussian distribution
ltx1gt x1 and ltx2gt x2
The probability of measuring two such measurements is
119875119903119900119887119909 1199091 1199092 = 119875119903119900119887119909 1199091 119875119903119900119887119909 1199092
= 1
12059011205902119890minus12059422 119908ℎ119890119903119890 1205942 equiv
1199091 minus 119883
1205901 2 +
1199092 minus 119883
1205902 2
To find the best value for χ find the maximum Prob or minimum χ2
Weighted Averages
Assume negligible systematic errors
Note χ2 or Chi squared is the sum of the squares of the deviations from the mean divided by
the corresponding uncertainty
Such methods are called ldquoMethods of Least Squaresrdquo They follow directly from the Principle of
Maximum Likelihood
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 50
Lecture 3 Slide 50
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
The probability of measuring two such measurements is
119875119903119900119887119909 1199091 1199092 = 119875119903119900119887119909 1199091 119875119903119900119887119909 1199092
= 1
12059011205902119890minus12059422 119908ℎ119890119903119890 1205942 equiv
1199091 minus 119883
1205901 2 +
1199092 minus 119883
1205902 2
To find the best value for χ find the maximum Prob or minimum χ2
Weighted Averages
This leads to 119909119882_119886119907119892 = 11990811199091+11990821199092
1199081+1199082=
119908119894 119909119894
119908119894 119908ℎ119890119903119890 119908119894 = 1
120590119894 2
Best Estimate of χ is from maximum probibility or minimum summation
Minimize Sum Solve for best estimate of χ Solve for derivative wrst χ set to 0
Note If w1=w2 we recover the standard result Xwavg= (12) (x1+x2)
Finally the width of a weighted average distribution is
120590119908119894119890119892 ℎ119905119890119889 119886119907119892 = 1
119908119894119894
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 51
Lecture 3 Slide 51
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Weighted Averages on Steroids
A very powerful method for combining data from different sources
with different methods and uncertainties (or indeed data of
related measured and calculated quantities) is Kalman filtering
The Kalman filter also known as
linear quadratic estimation (LQE)
is an algorithm that uses a series
of measurements observed over
time containing noise (random
variations) and other
inaccuracies and produces
estimates of unknown variables
that tend to be more precise than
those based on a single
measurement alone
The Kalman filter keeps track of the estimated state of the system and the variance or uncertainty of the
estimate The estimate is updated using a state transition model and measurements xk|k-1 denotes the
estimate of the systems state at time step k before the kth measurement yk has been taken into account Pk|k-1
is the corresponding uncertainty --Wikipedia 2013
Ludger Scherliess of USU Physics is a world expert at using Kalman filtering for the assimilation
of satellite and ground-based data and the USU GAMES model to predict space weather
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 52
Lecture 3 Slide 52
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Rejecting Data
Chauvenetrsquos Criterion
References Taylor Ch 6
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 53
Lecture 3 Slide 53
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly
ldquounreasonablerdquo data point from a set of randomly distributed
measurements
bull Never
bull Whenever it makes things look better
bull Chauvenetrsquos criterion provides a (quantitative) compromise
Rejecting Data
What is a good criteria for
rejecting data
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 54
Lecture 3 Slide 54
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly
ldquounreasonablerdquo data point from a set of randomly distributed
measurements
bull Never
Rejecting Data
Zallenrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 55
Lecture 3 Slide 55
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly
ldquounreasonablerdquo data point from a set of randomly distributed
measurements
bull Whenever it makes things look better
Rejecting Data
Disneyrsquos Criterion
Disneyrsquos First Law
Wishing will make it so
Disneyrsquos Second Law
Dreams are more colorful than reality
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 56
Lecture 3 Slide 56
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Data may be rejected if the expected number of measurements at
least as deviant as the suspect measurement is less than 50
Consider a set of N measurements of a single quantity
x1x2 helliphellipxN
Calculate ltxgt and σx and then determine the fractional deviations from the mean of all
the points
For the suspect point(s) xsuspect find the probability of such a point occurring in N
measurements
n = (expected number as deviant as xsuspect)
= N Prob(outside xsuspectσx)
Rejecting Data
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 57
Lecture 3 Slide 57
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Function of Gaussian Distribution
Area under curve
(probability that
ndashtσltxlt+tσ) is given
in Table at right
Probable Error (probability that
ndash067σltxlt+067σ) is 50
Error Function (probability that ndashtσltxlt+tσ )
Error Function Tabulated values-see App A
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 58
Lecture 3 Slide 58
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 59
Lecture 3 Slide 59
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashExample 1
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 60
Lecture 3 Slide 60
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Details (1)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 61
Lecture 3 Slide 61
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashDetails (2)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 62
Lecture 3 Slide 62
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos
Criterionmdash
Details (3)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 63
Lecture 3 Slide 63
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos
Criterionmdash
Example 2
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 64
Lecture 3 Slide 64
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Summary of Probability Theory
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 65
Lecture 3 Slide 65
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of
Probability
Theory-I
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 66
Lecture 3 Slide 66
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of
Probability
Theory-II
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 67
Lecture 3 Slide 67
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-III
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 68
Lecture 3 Slide 68
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-IV
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 29
Lecture 3 Slide 29
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
When is mean x not Xbest
Answer When the
distribution is has
more than one peak
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 30
Lecture 3 Slide 30
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
The Gaussian Distribution
Function and Its Relation to
Errors
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 31
Lecture 3 Slide 31
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
A Review of Probabilities in Combination
Probability of a data set of N like measurements (x1x2hellipxN)
P (x1x2hellipxN) = P(x)1P(x2)hellipP(xN)
1 head AND 1 Four
P(H4) = P(H) P(4)
1 head OR 1 Four
P(H4) = P(H) + P(4) - P(H and 4) (true for a ldquonon-mutually exclusiverdquo events)
NOT 1 Six
P(NOT 6) = 1 - P(6)
1 Six OR 1 Four
P(64) = P(6) + P(4) (true for a ldquomutually exclusiverdquo single role)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 32
Lecture 3 Slide 32
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
The Gaussian Distribution Function and Its Relation to Errors
We will use the Gaussian distribution as applied to random
variables to develop the mathematical basis for bull Mean
bull Standard deviation
bull Standard deviation of the mean (SDOM)
bull Moments and expectation values
bull Error propagation formulas
bull Addition of errors in quadrature (for independent and random
measurements)
bull Numerical values for confidence limits (t-test)
bull Principle of maximal likelihood
bull Central limit theorem
bull Weighted distributions and Chi squared
bull Schwartz inequality (ie the uncertainty principle) (next lecture)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 33
Lecture 3 Slide 33
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider the Gaussian distribution function
Use the normalization condition to
evaluate the normalization constant
(see Taylor p 132)
The mean Ẋ is the first moment
of the Gaussian distribution
function (see Taylor p 134)
The standard deviation σx is the standard
deviation of the mean of the Gaussian
distribution function (see Taylor p 143)
Gaussian Distribution Moments
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 34
Lecture 3 Slide 34
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Standard Deviation of Gaussian Distribution
Area under curve
(probability that
ndashσltxlt+σ) is 68
5 or ~2σ ldquoSignificantrdquo
1 or ~3σ ldquoHighly Significantrdquo
1σ ldquoWithin errorsrdquo
5 ppm or ~5σ ldquoValid for HEPrdquo
See Sec 106 Testing of
Hypotheses
More complete Table in App A and B
Ah thatrsquos
highly
significant
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 35
Lecture 3 Slide 35
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Function of Gaussian Distribution
Area under curve
(probability that
ndashtσltxlt+tσ) is given
in Table at right
Probable Error (probability that
ndash067σltxlt+067σ) is 50
Error Function (probability that ndashtσltxlt+tσ )
Error Function Tabulated values-see App A
Complementary Error Function
(probability that ndashxlt-tσ AND xgt+tσ )
Prob(x outside tσ) = 1 - Prob(x within tσ)
More complete Table in App A and B
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 36
Lecture 3 Slide 36
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Useful Points on Gaussian Distribution
Full Width at Half Maximum FWHM
(See Prob 512)
Points of Inflection Occur at plusmnσ
(See Prob 513)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 37
Lecture 3 Slide 37
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Analysis and Gaussian Distribution
Adding a Constant Multiplying by a Constant
XX+A XBX and σB σ
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 38
Lecture 3 Slide 38
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Sum of Two Variables
Consider the derived quantity
Z=X + Y
(with X=0 and Y=0)
Error Propagation Addition
Error in Z
Multiple two probabilities 119875119903119900119887 119909 119910 = 119875119903119900119887 119909 ∙ 119875119903119900119887 119909 prop 119890119909119901 minus
1
2
1199092
1205901199092+
1199102
1205901199102
prop 119890119909119901 minus1
2
(119909+119910)2
1205901199092+120590119910
2 minus1199112
prop 119890119909119901 minus1
2
(119909+119910)2
1205901199092+120590119910
2 119890119909119901 minus
1199112
2
ICBST (Eq 553)
X+YZ and σx2 + σy
2 σz
2
(addition in quadrature for random independent variables)
Integrates to radic2π
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 39
Lecture 3 Slide 39
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Error Propagation
General formula for error propagation see [Taylor Secs 35 and 39]
Uncertainty as a function of one variable [Taylor Sec 35]
1 Consider a graphical method of estimating error
a) Consider an arbitaray function q(x)
b) Plot q(x) vs x
c) On the graph label
(1) qbest = q(xbest)
(2) qhi = q(xbest + δx)
(3) qlow = q(xbest- δx)
d) Making a linear approximation
x
qqxslopeqq
x
qqxslopeqq
bestbestlow
bestbesthi
e) Therefore
xx
Note the absolute value
How do we determine the error of a derived quantity Z(XYhellip) from errors in XYhellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 40
Lecture 3 Slide 40
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Error Propagation
General formula for uncertainty of a function of one variable
xx
[Taylor Eq 323]
Can you now derive for specific rules of error propagation
1 Addition and Subtraction [Taylor p 49]
2 Multiplication and Division [Taylor p 51]
3 Multiplication by a constant (exact number) [Taylor p 54]
4 Exponentiation (powers) [Taylor p 56]
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 41
Lecture 3 Slide 41
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Multiple Variables
Uncertainty of a function of multiple variables [Taylor Sec 311]
1 It can easily (no really) be shown that (see Taylor Sec 311) for a
function of several variables
)(
z
z
qy
y
qx
x
qzyxq
[Taylor Eq 347]
2 More correctly it can be shown that (see Taylor Sec 311) for a
function of several variables
)(
z
z
qy
y
qx
x
qzyxq
[Taylor Eq 347]
where the equals sign represents an upper bound as discussed above
3 For a function of several independent and random variables
)(
222
z
z
qy
y
qx
x
qzyxq
[Taylor Eq 348]
Again the proof is left for Ch 5
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 42
Lecture 3 Slide 42
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Product of Two Variables
Consider the arbitrary derived quantity
q(xy) of two independent random variables x and y
Expand q(xy) in a Taylor series about the expected values of x and y
(ie at points near X and Y)
Error Propagation General Case
119902 119909 119910 = 119902 119883 119884 + 120597119902
120597119909
119883 119909 minus 119883 +
120597119902
120597119910
119884
(119910 minus 119884)
Fixed shifts peak of distribution
Fixed Distribution centered at X with width σX
120575119902 119909 119910 = 120590119902 = 119902 119883 119884 + 120597119902
120597119909
119883120590119909
2
+ 120597119902
120597119910
119884
120590119910
2
0
How do we determine the error of a derived quantity Z(XYhellip) from errors in XYhellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 43
Lecture 3 Slide 43
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
SDOM of Gaussian Distribution
Standard Deviation of the Mean
The SDOM decreases as the square root of the number of measurements
That is the relative width σẊ of the distribution gets narrower as more
measurements are made
Each measurement
has similar σXi= σẊ
and similar partial
derivatives
Thushellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 44
Lecture 3 Slide 44
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Two Key Theorems from Probability
Central Limit Theorem
For random independent measurements (each with a well-define
expectation value and well-defined variance) the arithmetic mean
(average) will be approximately normally distributed
Principle of Maximum Likelihood
Given the N observed measurements x1 x2hellipxN the best estimates
for Ẋ and σ are those values for which the observed x1 x2hellipxN are
most likely
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 45
Lecture 3 Slide 45
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Mean of Gaussian Distribution as ldquoBest Estimaterdquo
Principle of Maximum Likelihood
Best Estimate of X is from maximum probability or minimum summation
Consider a data set
Each randomly distributed with
x1 x2 x3 hellipxN
To find the most likely value of the mean (the best estimate of ẋ)
find X that yields the highest probability for the data set
The combined probability for the full data set is the product
Minimize
Sum
Solve for
derivative wrst
X set to 0
Best
estimate
of X
prop1
120590119890minus(1199091minus119883)2 2120590 times
1
120590119890minus(1199092minus119883)2 2120590 times hellip times
1
120590119890minus(119909119873minus119883)2 2120590 =
1
120590119873119890 minus(119909119894minus119883)2 2120590
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 46
Lecture 3 Slide 46
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Uncertainty of ldquoBest Estimatesrdquo of Gaussian Distribution Principle of Maximum Likelihood
Best Estimate of X is from maximum probability or minimum summation
Consider a data set x1 x2 x3 hellipxN
To find the most likely value of the standard deviation (the best estimate of
the width of the x distribution) find σx that yields the highest probability for
the data set
The combined probability for the full data set is the product
Minimize
Sum
Solve for
derivative
wrst X set to 0
Best
estimate
of X
Best Estimate of σ is from maximum probability or minimum summation
Minimize
Sum
Best
estimate
of σ
Solve for
derivative
wrst σ set to 0
See
Prob 526
prop1
120590119890minus(1199091minus119883)2 2120590 times
1
120590119890minus(1199092minus119883)2 2120590 times hellip times
1
120590119890minus(119909119873minus119883)2 2120590 =
1
120590119873119890 minus(119909119894minus119883)2 2120590
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 47
Lecture 3 Slide 47
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Combining Data Sets
Weighted Averages
References Taylor Ch 7
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 48
Lecture 3 Slide 48
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Weighted Averages
Question How can we properly combine two or more separate
independent measurements of the same randomly distributed
quantity to determine a best combined value with uncertainty
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 49
Lecture 3 Slide 49
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider two measurements of the same quantity described by a random Gaussian distribution
ltx1gt x1 and ltx2gt x2
The probability of measuring two such measurements is
119875119903119900119887119909 1199091 1199092 = 119875119903119900119887119909 1199091 119875119903119900119887119909 1199092
= 1
12059011205902119890minus12059422 119908ℎ119890119903119890 1205942 equiv
1199091 minus 119883
1205901 2 +
1199092 minus 119883
1205902 2
To find the best value for χ find the maximum Prob or minimum χ2
Weighted Averages
Assume negligible systematic errors
Note χ2 or Chi squared is the sum of the squares of the deviations from the mean divided by
the corresponding uncertainty
Such methods are called ldquoMethods of Least Squaresrdquo They follow directly from the Principle of
Maximum Likelihood
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 50
Lecture 3 Slide 50
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
The probability of measuring two such measurements is
119875119903119900119887119909 1199091 1199092 = 119875119903119900119887119909 1199091 119875119903119900119887119909 1199092
= 1
12059011205902119890minus12059422 119908ℎ119890119903119890 1205942 equiv
1199091 minus 119883
1205901 2 +
1199092 minus 119883
1205902 2
To find the best value for χ find the maximum Prob or minimum χ2
Weighted Averages
This leads to 119909119882_119886119907119892 = 11990811199091+11990821199092
1199081+1199082=
119908119894 119909119894
119908119894 119908ℎ119890119903119890 119908119894 = 1
120590119894 2
Best Estimate of χ is from maximum probibility or minimum summation
Minimize Sum Solve for best estimate of χ Solve for derivative wrst χ set to 0
Note If w1=w2 we recover the standard result Xwavg= (12) (x1+x2)
Finally the width of a weighted average distribution is
120590119908119894119890119892 ℎ119905119890119889 119886119907119892 = 1
119908119894119894
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 51
Lecture 3 Slide 51
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Weighted Averages on Steroids
A very powerful method for combining data from different sources
with different methods and uncertainties (or indeed data of
related measured and calculated quantities) is Kalman filtering
The Kalman filter also known as
linear quadratic estimation (LQE)
is an algorithm that uses a series
of measurements observed over
time containing noise (random
variations) and other
inaccuracies and produces
estimates of unknown variables
that tend to be more precise than
those based on a single
measurement alone
The Kalman filter keeps track of the estimated state of the system and the variance or uncertainty of the
estimate The estimate is updated using a state transition model and measurements xk|k-1 denotes the
estimate of the systems state at time step k before the kth measurement yk has been taken into account Pk|k-1
is the corresponding uncertainty --Wikipedia 2013
Ludger Scherliess of USU Physics is a world expert at using Kalman filtering for the assimilation
of satellite and ground-based data and the USU GAMES model to predict space weather
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 52
Lecture 3 Slide 52
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Rejecting Data
Chauvenetrsquos Criterion
References Taylor Ch 6
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 53
Lecture 3 Slide 53
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly
ldquounreasonablerdquo data point from a set of randomly distributed
measurements
bull Never
bull Whenever it makes things look better
bull Chauvenetrsquos criterion provides a (quantitative) compromise
Rejecting Data
What is a good criteria for
rejecting data
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 54
Lecture 3 Slide 54
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly
ldquounreasonablerdquo data point from a set of randomly distributed
measurements
bull Never
Rejecting Data
Zallenrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 55
Lecture 3 Slide 55
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly
ldquounreasonablerdquo data point from a set of randomly distributed
measurements
bull Whenever it makes things look better
Rejecting Data
Disneyrsquos Criterion
Disneyrsquos First Law
Wishing will make it so
Disneyrsquos Second Law
Dreams are more colorful than reality
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 56
Lecture 3 Slide 56
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Data may be rejected if the expected number of measurements at
least as deviant as the suspect measurement is less than 50
Consider a set of N measurements of a single quantity
x1x2 helliphellipxN
Calculate ltxgt and σx and then determine the fractional deviations from the mean of all
the points
For the suspect point(s) xsuspect find the probability of such a point occurring in N
measurements
n = (expected number as deviant as xsuspect)
= N Prob(outside xsuspectσx)
Rejecting Data
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 57
Lecture 3 Slide 57
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Function of Gaussian Distribution
Area under curve
(probability that
ndashtσltxlt+tσ) is given
in Table at right
Probable Error (probability that
ndash067σltxlt+067σ) is 50
Error Function (probability that ndashtσltxlt+tσ )
Error Function Tabulated values-see App A
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 58
Lecture 3 Slide 58
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 59
Lecture 3 Slide 59
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashExample 1
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 60
Lecture 3 Slide 60
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Details (1)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 61
Lecture 3 Slide 61
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashDetails (2)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 62
Lecture 3 Slide 62
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos
Criterionmdash
Details (3)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 63
Lecture 3 Slide 63
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos
Criterionmdash
Example 2
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 64
Lecture 3 Slide 64
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Summary of Probability Theory
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 65
Lecture 3 Slide 65
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of
Probability
Theory-I
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 66
Lecture 3 Slide 66
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of
Probability
Theory-II
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 67
Lecture 3 Slide 67
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-III
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 68
Lecture 3 Slide 68
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-IV
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 30
Lecture 3 Slide 30
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
The Gaussian Distribution
Function and Its Relation to
Errors
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 31
Lecture 3 Slide 31
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
A Review of Probabilities in Combination
Probability of a data set of N like measurements (x1x2hellipxN)
P (x1x2hellipxN) = P(x)1P(x2)hellipP(xN)
1 head AND 1 Four
P(H4) = P(H) P(4)
1 head OR 1 Four
P(H4) = P(H) + P(4) - P(H and 4) (true for a ldquonon-mutually exclusiverdquo events)
NOT 1 Six
P(NOT 6) = 1 - P(6)
1 Six OR 1 Four
P(64) = P(6) + P(4) (true for a ldquomutually exclusiverdquo single role)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 32
Lecture 3 Slide 32
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
The Gaussian Distribution Function and Its Relation to Errors
We will use the Gaussian distribution as applied to random
variables to develop the mathematical basis for bull Mean
bull Standard deviation
bull Standard deviation of the mean (SDOM)
bull Moments and expectation values
bull Error propagation formulas
bull Addition of errors in quadrature (for independent and random
measurements)
bull Numerical values for confidence limits (t-test)
bull Principle of maximal likelihood
bull Central limit theorem
bull Weighted distributions and Chi squared
bull Schwartz inequality (ie the uncertainty principle) (next lecture)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 33
Lecture 3 Slide 33
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider the Gaussian distribution function
Use the normalization condition to
evaluate the normalization constant
(see Taylor p 132)
The mean Ẋ is the first moment
of the Gaussian distribution
function (see Taylor p 134)
The standard deviation σx is the standard
deviation of the mean of the Gaussian
distribution function (see Taylor p 143)
Gaussian Distribution Moments
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 34
Lecture 3 Slide 34
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Standard Deviation of Gaussian Distribution
Area under curve
(probability that
ndashσltxlt+σ) is 68
5 or ~2σ ldquoSignificantrdquo
1 or ~3σ ldquoHighly Significantrdquo
1σ ldquoWithin errorsrdquo
5 ppm or ~5σ ldquoValid for HEPrdquo
See Sec 106 Testing of
Hypotheses
More complete Table in App A and B
Ah thatrsquos
highly
significant
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 35
Lecture 3 Slide 35
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Function of Gaussian Distribution
Area under curve
(probability that
ndashtσltxlt+tσ) is given
in Table at right
Probable Error (probability that
ndash067σltxlt+067σ) is 50
Error Function (probability that ndashtσltxlt+tσ )
Error Function Tabulated values-see App A
Complementary Error Function
(probability that ndashxlt-tσ AND xgt+tσ )
Prob(x outside tσ) = 1 - Prob(x within tσ)
More complete Table in App A and B
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 36
Lecture 3 Slide 36
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Useful Points on Gaussian Distribution
Full Width at Half Maximum FWHM
(See Prob 512)
Points of Inflection Occur at plusmnσ
(See Prob 513)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 37
Lecture 3 Slide 37
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Analysis and Gaussian Distribution
Adding a Constant Multiplying by a Constant
XX+A XBX and σB σ
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 38
Lecture 3 Slide 38
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Sum of Two Variables
Consider the derived quantity
Z=X + Y
(with X=0 and Y=0)
Error Propagation Addition
Error in Z
Multiple two probabilities 119875119903119900119887 119909 119910 = 119875119903119900119887 119909 ∙ 119875119903119900119887 119909 prop 119890119909119901 minus
1
2
1199092
1205901199092+
1199102
1205901199102
prop 119890119909119901 minus1
2
(119909+119910)2
1205901199092+120590119910
2 minus1199112
prop 119890119909119901 minus1
2
(119909+119910)2
1205901199092+120590119910
2 119890119909119901 minus
1199112
2
ICBST (Eq 553)
X+YZ and σx2 + σy
2 σz
2
(addition in quadrature for random independent variables)
Integrates to radic2π
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 39
Lecture 3 Slide 39
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Error Propagation
General formula for error propagation see [Taylor Secs 35 and 39]
Uncertainty as a function of one variable [Taylor Sec 35]
1 Consider a graphical method of estimating error
a) Consider an arbitaray function q(x)
b) Plot q(x) vs x
c) On the graph label
(1) qbest = q(xbest)
(2) qhi = q(xbest + δx)
(3) qlow = q(xbest- δx)
d) Making a linear approximation
x
qqxslopeqq
x
qqxslopeqq
bestbestlow
bestbesthi
e) Therefore
xx
Note the absolute value
How do we determine the error of a derived quantity Z(XYhellip) from errors in XYhellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 40
Lecture 3 Slide 40
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Error Propagation
General formula for uncertainty of a function of one variable
xx
[Taylor Eq 323]
Can you now derive for specific rules of error propagation
1 Addition and Subtraction [Taylor p 49]
2 Multiplication and Division [Taylor p 51]
3 Multiplication by a constant (exact number) [Taylor p 54]
4 Exponentiation (powers) [Taylor p 56]
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 41
Lecture 3 Slide 41
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Multiple Variables
Uncertainty of a function of multiple variables [Taylor Sec 311]
1 It can easily (no really) be shown that (see Taylor Sec 311) for a
function of several variables
)(
z
z
qy
y
qx
x
qzyxq
[Taylor Eq 347]
2 More correctly it can be shown that (see Taylor Sec 311) for a
function of several variables
)(
z
z
qy
y
qx
x
qzyxq
[Taylor Eq 347]
where the equals sign represents an upper bound as discussed above
3 For a function of several independent and random variables
)(
222
z
z
qy
y
qx
x
qzyxq
[Taylor Eq 348]
Again the proof is left for Ch 5
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 42
Lecture 3 Slide 42
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Product of Two Variables
Consider the arbitrary derived quantity
q(xy) of two independent random variables x and y
Expand q(xy) in a Taylor series about the expected values of x and y
(ie at points near X and Y)
Error Propagation General Case
119902 119909 119910 = 119902 119883 119884 + 120597119902
120597119909
119883 119909 minus 119883 +
120597119902
120597119910
119884
(119910 minus 119884)
Fixed shifts peak of distribution
Fixed Distribution centered at X with width σX
120575119902 119909 119910 = 120590119902 = 119902 119883 119884 + 120597119902
120597119909
119883120590119909
2
+ 120597119902
120597119910
119884
120590119910
2
0
How do we determine the error of a derived quantity Z(XYhellip) from errors in XYhellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 43
Lecture 3 Slide 43
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
SDOM of Gaussian Distribution
Standard Deviation of the Mean
The SDOM decreases as the square root of the number of measurements
That is the relative width σẊ of the distribution gets narrower as more
measurements are made
Each measurement
has similar σXi= σẊ
and similar partial
derivatives
Thushellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 44
Lecture 3 Slide 44
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Two Key Theorems from Probability
Central Limit Theorem
For random independent measurements (each with a well-define
expectation value and well-defined variance) the arithmetic mean
(average) will be approximately normally distributed
Principle of Maximum Likelihood
Given the N observed measurements x1 x2hellipxN the best estimates
for Ẋ and σ are those values for which the observed x1 x2hellipxN are
most likely
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 45
Lecture 3 Slide 45
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Mean of Gaussian Distribution as ldquoBest Estimaterdquo
Principle of Maximum Likelihood
Best Estimate of X is from maximum probability or minimum summation
Consider a data set
Each randomly distributed with
x1 x2 x3 hellipxN
To find the most likely value of the mean (the best estimate of ẋ)
find X that yields the highest probability for the data set
The combined probability for the full data set is the product
Minimize
Sum
Solve for
derivative wrst
X set to 0
Best
estimate
of X
prop1
120590119890minus(1199091minus119883)2 2120590 times
1
120590119890minus(1199092minus119883)2 2120590 times hellip times
1
120590119890minus(119909119873minus119883)2 2120590 =
1
120590119873119890 minus(119909119894minus119883)2 2120590
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 46
Lecture 3 Slide 46
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Uncertainty of ldquoBest Estimatesrdquo of Gaussian Distribution Principle of Maximum Likelihood
Best Estimate of X is from maximum probability or minimum summation
Consider a data set x1 x2 x3 hellipxN
To find the most likely value of the standard deviation (the best estimate of
the width of the x distribution) find σx that yields the highest probability for
the data set
The combined probability for the full data set is the product
Minimize
Sum
Solve for
derivative
wrst X set to 0
Best
estimate
of X
Best Estimate of σ is from maximum probability or minimum summation
Minimize
Sum
Best
estimate
of σ
Solve for
derivative
wrst σ set to 0
See
Prob 526
prop1
120590119890minus(1199091minus119883)2 2120590 times
1
120590119890minus(1199092minus119883)2 2120590 times hellip times
1
120590119890minus(119909119873minus119883)2 2120590 =
1
120590119873119890 minus(119909119894minus119883)2 2120590
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 47
Lecture 3 Slide 47
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Combining Data Sets
Weighted Averages
References Taylor Ch 7
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 48
Lecture 3 Slide 48
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Weighted Averages
Question How can we properly combine two or more separate
independent measurements of the same randomly distributed
quantity to determine a best combined value with uncertainty
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 49
Lecture 3 Slide 49
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider two measurements of the same quantity described by a random Gaussian distribution
ltx1gt x1 and ltx2gt x2
The probability of measuring two such measurements is
119875119903119900119887119909 1199091 1199092 = 119875119903119900119887119909 1199091 119875119903119900119887119909 1199092
= 1
12059011205902119890minus12059422 119908ℎ119890119903119890 1205942 equiv
1199091 minus 119883
1205901 2 +
1199092 minus 119883
1205902 2
To find the best value for χ find the maximum Prob or minimum χ2
Weighted Averages
Assume negligible systematic errors
Note χ2 or Chi squared is the sum of the squares of the deviations from the mean divided by
the corresponding uncertainty
Such methods are called ldquoMethods of Least Squaresrdquo They follow directly from the Principle of
Maximum Likelihood
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 50
Lecture 3 Slide 50
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
The probability of measuring two such measurements is
119875119903119900119887119909 1199091 1199092 = 119875119903119900119887119909 1199091 119875119903119900119887119909 1199092
= 1
12059011205902119890minus12059422 119908ℎ119890119903119890 1205942 equiv
1199091 minus 119883
1205901 2 +
1199092 minus 119883
1205902 2
To find the best value for χ find the maximum Prob or minimum χ2
Weighted Averages
This leads to 119909119882_119886119907119892 = 11990811199091+11990821199092
1199081+1199082=
119908119894 119909119894
119908119894 119908ℎ119890119903119890 119908119894 = 1
120590119894 2
Best Estimate of χ is from maximum probibility or minimum summation
Minimize Sum Solve for best estimate of χ Solve for derivative wrst χ set to 0
Note If w1=w2 we recover the standard result Xwavg= (12) (x1+x2)
Finally the width of a weighted average distribution is
120590119908119894119890119892 ℎ119905119890119889 119886119907119892 = 1
119908119894119894
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 51
Lecture 3 Slide 51
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Weighted Averages on Steroids
A very powerful method for combining data from different sources
with different methods and uncertainties (or indeed data of
related measured and calculated quantities) is Kalman filtering
The Kalman filter also known as
linear quadratic estimation (LQE)
is an algorithm that uses a series
of measurements observed over
time containing noise (random
variations) and other
inaccuracies and produces
estimates of unknown variables
that tend to be more precise than
those based on a single
measurement alone
The Kalman filter keeps track of the estimated state of the system and the variance or uncertainty of the
estimate The estimate is updated using a state transition model and measurements xk|k-1 denotes the
estimate of the systems state at time step k before the kth measurement yk has been taken into account Pk|k-1
is the corresponding uncertainty --Wikipedia 2013
Ludger Scherliess of USU Physics is a world expert at using Kalman filtering for the assimilation
of satellite and ground-based data and the USU GAMES model to predict space weather
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 52
Lecture 3 Slide 52
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Rejecting Data
Chauvenetrsquos Criterion
References Taylor Ch 6
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 53
Lecture 3 Slide 53
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly
ldquounreasonablerdquo data point from a set of randomly distributed
measurements
bull Never
bull Whenever it makes things look better
bull Chauvenetrsquos criterion provides a (quantitative) compromise
Rejecting Data
What is a good criteria for
rejecting data
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 54
Lecture 3 Slide 54
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly
ldquounreasonablerdquo data point from a set of randomly distributed
measurements
bull Never
Rejecting Data
Zallenrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 55
Lecture 3 Slide 55
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly
ldquounreasonablerdquo data point from a set of randomly distributed
measurements
bull Whenever it makes things look better
Rejecting Data
Disneyrsquos Criterion
Disneyrsquos First Law
Wishing will make it so
Disneyrsquos Second Law
Dreams are more colorful than reality
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 56
Lecture 3 Slide 56
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Data may be rejected if the expected number of measurements at
least as deviant as the suspect measurement is less than 50
Consider a set of N measurements of a single quantity
x1x2 helliphellipxN
Calculate ltxgt and σx and then determine the fractional deviations from the mean of all
the points
For the suspect point(s) xsuspect find the probability of such a point occurring in N
measurements
n = (expected number as deviant as xsuspect)
= N Prob(outside xsuspectσx)
Rejecting Data
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 57
Lecture 3 Slide 57
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Function of Gaussian Distribution
Area under curve
(probability that
ndashtσltxlt+tσ) is given
in Table at right
Probable Error (probability that
ndash067σltxlt+067σ) is 50
Error Function (probability that ndashtσltxlt+tσ )
Error Function Tabulated values-see App A
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 58
Lecture 3 Slide 58
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 59
Lecture 3 Slide 59
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashExample 1
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 60
Lecture 3 Slide 60
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Details (1)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 61
Lecture 3 Slide 61
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashDetails (2)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 62
Lecture 3 Slide 62
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos
Criterionmdash
Details (3)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 63
Lecture 3 Slide 63
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos
Criterionmdash
Example 2
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 64
Lecture 3 Slide 64
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Summary of Probability Theory
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 65
Lecture 3 Slide 65
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of
Probability
Theory-I
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 66
Lecture 3 Slide 66
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of
Probability
Theory-II
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 67
Lecture 3 Slide 67
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-III
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 68
Lecture 3 Slide 68
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-IV
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 31
Lecture 3 Slide 31
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
A Review of Probabilities in Combination
Probability of a data set of N like measurements (x1x2hellipxN)
P (x1x2hellipxN) = P(x)1P(x2)hellipP(xN)
1 head AND 1 Four
P(H4) = P(H) P(4)
1 head OR 1 Four
P(H4) = P(H) + P(4) - P(H and 4) (true for a ldquonon-mutually exclusiverdquo events)
NOT 1 Six
P(NOT 6) = 1 - P(6)
1 Six OR 1 Four
P(64) = P(6) + P(4) (true for a ldquomutually exclusiverdquo single role)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 32
Lecture 3 Slide 32
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
The Gaussian Distribution Function and Its Relation to Errors
We will use the Gaussian distribution as applied to random
variables to develop the mathematical basis for bull Mean
bull Standard deviation
bull Standard deviation of the mean (SDOM)
bull Moments and expectation values
bull Error propagation formulas
bull Addition of errors in quadrature (for independent and random
measurements)
bull Numerical values for confidence limits (t-test)
bull Principle of maximal likelihood
bull Central limit theorem
bull Weighted distributions and Chi squared
bull Schwartz inequality (ie the uncertainty principle) (next lecture)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 33
Lecture 3 Slide 33
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider the Gaussian distribution function
Use the normalization condition to
evaluate the normalization constant
(see Taylor p 132)
The mean Ẋ is the first moment
of the Gaussian distribution
function (see Taylor p 134)
The standard deviation σx is the standard
deviation of the mean of the Gaussian
distribution function (see Taylor p 143)
Gaussian Distribution Moments
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 34
Lecture 3 Slide 34
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Standard Deviation of Gaussian Distribution
Area under curve
(probability that
ndashσltxlt+σ) is 68
5 or ~2σ ldquoSignificantrdquo
1 or ~3σ ldquoHighly Significantrdquo
1σ ldquoWithin errorsrdquo
5 ppm or ~5σ ldquoValid for HEPrdquo
See Sec 106 Testing of
Hypotheses
More complete Table in App A and B
Ah thatrsquos
highly
significant
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 35
Lecture 3 Slide 35
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Function of Gaussian Distribution
Area under curve
(probability that
ndashtσltxlt+tσ) is given
in Table at right
Probable Error (probability that
ndash067σltxlt+067σ) is 50
Error Function (probability that ndashtσltxlt+tσ )
Error Function Tabulated values-see App A
Complementary Error Function
(probability that ndashxlt-tσ AND xgt+tσ )
Prob(x outside tσ) = 1 - Prob(x within tσ)
More complete Table in App A and B
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 36
Lecture 3 Slide 36
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Useful Points on Gaussian Distribution
Full Width at Half Maximum FWHM
(See Prob 512)
Points of Inflection Occur at plusmnσ
(See Prob 513)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 37
Lecture 3 Slide 37
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Analysis and Gaussian Distribution
Adding a Constant Multiplying by a Constant
XX+A XBX and σB σ
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 38
Lecture 3 Slide 38
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Sum of Two Variables
Consider the derived quantity
Z=X + Y
(with X=0 and Y=0)
Error Propagation Addition
Error in Z
Multiple two probabilities 119875119903119900119887 119909 119910 = 119875119903119900119887 119909 ∙ 119875119903119900119887 119909 prop 119890119909119901 minus
1
2
1199092
1205901199092+
1199102
1205901199102
prop 119890119909119901 minus1
2
(119909+119910)2
1205901199092+120590119910
2 minus1199112
prop 119890119909119901 minus1
2
(119909+119910)2
1205901199092+120590119910
2 119890119909119901 minus
1199112
2
ICBST (Eq 553)
X+YZ and σx2 + σy
2 σz
2
(addition in quadrature for random independent variables)
Integrates to radic2π
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 39
Lecture 3 Slide 39
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Error Propagation
General formula for error propagation see [Taylor Secs 35 and 39]
Uncertainty as a function of one variable [Taylor Sec 35]
1 Consider a graphical method of estimating error
a) Consider an arbitaray function q(x)
b) Plot q(x) vs x
c) On the graph label
(1) qbest = q(xbest)
(2) qhi = q(xbest + δx)
(3) qlow = q(xbest- δx)
d) Making a linear approximation
x
qqxslopeqq
x
qqxslopeqq
bestbestlow
bestbesthi
e) Therefore
xx
Note the absolute value
How do we determine the error of a derived quantity Z(XYhellip) from errors in XYhellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 40
Lecture 3 Slide 40
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Error Propagation
General formula for uncertainty of a function of one variable
xx
[Taylor Eq 323]
Can you now derive for specific rules of error propagation
1 Addition and Subtraction [Taylor p 49]
2 Multiplication and Division [Taylor p 51]
3 Multiplication by a constant (exact number) [Taylor p 54]
4 Exponentiation (powers) [Taylor p 56]
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 41
Lecture 3 Slide 41
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Multiple Variables
Uncertainty of a function of multiple variables [Taylor Sec 311]
1 It can easily (no really) be shown that (see Taylor Sec 311) for a
function of several variables
)(
z
z
qy
y
qx
x
qzyxq
[Taylor Eq 347]
2 More correctly it can be shown that (see Taylor Sec 311) for a
function of several variables
)(
z
z
qy
y
qx
x
qzyxq
[Taylor Eq 347]
where the equals sign represents an upper bound as discussed above
3 For a function of several independent and random variables
)(
222
z
z
qy
y
qx
x
qzyxq
[Taylor Eq 348]
Again the proof is left for Ch 5
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 42
Lecture 3 Slide 42
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Product of Two Variables
Consider the arbitrary derived quantity
q(xy) of two independent random variables x and y
Expand q(xy) in a Taylor series about the expected values of x and y
(ie at points near X and Y)
Error Propagation General Case
119902 119909 119910 = 119902 119883 119884 + 120597119902
120597119909
119883 119909 minus 119883 +
120597119902
120597119910
119884
(119910 minus 119884)
Fixed shifts peak of distribution
Fixed Distribution centered at X with width σX
120575119902 119909 119910 = 120590119902 = 119902 119883 119884 + 120597119902
120597119909
119883120590119909
2
+ 120597119902
120597119910
119884
120590119910
2
0
How do we determine the error of a derived quantity Z(XYhellip) from errors in XYhellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 43
Lecture 3 Slide 43
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
SDOM of Gaussian Distribution
Standard Deviation of the Mean
The SDOM decreases as the square root of the number of measurements
That is the relative width σẊ of the distribution gets narrower as more
measurements are made
Each measurement
has similar σXi= σẊ
and similar partial
derivatives
Thushellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 44
Lecture 3 Slide 44
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Two Key Theorems from Probability
Central Limit Theorem
For random independent measurements (each with a well-define
expectation value and well-defined variance) the arithmetic mean
(average) will be approximately normally distributed
Principle of Maximum Likelihood
Given the N observed measurements x1 x2hellipxN the best estimates
for Ẋ and σ are those values for which the observed x1 x2hellipxN are
most likely
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 45
Lecture 3 Slide 45
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Mean of Gaussian Distribution as ldquoBest Estimaterdquo
Principle of Maximum Likelihood
Best Estimate of X is from maximum probability or minimum summation
Consider a data set
Each randomly distributed with
x1 x2 x3 hellipxN
To find the most likely value of the mean (the best estimate of ẋ)
find X that yields the highest probability for the data set
The combined probability for the full data set is the product
Minimize
Sum
Solve for
derivative wrst
X set to 0
Best
estimate
of X
prop1
120590119890minus(1199091minus119883)2 2120590 times
1
120590119890minus(1199092minus119883)2 2120590 times hellip times
1
120590119890minus(119909119873minus119883)2 2120590 =
1
120590119873119890 minus(119909119894minus119883)2 2120590
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 46
Lecture 3 Slide 46
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Uncertainty of ldquoBest Estimatesrdquo of Gaussian Distribution Principle of Maximum Likelihood
Best Estimate of X is from maximum probability or minimum summation
Consider a data set x1 x2 x3 hellipxN
To find the most likely value of the standard deviation (the best estimate of
the width of the x distribution) find σx that yields the highest probability for
the data set
The combined probability for the full data set is the product
Minimize
Sum
Solve for
derivative
wrst X set to 0
Best
estimate
of X
Best Estimate of σ is from maximum probability or minimum summation
Minimize
Sum
Best
estimate
of σ
Solve for
derivative
wrst σ set to 0
See
Prob 526
prop1
120590119890minus(1199091minus119883)2 2120590 times
1
120590119890minus(1199092minus119883)2 2120590 times hellip times
1
120590119890minus(119909119873minus119883)2 2120590 =
1
120590119873119890 minus(119909119894minus119883)2 2120590
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 47
Lecture 3 Slide 47
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Combining Data Sets
Weighted Averages
References Taylor Ch 7
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 48
Lecture 3 Slide 48
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Weighted Averages
Question How can we properly combine two or more separate
independent measurements of the same randomly distributed
quantity to determine a best combined value with uncertainty
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 49
Lecture 3 Slide 49
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider two measurements of the same quantity described by a random Gaussian distribution
ltx1gt x1 and ltx2gt x2
The probability of measuring two such measurements is
119875119903119900119887119909 1199091 1199092 = 119875119903119900119887119909 1199091 119875119903119900119887119909 1199092
= 1
12059011205902119890minus12059422 119908ℎ119890119903119890 1205942 equiv
1199091 minus 119883
1205901 2 +
1199092 minus 119883
1205902 2
To find the best value for χ find the maximum Prob or minimum χ2
Weighted Averages
Assume negligible systematic errors
Note χ2 or Chi squared is the sum of the squares of the deviations from the mean divided by
the corresponding uncertainty
Such methods are called ldquoMethods of Least Squaresrdquo They follow directly from the Principle of
Maximum Likelihood
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 50
Lecture 3 Slide 50
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
The probability of measuring two such measurements is
119875119903119900119887119909 1199091 1199092 = 119875119903119900119887119909 1199091 119875119903119900119887119909 1199092
= 1
12059011205902119890minus12059422 119908ℎ119890119903119890 1205942 equiv
1199091 minus 119883
1205901 2 +
1199092 minus 119883
1205902 2
To find the best value for χ find the maximum Prob or minimum χ2
Weighted Averages
This leads to 119909119882_119886119907119892 = 11990811199091+11990821199092
1199081+1199082=
119908119894 119909119894
119908119894 119908ℎ119890119903119890 119908119894 = 1
120590119894 2
Best Estimate of χ is from maximum probibility or minimum summation
Minimize Sum Solve for best estimate of χ Solve for derivative wrst χ set to 0
Note If w1=w2 we recover the standard result Xwavg= (12) (x1+x2)
Finally the width of a weighted average distribution is
120590119908119894119890119892 ℎ119905119890119889 119886119907119892 = 1
119908119894119894
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 51
Lecture 3 Slide 51
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Weighted Averages on Steroids
A very powerful method for combining data from different sources
with different methods and uncertainties (or indeed data of
related measured and calculated quantities) is Kalman filtering
The Kalman filter also known as
linear quadratic estimation (LQE)
is an algorithm that uses a series
of measurements observed over
time containing noise (random
variations) and other
inaccuracies and produces
estimates of unknown variables
that tend to be more precise than
those based on a single
measurement alone
The Kalman filter keeps track of the estimated state of the system and the variance or uncertainty of the
estimate The estimate is updated using a state transition model and measurements xk|k-1 denotes the
estimate of the systems state at time step k before the kth measurement yk has been taken into account Pk|k-1
is the corresponding uncertainty --Wikipedia 2013
Ludger Scherliess of USU Physics is a world expert at using Kalman filtering for the assimilation
of satellite and ground-based data and the USU GAMES model to predict space weather
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 52
Lecture 3 Slide 52
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Rejecting Data
Chauvenetrsquos Criterion
References Taylor Ch 6
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 53
Lecture 3 Slide 53
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly
ldquounreasonablerdquo data point from a set of randomly distributed
measurements
bull Never
bull Whenever it makes things look better
bull Chauvenetrsquos criterion provides a (quantitative) compromise
Rejecting Data
What is a good criteria for
rejecting data
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 54
Lecture 3 Slide 54
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly
ldquounreasonablerdquo data point from a set of randomly distributed
measurements
bull Never
Rejecting Data
Zallenrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 55
Lecture 3 Slide 55
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly
ldquounreasonablerdquo data point from a set of randomly distributed
measurements
bull Whenever it makes things look better
Rejecting Data
Disneyrsquos Criterion
Disneyrsquos First Law
Wishing will make it so
Disneyrsquos Second Law
Dreams are more colorful than reality
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 56
Lecture 3 Slide 56
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Data may be rejected if the expected number of measurements at
least as deviant as the suspect measurement is less than 50
Consider a set of N measurements of a single quantity
x1x2 helliphellipxN
Calculate ltxgt and σx and then determine the fractional deviations from the mean of all
the points
For the suspect point(s) xsuspect find the probability of such a point occurring in N
measurements
n = (expected number as deviant as xsuspect)
= N Prob(outside xsuspectσx)
Rejecting Data
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 57
Lecture 3 Slide 57
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Function of Gaussian Distribution
Area under curve
(probability that
ndashtσltxlt+tσ) is given
in Table at right
Probable Error (probability that
ndash067σltxlt+067σ) is 50
Error Function (probability that ndashtσltxlt+tσ )
Error Function Tabulated values-see App A
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 58
Lecture 3 Slide 58
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 59
Lecture 3 Slide 59
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashExample 1
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 60
Lecture 3 Slide 60
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Details (1)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 61
Lecture 3 Slide 61
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashDetails (2)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 62
Lecture 3 Slide 62
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos
Criterionmdash
Details (3)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 63
Lecture 3 Slide 63
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos
Criterionmdash
Example 2
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 64
Lecture 3 Slide 64
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Summary of Probability Theory
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 65
Lecture 3 Slide 65
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of
Probability
Theory-I
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 66
Lecture 3 Slide 66
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of
Probability
Theory-II
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 67
Lecture 3 Slide 67
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-III
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 68
Lecture 3 Slide 68
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-IV
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 32
Lecture 3 Slide 32
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
The Gaussian Distribution Function and Its Relation to Errors
We will use the Gaussian distribution as applied to random
variables to develop the mathematical basis for bull Mean
bull Standard deviation
bull Standard deviation of the mean (SDOM)
bull Moments and expectation values
bull Error propagation formulas
bull Addition of errors in quadrature (for independent and random
measurements)
bull Numerical values for confidence limits (t-test)
bull Principle of maximal likelihood
bull Central limit theorem
bull Weighted distributions and Chi squared
bull Schwartz inequality (ie the uncertainty principle) (next lecture)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 33
Lecture 3 Slide 33
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider the Gaussian distribution function
Use the normalization condition to
evaluate the normalization constant
(see Taylor p 132)
The mean Ẋ is the first moment
of the Gaussian distribution
function (see Taylor p 134)
The standard deviation σx is the standard
deviation of the mean of the Gaussian
distribution function (see Taylor p 143)
Gaussian Distribution Moments
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 34
Lecture 3 Slide 34
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Standard Deviation of Gaussian Distribution
Area under curve
(probability that
ndashσltxlt+σ) is 68
5 or ~2σ ldquoSignificantrdquo
1 or ~3σ ldquoHighly Significantrdquo
1σ ldquoWithin errorsrdquo
5 ppm or ~5σ ldquoValid for HEPrdquo
See Sec 106 Testing of
Hypotheses
More complete Table in App A and B
Ah thatrsquos
highly
significant
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 35
Lecture 3 Slide 35
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Function of Gaussian Distribution
Area under curve
(probability that
ndashtσltxlt+tσ) is given
in Table at right
Probable Error (probability that
ndash067σltxlt+067σ) is 50
Error Function (probability that ndashtσltxlt+tσ )
Error Function Tabulated values-see App A
Complementary Error Function
(probability that ndashxlt-tσ AND xgt+tσ )
Prob(x outside tσ) = 1 - Prob(x within tσ)
More complete Table in App A and B
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 36
Lecture 3 Slide 36
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Useful Points on Gaussian Distribution
Full Width at Half Maximum FWHM
(See Prob 512)
Points of Inflection Occur at plusmnσ
(See Prob 513)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 37
Lecture 3 Slide 37
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Analysis and Gaussian Distribution
Adding a Constant Multiplying by a Constant
XX+A XBX and σB σ
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 38
Lecture 3 Slide 38
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Sum of Two Variables
Consider the derived quantity
Z=X + Y
(with X=0 and Y=0)
Error Propagation Addition
Error in Z
Multiple two probabilities 119875119903119900119887 119909 119910 = 119875119903119900119887 119909 ∙ 119875119903119900119887 119909 prop 119890119909119901 minus
1
2
1199092
1205901199092+
1199102
1205901199102
prop 119890119909119901 minus1
2
(119909+119910)2
1205901199092+120590119910
2 minus1199112
prop 119890119909119901 minus1
2
(119909+119910)2
1205901199092+120590119910
2 119890119909119901 minus
1199112
2
ICBST (Eq 553)
X+YZ and σx2 + σy
2 σz
2
(addition in quadrature for random independent variables)
Integrates to radic2π
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 39
Lecture 3 Slide 39
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Error Propagation
General formula for error propagation see [Taylor Secs 35 and 39]
Uncertainty as a function of one variable [Taylor Sec 35]
1 Consider a graphical method of estimating error
a) Consider an arbitaray function q(x)
b) Plot q(x) vs x
c) On the graph label
(1) qbest = q(xbest)
(2) qhi = q(xbest + δx)
(3) qlow = q(xbest- δx)
d) Making a linear approximation
x
qqxslopeqq
x
qqxslopeqq
bestbestlow
bestbesthi
e) Therefore
xx
Note the absolute value
How do we determine the error of a derived quantity Z(XYhellip) from errors in XYhellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 40
Lecture 3 Slide 40
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Error Propagation
General formula for uncertainty of a function of one variable
xx
[Taylor Eq 323]
Can you now derive for specific rules of error propagation
1 Addition and Subtraction [Taylor p 49]
2 Multiplication and Division [Taylor p 51]
3 Multiplication by a constant (exact number) [Taylor p 54]
4 Exponentiation (powers) [Taylor p 56]
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 41
Lecture 3 Slide 41
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Multiple Variables
Uncertainty of a function of multiple variables [Taylor Sec 311]
1 It can easily (no really) be shown that (see Taylor Sec 311) for a
function of several variables
)(
z
z
qy
y
qx
x
qzyxq
[Taylor Eq 347]
2 More correctly it can be shown that (see Taylor Sec 311) for a
function of several variables
)(
z
z
qy
y
qx
x
qzyxq
[Taylor Eq 347]
where the equals sign represents an upper bound as discussed above
3 For a function of several independent and random variables
)(
222
z
z
qy
y
qx
x
qzyxq
[Taylor Eq 348]
Again the proof is left for Ch 5
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 42
Lecture 3 Slide 42
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Product of Two Variables
Consider the arbitrary derived quantity
q(xy) of two independent random variables x and y
Expand q(xy) in a Taylor series about the expected values of x and y
(ie at points near X and Y)
Error Propagation General Case
119902 119909 119910 = 119902 119883 119884 + 120597119902
120597119909
119883 119909 minus 119883 +
120597119902
120597119910
119884
(119910 minus 119884)
Fixed shifts peak of distribution
Fixed Distribution centered at X with width σX
120575119902 119909 119910 = 120590119902 = 119902 119883 119884 + 120597119902
120597119909
119883120590119909
2
+ 120597119902
120597119910
119884
120590119910
2
0
How do we determine the error of a derived quantity Z(XYhellip) from errors in XYhellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 43
Lecture 3 Slide 43
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
SDOM of Gaussian Distribution
Standard Deviation of the Mean
The SDOM decreases as the square root of the number of measurements
That is the relative width σẊ of the distribution gets narrower as more
measurements are made
Each measurement
has similar σXi= σẊ
and similar partial
derivatives
Thushellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 44
Lecture 3 Slide 44
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Two Key Theorems from Probability
Central Limit Theorem
For random independent measurements (each with a well-define
expectation value and well-defined variance) the arithmetic mean
(average) will be approximately normally distributed
Principle of Maximum Likelihood
Given the N observed measurements x1 x2hellipxN the best estimates
for Ẋ and σ are those values for which the observed x1 x2hellipxN are
most likely
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 45
Lecture 3 Slide 45
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Mean of Gaussian Distribution as ldquoBest Estimaterdquo
Principle of Maximum Likelihood
Best Estimate of X is from maximum probability or minimum summation
Consider a data set
Each randomly distributed with
x1 x2 x3 hellipxN
To find the most likely value of the mean (the best estimate of ẋ)
find X that yields the highest probability for the data set
The combined probability for the full data set is the product
Minimize
Sum
Solve for
derivative wrst
X set to 0
Best
estimate
of X
prop1
120590119890minus(1199091minus119883)2 2120590 times
1
120590119890minus(1199092minus119883)2 2120590 times hellip times
1
120590119890minus(119909119873minus119883)2 2120590 =
1
120590119873119890 minus(119909119894minus119883)2 2120590
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 46
Lecture 3 Slide 46
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Uncertainty of ldquoBest Estimatesrdquo of Gaussian Distribution Principle of Maximum Likelihood
Best Estimate of X is from maximum probability or minimum summation
Consider a data set x1 x2 x3 hellipxN
To find the most likely value of the standard deviation (the best estimate of
the width of the x distribution) find σx that yields the highest probability for
the data set
The combined probability for the full data set is the product
Minimize
Sum
Solve for
derivative
wrst X set to 0
Best
estimate
of X
Best Estimate of σ is from maximum probability or minimum summation
Minimize
Sum
Best
estimate
of σ
Solve for
derivative
wrst σ set to 0
See
Prob 526
prop1
120590119890minus(1199091minus119883)2 2120590 times
1
120590119890minus(1199092minus119883)2 2120590 times hellip times
1
120590119890minus(119909119873minus119883)2 2120590 =
1
120590119873119890 minus(119909119894minus119883)2 2120590
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 47
Lecture 3 Slide 47
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Combining Data Sets
Weighted Averages
References Taylor Ch 7
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 48
Lecture 3 Slide 48
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Weighted Averages
Question How can we properly combine two or more separate
independent measurements of the same randomly distributed
quantity to determine a best combined value with uncertainty
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 49
Lecture 3 Slide 49
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider two measurements of the same quantity described by a random Gaussian distribution
ltx1gt x1 and ltx2gt x2
The probability of measuring two such measurements is
119875119903119900119887119909 1199091 1199092 = 119875119903119900119887119909 1199091 119875119903119900119887119909 1199092
= 1
12059011205902119890minus12059422 119908ℎ119890119903119890 1205942 equiv
1199091 minus 119883
1205901 2 +
1199092 minus 119883
1205902 2
To find the best value for χ find the maximum Prob or minimum χ2
Weighted Averages
Assume negligible systematic errors
Note χ2 or Chi squared is the sum of the squares of the deviations from the mean divided by
the corresponding uncertainty
Such methods are called ldquoMethods of Least Squaresrdquo They follow directly from the Principle of
Maximum Likelihood
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 50
Lecture 3 Slide 50
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
The probability of measuring two such measurements is
119875119903119900119887119909 1199091 1199092 = 119875119903119900119887119909 1199091 119875119903119900119887119909 1199092
= 1
12059011205902119890minus12059422 119908ℎ119890119903119890 1205942 equiv
1199091 minus 119883
1205901 2 +
1199092 minus 119883
1205902 2
To find the best value for χ find the maximum Prob or minimum χ2
Weighted Averages
This leads to 119909119882_119886119907119892 = 11990811199091+11990821199092
1199081+1199082=
119908119894 119909119894
119908119894 119908ℎ119890119903119890 119908119894 = 1
120590119894 2
Best Estimate of χ is from maximum probibility or minimum summation
Minimize Sum Solve for best estimate of χ Solve for derivative wrst χ set to 0
Note If w1=w2 we recover the standard result Xwavg= (12) (x1+x2)
Finally the width of a weighted average distribution is
120590119908119894119890119892 ℎ119905119890119889 119886119907119892 = 1
119908119894119894
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 51
Lecture 3 Slide 51
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Weighted Averages on Steroids
A very powerful method for combining data from different sources
with different methods and uncertainties (or indeed data of
related measured and calculated quantities) is Kalman filtering
The Kalman filter also known as
linear quadratic estimation (LQE)
is an algorithm that uses a series
of measurements observed over
time containing noise (random
variations) and other
inaccuracies and produces
estimates of unknown variables
that tend to be more precise than
those based on a single
measurement alone
The Kalman filter keeps track of the estimated state of the system and the variance or uncertainty of the
estimate The estimate is updated using a state transition model and measurements xk|k-1 denotes the
estimate of the systems state at time step k before the kth measurement yk has been taken into account Pk|k-1
is the corresponding uncertainty --Wikipedia 2013
Ludger Scherliess of USU Physics is a world expert at using Kalman filtering for the assimilation
of satellite and ground-based data and the USU GAMES model to predict space weather
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 52
Lecture 3 Slide 52
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Rejecting Data
Chauvenetrsquos Criterion
References Taylor Ch 6
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 53
Lecture 3 Slide 53
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly
ldquounreasonablerdquo data point from a set of randomly distributed
measurements
bull Never
bull Whenever it makes things look better
bull Chauvenetrsquos criterion provides a (quantitative) compromise
Rejecting Data
What is a good criteria for
rejecting data
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 54
Lecture 3 Slide 54
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly
ldquounreasonablerdquo data point from a set of randomly distributed
measurements
bull Never
Rejecting Data
Zallenrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 55
Lecture 3 Slide 55
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly
ldquounreasonablerdquo data point from a set of randomly distributed
measurements
bull Whenever it makes things look better
Rejecting Data
Disneyrsquos Criterion
Disneyrsquos First Law
Wishing will make it so
Disneyrsquos Second Law
Dreams are more colorful than reality
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 56
Lecture 3 Slide 56
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Data may be rejected if the expected number of measurements at
least as deviant as the suspect measurement is less than 50
Consider a set of N measurements of a single quantity
x1x2 helliphellipxN
Calculate ltxgt and σx and then determine the fractional deviations from the mean of all
the points
For the suspect point(s) xsuspect find the probability of such a point occurring in N
measurements
n = (expected number as deviant as xsuspect)
= N Prob(outside xsuspectσx)
Rejecting Data
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 57
Lecture 3 Slide 57
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Function of Gaussian Distribution
Area under curve
(probability that
ndashtσltxlt+tσ) is given
in Table at right
Probable Error (probability that
ndash067σltxlt+067σ) is 50
Error Function (probability that ndashtσltxlt+tσ )
Error Function Tabulated values-see App A
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 58
Lecture 3 Slide 58
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 59
Lecture 3 Slide 59
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashExample 1
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 60
Lecture 3 Slide 60
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Details (1)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 61
Lecture 3 Slide 61
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashDetails (2)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 62
Lecture 3 Slide 62
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos
Criterionmdash
Details (3)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 63
Lecture 3 Slide 63
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos
Criterionmdash
Example 2
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 64
Lecture 3 Slide 64
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Summary of Probability Theory
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 65
Lecture 3 Slide 65
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of
Probability
Theory-I
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 66
Lecture 3 Slide 66
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of
Probability
Theory-II
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 67
Lecture 3 Slide 67
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-III
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 68
Lecture 3 Slide 68
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-IV
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 33
Lecture 3 Slide 33
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider the Gaussian distribution function
Use the normalization condition to
evaluate the normalization constant
(see Taylor p 132)
The mean Ẋ is the first moment
of the Gaussian distribution
function (see Taylor p 134)
The standard deviation σx is the standard
deviation of the mean of the Gaussian
distribution function (see Taylor p 143)
Gaussian Distribution Moments
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 34
Lecture 3 Slide 34
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Standard Deviation of Gaussian Distribution
Area under curve
(probability that
ndashσltxlt+σ) is 68
5 or ~2σ ldquoSignificantrdquo
1 or ~3σ ldquoHighly Significantrdquo
1σ ldquoWithin errorsrdquo
5 ppm or ~5σ ldquoValid for HEPrdquo
See Sec 106 Testing of
Hypotheses
More complete Table in App A and B
Ah thatrsquos
highly
significant
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 35
Lecture 3 Slide 35
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Function of Gaussian Distribution
Area under curve
(probability that
ndashtσltxlt+tσ) is given
in Table at right
Probable Error (probability that
ndash067σltxlt+067σ) is 50
Error Function (probability that ndashtσltxlt+tσ )
Error Function Tabulated values-see App A
Complementary Error Function
(probability that ndashxlt-tσ AND xgt+tσ )
Prob(x outside tσ) = 1 - Prob(x within tσ)
More complete Table in App A and B
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 36
Lecture 3 Slide 36
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Useful Points on Gaussian Distribution
Full Width at Half Maximum FWHM
(See Prob 512)
Points of Inflection Occur at plusmnσ
(See Prob 513)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 37
Lecture 3 Slide 37
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Analysis and Gaussian Distribution
Adding a Constant Multiplying by a Constant
XX+A XBX and σB σ
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 38
Lecture 3 Slide 38
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Sum of Two Variables
Consider the derived quantity
Z=X + Y
(with X=0 and Y=0)
Error Propagation Addition
Error in Z
Multiple two probabilities 119875119903119900119887 119909 119910 = 119875119903119900119887 119909 ∙ 119875119903119900119887 119909 prop 119890119909119901 minus
1
2
1199092
1205901199092+
1199102
1205901199102
prop 119890119909119901 minus1
2
(119909+119910)2
1205901199092+120590119910
2 minus1199112
prop 119890119909119901 minus1
2
(119909+119910)2
1205901199092+120590119910
2 119890119909119901 minus
1199112
2
ICBST (Eq 553)
X+YZ and σx2 + σy
2 σz
2
(addition in quadrature for random independent variables)
Integrates to radic2π
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 39
Lecture 3 Slide 39
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Error Propagation
General formula for error propagation see [Taylor Secs 35 and 39]
Uncertainty as a function of one variable [Taylor Sec 35]
1 Consider a graphical method of estimating error
a) Consider an arbitaray function q(x)
b) Plot q(x) vs x
c) On the graph label
(1) qbest = q(xbest)
(2) qhi = q(xbest + δx)
(3) qlow = q(xbest- δx)
d) Making a linear approximation
x
qqxslopeqq
x
qqxslopeqq
bestbestlow
bestbesthi
e) Therefore
xx
Note the absolute value
How do we determine the error of a derived quantity Z(XYhellip) from errors in XYhellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 40
Lecture 3 Slide 40
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Error Propagation
General formula for uncertainty of a function of one variable
xx
[Taylor Eq 323]
Can you now derive for specific rules of error propagation
1 Addition and Subtraction [Taylor p 49]
2 Multiplication and Division [Taylor p 51]
3 Multiplication by a constant (exact number) [Taylor p 54]
4 Exponentiation (powers) [Taylor p 56]
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 41
Lecture 3 Slide 41
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Multiple Variables
Uncertainty of a function of multiple variables [Taylor Sec 311]
1 It can easily (no really) be shown that (see Taylor Sec 311) for a
function of several variables
)(
z
z
qy
y
qx
x
qzyxq
[Taylor Eq 347]
2 More correctly it can be shown that (see Taylor Sec 311) for a
function of several variables
)(
z
z
qy
y
qx
x
qzyxq
[Taylor Eq 347]
where the equals sign represents an upper bound as discussed above
3 For a function of several independent and random variables
)(
222
z
z
qy
y
qx
x
qzyxq
[Taylor Eq 348]
Again the proof is left for Ch 5
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 42
Lecture 3 Slide 42
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Product of Two Variables
Consider the arbitrary derived quantity
q(xy) of two independent random variables x and y
Expand q(xy) in a Taylor series about the expected values of x and y
(ie at points near X and Y)
Error Propagation General Case
119902 119909 119910 = 119902 119883 119884 + 120597119902
120597119909
119883 119909 minus 119883 +
120597119902
120597119910
119884
(119910 minus 119884)
Fixed shifts peak of distribution
Fixed Distribution centered at X with width σX
120575119902 119909 119910 = 120590119902 = 119902 119883 119884 + 120597119902
120597119909
119883120590119909
2
+ 120597119902
120597119910
119884
120590119910
2
0
How do we determine the error of a derived quantity Z(XYhellip) from errors in XYhellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 43
Lecture 3 Slide 43
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
SDOM of Gaussian Distribution
Standard Deviation of the Mean
The SDOM decreases as the square root of the number of measurements
That is the relative width σẊ of the distribution gets narrower as more
measurements are made
Each measurement
has similar σXi= σẊ
and similar partial
derivatives
Thushellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 44
Lecture 3 Slide 44
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Two Key Theorems from Probability
Central Limit Theorem
For random independent measurements (each with a well-define
expectation value and well-defined variance) the arithmetic mean
(average) will be approximately normally distributed
Principle of Maximum Likelihood
Given the N observed measurements x1 x2hellipxN the best estimates
for Ẋ and σ are those values for which the observed x1 x2hellipxN are
most likely
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 45
Lecture 3 Slide 45
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Mean of Gaussian Distribution as ldquoBest Estimaterdquo
Principle of Maximum Likelihood
Best Estimate of X is from maximum probability or minimum summation
Consider a data set
Each randomly distributed with
x1 x2 x3 hellipxN
To find the most likely value of the mean (the best estimate of ẋ)
find X that yields the highest probability for the data set
The combined probability for the full data set is the product
Minimize
Sum
Solve for
derivative wrst
X set to 0
Best
estimate
of X
prop1
120590119890minus(1199091minus119883)2 2120590 times
1
120590119890minus(1199092minus119883)2 2120590 times hellip times
1
120590119890minus(119909119873minus119883)2 2120590 =
1
120590119873119890 minus(119909119894minus119883)2 2120590
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 46
Lecture 3 Slide 46
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Uncertainty of ldquoBest Estimatesrdquo of Gaussian Distribution Principle of Maximum Likelihood
Best Estimate of X is from maximum probability or minimum summation
Consider a data set x1 x2 x3 hellipxN
To find the most likely value of the standard deviation (the best estimate of
the width of the x distribution) find σx that yields the highest probability for
the data set
The combined probability for the full data set is the product
Minimize
Sum
Solve for
derivative
wrst X set to 0
Best
estimate
of X
Best Estimate of σ is from maximum probability or minimum summation
Minimize
Sum
Best
estimate
of σ
Solve for
derivative
wrst σ set to 0
See
Prob 526
prop1
120590119890minus(1199091minus119883)2 2120590 times
1
120590119890minus(1199092minus119883)2 2120590 times hellip times
1
120590119890minus(119909119873minus119883)2 2120590 =
1
120590119873119890 minus(119909119894minus119883)2 2120590
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 47
Lecture 3 Slide 47
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Combining Data Sets
Weighted Averages
References Taylor Ch 7
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 48
Lecture 3 Slide 48
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Weighted Averages
Question How can we properly combine two or more separate
independent measurements of the same randomly distributed
quantity to determine a best combined value with uncertainty
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 49
Lecture 3 Slide 49
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider two measurements of the same quantity described by a random Gaussian distribution
ltx1gt x1 and ltx2gt x2
The probability of measuring two such measurements is
119875119903119900119887119909 1199091 1199092 = 119875119903119900119887119909 1199091 119875119903119900119887119909 1199092
= 1
12059011205902119890minus12059422 119908ℎ119890119903119890 1205942 equiv
1199091 minus 119883
1205901 2 +
1199092 minus 119883
1205902 2
To find the best value for χ find the maximum Prob or minimum χ2
Weighted Averages
Assume negligible systematic errors
Note χ2 or Chi squared is the sum of the squares of the deviations from the mean divided by
the corresponding uncertainty
Such methods are called ldquoMethods of Least Squaresrdquo They follow directly from the Principle of
Maximum Likelihood
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 50
Lecture 3 Slide 50
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
The probability of measuring two such measurements is
119875119903119900119887119909 1199091 1199092 = 119875119903119900119887119909 1199091 119875119903119900119887119909 1199092
= 1
12059011205902119890minus12059422 119908ℎ119890119903119890 1205942 equiv
1199091 minus 119883
1205901 2 +
1199092 minus 119883
1205902 2
To find the best value for χ find the maximum Prob or minimum χ2
Weighted Averages
This leads to 119909119882_119886119907119892 = 11990811199091+11990821199092
1199081+1199082=
119908119894 119909119894
119908119894 119908ℎ119890119903119890 119908119894 = 1
120590119894 2
Best Estimate of χ is from maximum probibility or minimum summation
Minimize Sum Solve for best estimate of χ Solve for derivative wrst χ set to 0
Note If w1=w2 we recover the standard result Xwavg= (12) (x1+x2)
Finally the width of a weighted average distribution is
120590119908119894119890119892 ℎ119905119890119889 119886119907119892 = 1
119908119894119894
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 51
Lecture 3 Slide 51
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Weighted Averages on Steroids
A very powerful method for combining data from different sources
with different methods and uncertainties (or indeed data of
related measured and calculated quantities) is Kalman filtering
The Kalman filter also known as
linear quadratic estimation (LQE)
is an algorithm that uses a series
of measurements observed over
time containing noise (random
variations) and other
inaccuracies and produces
estimates of unknown variables
that tend to be more precise than
those based on a single
measurement alone
The Kalman filter keeps track of the estimated state of the system and the variance or uncertainty of the
estimate The estimate is updated using a state transition model and measurements xk|k-1 denotes the
estimate of the systems state at time step k before the kth measurement yk has been taken into account Pk|k-1
is the corresponding uncertainty --Wikipedia 2013
Ludger Scherliess of USU Physics is a world expert at using Kalman filtering for the assimilation
of satellite and ground-based data and the USU GAMES model to predict space weather
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 52
Lecture 3 Slide 52
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Rejecting Data
Chauvenetrsquos Criterion
References Taylor Ch 6
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 53
Lecture 3 Slide 53
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly
ldquounreasonablerdquo data point from a set of randomly distributed
measurements
bull Never
bull Whenever it makes things look better
bull Chauvenetrsquos criterion provides a (quantitative) compromise
Rejecting Data
What is a good criteria for
rejecting data
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 54
Lecture 3 Slide 54
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly
ldquounreasonablerdquo data point from a set of randomly distributed
measurements
bull Never
Rejecting Data
Zallenrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 55
Lecture 3 Slide 55
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly
ldquounreasonablerdquo data point from a set of randomly distributed
measurements
bull Whenever it makes things look better
Rejecting Data
Disneyrsquos Criterion
Disneyrsquos First Law
Wishing will make it so
Disneyrsquos Second Law
Dreams are more colorful than reality
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 56
Lecture 3 Slide 56
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Data may be rejected if the expected number of measurements at
least as deviant as the suspect measurement is less than 50
Consider a set of N measurements of a single quantity
x1x2 helliphellipxN
Calculate ltxgt and σx and then determine the fractional deviations from the mean of all
the points
For the suspect point(s) xsuspect find the probability of such a point occurring in N
measurements
n = (expected number as deviant as xsuspect)
= N Prob(outside xsuspectσx)
Rejecting Data
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 57
Lecture 3 Slide 57
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Function of Gaussian Distribution
Area under curve
(probability that
ndashtσltxlt+tσ) is given
in Table at right
Probable Error (probability that
ndash067σltxlt+067σ) is 50
Error Function (probability that ndashtσltxlt+tσ )
Error Function Tabulated values-see App A
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 58
Lecture 3 Slide 58
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 59
Lecture 3 Slide 59
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashExample 1
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 60
Lecture 3 Slide 60
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Details (1)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 61
Lecture 3 Slide 61
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashDetails (2)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 62
Lecture 3 Slide 62
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos
Criterionmdash
Details (3)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 63
Lecture 3 Slide 63
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos
Criterionmdash
Example 2
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 64
Lecture 3 Slide 64
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Summary of Probability Theory
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 65
Lecture 3 Slide 65
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of
Probability
Theory-I
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 66
Lecture 3 Slide 66
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of
Probability
Theory-II
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 67
Lecture 3 Slide 67
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-III
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 68
Lecture 3 Slide 68
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-IV
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 34
Lecture 3 Slide 34
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Standard Deviation of Gaussian Distribution
Area under curve
(probability that
ndashσltxlt+σ) is 68
5 or ~2σ ldquoSignificantrdquo
1 or ~3σ ldquoHighly Significantrdquo
1σ ldquoWithin errorsrdquo
5 ppm or ~5σ ldquoValid for HEPrdquo
See Sec 106 Testing of
Hypotheses
More complete Table in App A and B
Ah thatrsquos
highly
significant
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 35
Lecture 3 Slide 35
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Function of Gaussian Distribution
Area under curve
(probability that
ndashtσltxlt+tσ) is given
in Table at right
Probable Error (probability that
ndash067σltxlt+067σ) is 50
Error Function (probability that ndashtσltxlt+tσ )
Error Function Tabulated values-see App A
Complementary Error Function
(probability that ndashxlt-tσ AND xgt+tσ )
Prob(x outside tσ) = 1 - Prob(x within tσ)
More complete Table in App A and B
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 36
Lecture 3 Slide 36
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Useful Points on Gaussian Distribution
Full Width at Half Maximum FWHM
(See Prob 512)
Points of Inflection Occur at plusmnσ
(See Prob 513)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 37
Lecture 3 Slide 37
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Analysis and Gaussian Distribution
Adding a Constant Multiplying by a Constant
XX+A XBX and σB σ
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 38
Lecture 3 Slide 38
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Sum of Two Variables
Consider the derived quantity
Z=X + Y
(with X=0 and Y=0)
Error Propagation Addition
Error in Z
Multiple two probabilities 119875119903119900119887 119909 119910 = 119875119903119900119887 119909 ∙ 119875119903119900119887 119909 prop 119890119909119901 minus
1
2
1199092
1205901199092+
1199102
1205901199102
prop 119890119909119901 minus1
2
(119909+119910)2
1205901199092+120590119910
2 minus1199112
prop 119890119909119901 minus1
2
(119909+119910)2
1205901199092+120590119910
2 119890119909119901 minus
1199112
2
ICBST (Eq 553)
X+YZ and σx2 + σy
2 σz
2
(addition in quadrature for random independent variables)
Integrates to radic2π
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 39
Lecture 3 Slide 39
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Error Propagation
General formula for error propagation see [Taylor Secs 35 and 39]
Uncertainty as a function of one variable [Taylor Sec 35]
1 Consider a graphical method of estimating error
a) Consider an arbitaray function q(x)
b) Plot q(x) vs x
c) On the graph label
(1) qbest = q(xbest)
(2) qhi = q(xbest + δx)
(3) qlow = q(xbest- δx)
d) Making a linear approximation
x
qqxslopeqq
x
qqxslopeqq
bestbestlow
bestbesthi
e) Therefore
xx
Note the absolute value
How do we determine the error of a derived quantity Z(XYhellip) from errors in XYhellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 40
Lecture 3 Slide 40
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Error Propagation
General formula for uncertainty of a function of one variable
xx
[Taylor Eq 323]
Can you now derive for specific rules of error propagation
1 Addition and Subtraction [Taylor p 49]
2 Multiplication and Division [Taylor p 51]
3 Multiplication by a constant (exact number) [Taylor p 54]
4 Exponentiation (powers) [Taylor p 56]
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 41
Lecture 3 Slide 41
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Multiple Variables
Uncertainty of a function of multiple variables [Taylor Sec 311]
1 It can easily (no really) be shown that (see Taylor Sec 311) for a
function of several variables
)(
z
z
qy
y
qx
x
qzyxq
[Taylor Eq 347]
2 More correctly it can be shown that (see Taylor Sec 311) for a
function of several variables
)(
z
z
qy
y
qx
x
qzyxq
[Taylor Eq 347]
where the equals sign represents an upper bound as discussed above
3 For a function of several independent and random variables
)(
222
z
z
qy
y
qx
x
qzyxq
[Taylor Eq 348]
Again the proof is left for Ch 5
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 42
Lecture 3 Slide 42
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Product of Two Variables
Consider the arbitrary derived quantity
q(xy) of two independent random variables x and y
Expand q(xy) in a Taylor series about the expected values of x and y
(ie at points near X and Y)
Error Propagation General Case
119902 119909 119910 = 119902 119883 119884 + 120597119902
120597119909
119883 119909 minus 119883 +
120597119902
120597119910
119884
(119910 minus 119884)
Fixed shifts peak of distribution
Fixed Distribution centered at X with width σX
120575119902 119909 119910 = 120590119902 = 119902 119883 119884 + 120597119902
120597119909
119883120590119909
2
+ 120597119902
120597119910
119884
120590119910
2
0
How do we determine the error of a derived quantity Z(XYhellip) from errors in XYhellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 43
Lecture 3 Slide 43
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
SDOM of Gaussian Distribution
Standard Deviation of the Mean
The SDOM decreases as the square root of the number of measurements
That is the relative width σẊ of the distribution gets narrower as more
measurements are made
Each measurement
has similar σXi= σẊ
and similar partial
derivatives
Thushellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 44
Lecture 3 Slide 44
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Two Key Theorems from Probability
Central Limit Theorem
For random independent measurements (each with a well-define
expectation value and well-defined variance) the arithmetic mean
(average) will be approximately normally distributed
Principle of Maximum Likelihood
Given the N observed measurements x1 x2hellipxN the best estimates
for Ẋ and σ are those values for which the observed x1 x2hellipxN are
most likely
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 45
Lecture 3 Slide 45
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Mean of Gaussian Distribution as ldquoBest Estimaterdquo
Principle of Maximum Likelihood
Best Estimate of X is from maximum probability or minimum summation
Consider a data set
Each randomly distributed with
x1 x2 x3 hellipxN
To find the most likely value of the mean (the best estimate of ẋ)
find X that yields the highest probability for the data set
The combined probability for the full data set is the product
Minimize
Sum
Solve for
derivative wrst
X set to 0
Best
estimate
of X
prop1
120590119890minus(1199091minus119883)2 2120590 times
1
120590119890minus(1199092minus119883)2 2120590 times hellip times
1
120590119890minus(119909119873minus119883)2 2120590 =
1
120590119873119890 minus(119909119894minus119883)2 2120590
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 46
Lecture 3 Slide 46
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Uncertainty of ldquoBest Estimatesrdquo of Gaussian Distribution Principle of Maximum Likelihood
Best Estimate of X is from maximum probability or minimum summation
Consider a data set x1 x2 x3 hellipxN
To find the most likely value of the standard deviation (the best estimate of
the width of the x distribution) find σx that yields the highest probability for
the data set
The combined probability for the full data set is the product
Minimize
Sum
Solve for
derivative
wrst X set to 0
Best
estimate
of X
Best Estimate of σ is from maximum probability or minimum summation
Minimize
Sum
Best
estimate
of σ
Solve for
derivative
wrst σ set to 0
See
Prob 526
prop1
120590119890minus(1199091minus119883)2 2120590 times
1
120590119890minus(1199092minus119883)2 2120590 times hellip times
1
120590119890minus(119909119873minus119883)2 2120590 =
1
120590119873119890 minus(119909119894minus119883)2 2120590
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 47
Lecture 3 Slide 47
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Combining Data Sets
Weighted Averages
References Taylor Ch 7
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 48
Lecture 3 Slide 48
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Weighted Averages
Question How can we properly combine two or more separate
independent measurements of the same randomly distributed
quantity to determine a best combined value with uncertainty
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 49
Lecture 3 Slide 49
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider two measurements of the same quantity described by a random Gaussian distribution
ltx1gt x1 and ltx2gt x2
The probability of measuring two such measurements is
119875119903119900119887119909 1199091 1199092 = 119875119903119900119887119909 1199091 119875119903119900119887119909 1199092
= 1
12059011205902119890minus12059422 119908ℎ119890119903119890 1205942 equiv
1199091 minus 119883
1205901 2 +
1199092 minus 119883
1205902 2
To find the best value for χ find the maximum Prob or minimum χ2
Weighted Averages
Assume negligible systematic errors
Note χ2 or Chi squared is the sum of the squares of the deviations from the mean divided by
the corresponding uncertainty
Such methods are called ldquoMethods of Least Squaresrdquo They follow directly from the Principle of
Maximum Likelihood
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 50
Lecture 3 Slide 50
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
The probability of measuring two such measurements is
119875119903119900119887119909 1199091 1199092 = 119875119903119900119887119909 1199091 119875119903119900119887119909 1199092
= 1
12059011205902119890minus12059422 119908ℎ119890119903119890 1205942 equiv
1199091 minus 119883
1205901 2 +
1199092 minus 119883
1205902 2
To find the best value for χ find the maximum Prob or minimum χ2
Weighted Averages
This leads to 119909119882_119886119907119892 = 11990811199091+11990821199092
1199081+1199082=
119908119894 119909119894
119908119894 119908ℎ119890119903119890 119908119894 = 1
120590119894 2
Best Estimate of χ is from maximum probibility or minimum summation
Minimize Sum Solve for best estimate of χ Solve for derivative wrst χ set to 0
Note If w1=w2 we recover the standard result Xwavg= (12) (x1+x2)
Finally the width of a weighted average distribution is
120590119908119894119890119892 ℎ119905119890119889 119886119907119892 = 1
119908119894119894
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 51
Lecture 3 Slide 51
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Weighted Averages on Steroids
A very powerful method for combining data from different sources
with different methods and uncertainties (or indeed data of
related measured and calculated quantities) is Kalman filtering
The Kalman filter also known as
linear quadratic estimation (LQE)
is an algorithm that uses a series
of measurements observed over
time containing noise (random
variations) and other
inaccuracies and produces
estimates of unknown variables
that tend to be more precise than
those based on a single
measurement alone
The Kalman filter keeps track of the estimated state of the system and the variance or uncertainty of the
estimate The estimate is updated using a state transition model and measurements xk|k-1 denotes the
estimate of the systems state at time step k before the kth measurement yk has been taken into account Pk|k-1
is the corresponding uncertainty --Wikipedia 2013
Ludger Scherliess of USU Physics is a world expert at using Kalman filtering for the assimilation
of satellite and ground-based data and the USU GAMES model to predict space weather
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 52
Lecture 3 Slide 52
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Rejecting Data
Chauvenetrsquos Criterion
References Taylor Ch 6
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 53
Lecture 3 Slide 53
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly
ldquounreasonablerdquo data point from a set of randomly distributed
measurements
bull Never
bull Whenever it makes things look better
bull Chauvenetrsquos criterion provides a (quantitative) compromise
Rejecting Data
What is a good criteria for
rejecting data
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 54
Lecture 3 Slide 54
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly
ldquounreasonablerdquo data point from a set of randomly distributed
measurements
bull Never
Rejecting Data
Zallenrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 55
Lecture 3 Slide 55
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly
ldquounreasonablerdquo data point from a set of randomly distributed
measurements
bull Whenever it makes things look better
Rejecting Data
Disneyrsquos Criterion
Disneyrsquos First Law
Wishing will make it so
Disneyrsquos Second Law
Dreams are more colorful than reality
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 56
Lecture 3 Slide 56
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Data may be rejected if the expected number of measurements at
least as deviant as the suspect measurement is less than 50
Consider a set of N measurements of a single quantity
x1x2 helliphellipxN
Calculate ltxgt and σx and then determine the fractional deviations from the mean of all
the points
For the suspect point(s) xsuspect find the probability of such a point occurring in N
measurements
n = (expected number as deviant as xsuspect)
= N Prob(outside xsuspectσx)
Rejecting Data
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 57
Lecture 3 Slide 57
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Function of Gaussian Distribution
Area under curve
(probability that
ndashtσltxlt+tσ) is given
in Table at right
Probable Error (probability that
ndash067σltxlt+067σ) is 50
Error Function (probability that ndashtσltxlt+tσ )
Error Function Tabulated values-see App A
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 58
Lecture 3 Slide 58
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 59
Lecture 3 Slide 59
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashExample 1
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 60
Lecture 3 Slide 60
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Details (1)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 61
Lecture 3 Slide 61
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashDetails (2)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 62
Lecture 3 Slide 62
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos
Criterionmdash
Details (3)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 63
Lecture 3 Slide 63
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos
Criterionmdash
Example 2
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 64
Lecture 3 Slide 64
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Summary of Probability Theory
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 65
Lecture 3 Slide 65
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of
Probability
Theory-I
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 66
Lecture 3 Slide 66
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of
Probability
Theory-II
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 67
Lecture 3 Slide 67
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-III
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 68
Lecture 3 Slide 68
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-IV
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 35
Lecture 3 Slide 35
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Function of Gaussian Distribution
Area under curve
(probability that
ndashtσltxlt+tσ) is given
in Table at right
Probable Error (probability that
ndash067σltxlt+067σ) is 50
Error Function (probability that ndashtσltxlt+tσ )
Error Function Tabulated values-see App A
Complementary Error Function
(probability that ndashxlt-tσ AND xgt+tσ )
Prob(x outside tσ) = 1 - Prob(x within tσ)
More complete Table in App A and B
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 36
Lecture 3 Slide 36
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Useful Points on Gaussian Distribution
Full Width at Half Maximum FWHM
(See Prob 512)
Points of Inflection Occur at plusmnσ
(See Prob 513)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 37
Lecture 3 Slide 37
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Analysis and Gaussian Distribution
Adding a Constant Multiplying by a Constant
XX+A XBX and σB σ
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 38
Lecture 3 Slide 38
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Sum of Two Variables
Consider the derived quantity
Z=X + Y
(with X=0 and Y=0)
Error Propagation Addition
Error in Z
Multiple two probabilities 119875119903119900119887 119909 119910 = 119875119903119900119887 119909 ∙ 119875119903119900119887 119909 prop 119890119909119901 minus
1
2
1199092
1205901199092+
1199102
1205901199102
prop 119890119909119901 minus1
2
(119909+119910)2
1205901199092+120590119910
2 minus1199112
prop 119890119909119901 minus1
2
(119909+119910)2
1205901199092+120590119910
2 119890119909119901 minus
1199112
2
ICBST (Eq 553)
X+YZ and σx2 + σy
2 σz
2
(addition in quadrature for random independent variables)
Integrates to radic2π
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 39
Lecture 3 Slide 39
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Error Propagation
General formula for error propagation see [Taylor Secs 35 and 39]
Uncertainty as a function of one variable [Taylor Sec 35]
1 Consider a graphical method of estimating error
a) Consider an arbitaray function q(x)
b) Plot q(x) vs x
c) On the graph label
(1) qbest = q(xbest)
(2) qhi = q(xbest + δx)
(3) qlow = q(xbest- δx)
d) Making a linear approximation
x
qqxslopeqq
x
qqxslopeqq
bestbestlow
bestbesthi
e) Therefore
xx
Note the absolute value
How do we determine the error of a derived quantity Z(XYhellip) from errors in XYhellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 40
Lecture 3 Slide 40
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Error Propagation
General formula for uncertainty of a function of one variable
xx
[Taylor Eq 323]
Can you now derive for specific rules of error propagation
1 Addition and Subtraction [Taylor p 49]
2 Multiplication and Division [Taylor p 51]
3 Multiplication by a constant (exact number) [Taylor p 54]
4 Exponentiation (powers) [Taylor p 56]
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 41
Lecture 3 Slide 41
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Multiple Variables
Uncertainty of a function of multiple variables [Taylor Sec 311]
1 It can easily (no really) be shown that (see Taylor Sec 311) for a
function of several variables
)(
z
z
qy
y
qx
x
qzyxq
[Taylor Eq 347]
2 More correctly it can be shown that (see Taylor Sec 311) for a
function of several variables
)(
z
z
qy
y
qx
x
qzyxq
[Taylor Eq 347]
where the equals sign represents an upper bound as discussed above
3 For a function of several independent and random variables
)(
222
z
z
qy
y
qx
x
qzyxq
[Taylor Eq 348]
Again the proof is left for Ch 5
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 42
Lecture 3 Slide 42
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Product of Two Variables
Consider the arbitrary derived quantity
q(xy) of two independent random variables x and y
Expand q(xy) in a Taylor series about the expected values of x and y
(ie at points near X and Y)
Error Propagation General Case
119902 119909 119910 = 119902 119883 119884 + 120597119902
120597119909
119883 119909 minus 119883 +
120597119902
120597119910
119884
(119910 minus 119884)
Fixed shifts peak of distribution
Fixed Distribution centered at X with width σX
120575119902 119909 119910 = 120590119902 = 119902 119883 119884 + 120597119902
120597119909
119883120590119909
2
+ 120597119902
120597119910
119884
120590119910
2
0
How do we determine the error of a derived quantity Z(XYhellip) from errors in XYhellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 43
Lecture 3 Slide 43
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
SDOM of Gaussian Distribution
Standard Deviation of the Mean
The SDOM decreases as the square root of the number of measurements
That is the relative width σẊ of the distribution gets narrower as more
measurements are made
Each measurement
has similar σXi= σẊ
and similar partial
derivatives
Thushellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 44
Lecture 3 Slide 44
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Two Key Theorems from Probability
Central Limit Theorem
For random independent measurements (each with a well-define
expectation value and well-defined variance) the arithmetic mean
(average) will be approximately normally distributed
Principle of Maximum Likelihood
Given the N observed measurements x1 x2hellipxN the best estimates
for Ẋ and σ are those values for which the observed x1 x2hellipxN are
most likely
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 45
Lecture 3 Slide 45
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Mean of Gaussian Distribution as ldquoBest Estimaterdquo
Principle of Maximum Likelihood
Best Estimate of X is from maximum probability or minimum summation
Consider a data set
Each randomly distributed with
x1 x2 x3 hellipxN
To find the most likely value of the mean (the best estimate of ẋ)
find X that yields the highest probability for the data set
The combined probability for the full data set is the product
Minimize
Sum
Solve for
derivative wrst
X set to 0
Best
estimate
of X
prop1
120590119890minus(1199091minus119883)2 2120590 times
1
120590119890minus(1199092minus119883)2 2120590 times hellip times
1
120590119890minus(119909119873minus119883)2 2120590 =
1
120590119873119890 minus(119909119894minus119883)2 2120590
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 46
Lecture 3 Slide 46
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Uncertainty of ldquoBest Estimatesrdquo of Gaussian Distribution Principle of Maximum Likelihood
Best Estimate of X is from maximum probability or minimum summation
Consider a data set x1 x2 x3 hellipxN
To find the most likely value of the standard deviation (the best estimate of
the width of the x distribution) find σx that yields the highest probability for
the data set
The combined probability for the full data set is the product
Minimize
Sum
Solve for
derivative
wrst X set to 0
Best
estimate
of X
Best Estimate of σ is from maximum probability or minimum summation
Minimize
Sum
Best
estimate
of σ
Solve for
derivative
wrst σ set to 0
See
Prob 526
prop1
120590119890minus(1199091minus119883)2 2120590 times
1
120590119890minus(1199092minus119883)2 2120590 times hellip times
1
120590119890minus(119909119873minus119883)2 2120590 =
1
120590119873119890 minus(119909119894minus119883)2 2120590
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 47
Lecture 3 Slide 47
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Combining Data Sets
Weighted Averages
References Taylor Ch 7
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 48
Lecture 3 Slide 48
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Weighted Averages
Question How can we properly combine two or more separate
independent measurements of the same randomly distributed
quantity to determine a best combined value with uncertainty
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 49
Lecture 3 Slide 49
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider two measurements of the same quantity described by a random Gaussian distribution
ltx1gt x1 and ltx2gt x2
The probability of measuring two such measurements is
119875119903119900119887119909 1199091 1199092 = 119875119903119900119887119909 1199091 119875119903119900119887119909 1199092
= 1
12059011205902119890minus12059422 119908ℎ119890119903119890 1205942 equiv
1199091 minus 119883
1205901 2 +
1199092 minus 119883
1205902 2
To find the best value for χ find the maximum Prob or minimum χ2
Weighted Averages
Assume negligible systematic errors
Note χ2 or Chi squared is the sum of the squares of the deviations from the mean divided by
the corresponding uncertainty
Such methods are called ldquoMethods of Least Squaresrdquo They follow directly from the Principle of
Maximum Likelihood
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 50
Lecture 3 Slide 50
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
The probability of measuring two such measurements is
119875119903119900119887119909 1199091 1199092 = 119875119903119900119887119909 1199091 119875119903119900119887119909 1199092
= 1
12059011205902119890minus12059422 119908ℎ119890119903119890 1205942 equiv
1199091 minus 119883
1205901 2 +
1199092 minus 119883
1205902 2
To find the best value for χ find the maximum Prob or minimum χ2
Weighted Averages
This leads to 119909119882_119886119907119892 = 11990811199091+11990821199092
1199081+1199082=
119908119894 119909119894
119908119894 119908ℎ119890119903119890 119908119894 = 1
120590119894 2
Best Estimate of χ is from maximum probibility or minimum summation
Minimize Sum Solve for best estimate of χ Solve for derivative wrst χ set to 0
Note If w1=w2 we recover the standard result Xwavg= (12) (x1+x2)
Finally the width of a weighted average distribution is
120590119908119894119890119892 ℎ119905119890119889 119886119907119892 = 1
119908119894119894
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 51
Lecture 3 Slide 51
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Weighted Averages on Steroids
A very powerful method for combining data from different sources
with different methods and uncertainties (or indeed data of
related measured and calculated quantities) is Kalman filtering
The Kalman filter also known as
linear quadratic estimation (LQE)
is an algorithm that uses a series
of measurements observed over
time containing noise (random
variations) and other
inaccuracies and produces
estimates of unknown variables
that tend to be more precise than
those based on a single
measurement alone
The Kalman filter keeps track of the estimated state of the system and the variance or uncertainty of the
estimate The estimate is updated using a state transition model and measurements xk|k-1 denotes the
estimate of the systems state at time step k before the kth measurement yk has been taken into account Pk|k-1
is the corresponding uncertainty --Wikipedia 2013
Ludger Scherliess of USU Physics is a world expert at using Kalman filtering for the assimilation
of satellite and ground-based data and the USU GAMES model to predict space weather
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 52
Lecture 3 Slide 52
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Rejecting Data
Chauvenetrsquos Criterion
References Taylor Ch 6
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 53
Lecture 3 Slide 53
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly
ldquounreasonablerdquo data point from a set of randomly distributed
measurements
bull Never
bull Whenever it makes things look better
bull Chauvenetrsquos criterion provides a (quantitative) compromise
Rejecting Data
What is a good criteria for
rejecting data
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 54
Lecture 3 Slide 54
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly
ldquounreasonablerdquo data point from a set of randomly distributed
measurements
bull Never
Rejecting Data
Zallenrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 55
Lecture 3 Slide 55
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly
ldquounreasonablerdquo data point from a set of randomly distributed
measurements
bull Whenever it makes things look better
Rejecting Data
Disneyrsquos Criterion
Disneyrsquos First Law
Wishing will make it so
Disneyrsquos Second Law
Dreams are more colorful than reality
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 56
Lecture 3 Slide 56
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Data may be rejected if the expected number of measurements at
least as deviant as the suspect measurement is less than 50
Consider a set of N measurements of a single quantity
x1x2 helliphellipxN
Calculate ltxgt and σx and then determine the fractional deviations from the mean of all
the points
For the suspect point(s) xsuspect find the probability of such a point occurring in N
measurements
n = (expected number as deviant as xsuspect)
= N Prob(outside xsuspectσx)
Rejecting Data
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 57
Lecture 3 Slide 57
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Function of Gaussian Distribution
Area under curve
(probability that
ndashtσltxlt+tσ) is given
in Table at right
Probable Error (probability that
ndash067σltxlt+067σ) is 50
Error Function (probability that ndashtσltxlt+tσ )
Error Function Tabulated values-see App A
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 58
Lecture 3 Slide 58
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 59
Lecture 3 Slide 59
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashExample 1
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 60
Lecture 3 Slide 60
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Details (1)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 61
Lecture 3 Slide 61
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashDetails (2)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 62
Lecture 3 Slide 62
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos
Criterionmdash
Details (3)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 63
Lecture 3 Slide 63
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos
Criterionmdash
Example 2
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 64
Lecture 3 Slide 64
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Summary of Probability Theory
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 65
Lecture 3 Slide 65
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of
Probability
Theory-I
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 66
Lecture 3 Slide 66
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of
Probability
Theory-II
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 67
Lecture 3 Slide 67
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-III
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 68
Lecture 3 Slide 68
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-IV
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 36
Lecture 3 Slide 36
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Useful Points on Gaussian Distribution
Full Width at Half Maximum FWHM
(See Prob 512)
Points of Inflection Occur at plusmnσ
(See Prob 513)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 37
Lecture 3 Slide 37
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Analysis and Gaussian Distribution
Adding a Constant Multiplying by a Constant
XX+A XBX and σB σ
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 38
Lecture 3 Slide 38
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Sum of Two Variables
Consider the derived quantity
Z=X + Y
(with X=0 and Y=0)
Error Propagation Addition
Error in Z
Multiple two probabilities 119875119903119900119887 119909 119910 = 119875119903119900119887 119909 ∙ 119875119903119900119887 119909 prop 119890119909119901 minus
1
2
1199092
1205901199092+
1199102
1205901199102
prop 119890119909119901 minus1
2
(119909+119910)2
1205901199092+120590119910
2 minus1199112
prop 119890119909119901 minus1
2
(119909+119910)2
1205901199092+120590119910
2 119890119909119901 minus
1199112
2
ICBST (Eq 553)
X+YZ and σx2 + σy
2 σz
2
(addition in quadrature for random independent variables)
Integrates to radic2π
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 39
Lecture 3 Slide 39
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Error Propagation
General formula for error propagation see [Taylor Secs 35 and 39]
Uncertainty as a function of one variable [Taylor Sec 35]
1 Consider a graphical method of estimating error
a) Consider an arbitaray function q(x)
b) Plot q(x) vs x
c) On the graph label
(1) qbest = q(xbest)
(2) qhi = q(xbest + δx)
(3) qlow = q(xbest- δx)
d) Making a linear approximation
x
qqxslopeqq
x
qqxslopeqq
bestbestlow
bestbesthi
e) Therefore
xx
Note the absolute value
How do we determine the error of a derived quantity Z(XYhellip) from errors in XYhellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 40
Lecture 3 Slide 40
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Error Propagation
General formula for uncertainty of a function of one variable
xx
[Taylor Eq 323]
Can you now derive for specific rules of error propagation
1 Addition and Subtraction [Taylor p 49]
2 Multiplication and Division [Taylor p 51]
3 Multiplication by a constant (exact number) [Taylor p 54]
4 Exponentiation (powers) [Taylor p 56]
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 41
Lecture 3 Slide 41
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Multiple Variables
Uncertainty of a function of multiple variables [Taylor Sec 311]
1 It can easily (no really) be shown that (see Taylor Sec 311) for a
function of several variables
)(
z
z
qy
y
qx
x
qzyxq
[Taylor Eq 347]
2 More correctly it can be shown that (see Taylor Sec 311) for a
function of several variables
)(
z
z
qy
y
qx
x
qzyxq
[Taylor Eq 347]
where the equals sign represents an upper bound as discussed above
3 For a function of several independent and random variables
)(
222
z
z
qy
y
qx
x
qzyxq
[Taylor Eq 348]
Again the proof is left for Ch 5
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 42
Lecture 3 Slide 42
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Product of Two Variables
Consider the arbitrary derived quantity
q(xy) of two independent random variables x and y
Expand q(xy) in a Taylor series about the expected values of x and y
(ie at points near X and Y)
Error Propagation General Case
119902 119909 119910 = 119902 119883 119884 + 120597119902
120597119909
119883 119909 minus 119883 +
120597119902
120597119910
119884
(119910 minus 119884)
Fixed shifts peak of distribution
Fixed Distribution centered at X with width σX
120575119902 119909 119910 = 120590119902 = 119902 119883 119884 + 120597119902
120597119909
119883120590119909
2
+ 120597119902
120597119910
119884
120590119910
2
0
How do we determine the error of a derived quantity Z(XYhellip) from errors in XYhellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 43
Lecture 3 Slide 43
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
SDOM of Gaussian Distribution
Standard Deviation of the Mean
The SDOM decreases as the square root of the number of measurements
That is the relative width σẊ of the distribution gets narrower as more
measurements are made
Each measurement
has similar σXi= σẊ
and similar partial
derivatives
Thushellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 44
Lecture 3 Slide 44
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Two Key Theorems from Probability
Central Limit Theorem
For random independent measurements (each with a well-define
expectation value and well-defined variance) the arithmetic mean
(average) will be approximately normally distributed
Principle of Maximum Likelihood
Given the N observed measurements x1 x2hellipxN the best estimates
for Ẋ and σ are those values for which the observed x1 x2hellipxN are
most likely
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 45
Lecture 3 Slide 45
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Mean of Gaussian Distribution as ldquoBest Estimaterdquo
Principle of Maximum Likelihood
Best Estimate of X is from maximum probability or minimum summation
Consider a data set
Each randomly distributed with
x1 x2 x3 hellipxN
To find the most likely value of the mean (the best estimate of ẋ)
find X that yields the highest probability for the data set
The combined probability for the full data set is the product
Minimize
Sum
Solve for
derivative wrst
X set to 0
Best
estimate
of X
prop1
120590119890minus(1199091minus119883)2 2120590 times
1
120590119890minus(1199092minus119883)2 2120590 times hellip times
1
120590119890minus(119909119873minus119883)2 2120590 =
1
120590119873119890 minus(119909119894minus119883)2 2120590
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 46
Lecture 3 Slide 46
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Uncertainty of ldquoBest Estimatesrdquo of Gaussian Distribution Principle of Maximum Likelihood
Best Estimate of X is from maximum probability or minimum summation
Consider a data set x1 x2 x3 hellipxN
To find the most likely value of the standard deviation (the best estimate of
the width of the x distribution) find σx that yields the highest probability for
the data set
The combined probability for the full data set is the product
Minimize
Sum
Solve for
derivative
wrst X set to 0
Best
estimate
of X
Best Estimate of σ is from maximum probability or minimum summation
Minimize
Sum
Best
estimate
of σ
Solve for
derivative
wrst σ set to 0
See
Prob 526
prop1
120590119890minus(1199091minus119883)2 2120590 times
1
120590119890minus(1199092minus119883)2 2120590 times hellip times
1
120590119890minus(119909119873minus119883)2 2120590 =
1
120590119873119890 minus(119909119894minus119883)2 2120590
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 47
Lecture 3 Slide 47
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Combining Data Sets
Weighted Averages
References Taylor Ch 7
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 48
Lecture 3 Slide 48
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Weighted Averages
Question How can we properly combine two or more separate
independent measurements of the same randomly distributed
quantity to determine a best combined value with uncertainty
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 49
Lecture 3 Slide 49
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider two measurements of the same quantity described by a random Gaussian distribution
ltx1gt x1 and ltx2gt x2
The probability of measuring two such measurements is
119875119903119900119887119909 1199091 1199092 = 119875119903119900119887119909 1199091 119875119903119900119887119909 1199092
= 1
12059011205902119890minus12059422 119908ℎ119890119903119890 1205942 equiv
1199091 minus 119883
1205901 2 +
1199092 minus 119883
1205902 2
To find the best value for χ find the maximum Prob or minimum χ2
Weighted Averages
Assume negligible systematic errors
Note χ2 or Chi squared is the sum of the squares of the deviations from the mean divided by
the corresponding uncertainty
Such methods are called ldquoMethods of Least Squaresrdquo They follow directly from the Principle of
Maximum Likelihood
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 50
Lecture 3 Slide 50
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
The probability of measuring two such measurements is
119875119903119900119887119909 1199091 1199092 = 119875119903119900119887119909 1199091 119875119903119900119887119909 1199092
= 1
12059011205902119890minus12059422 119908ℎ119890119903119890 1205942 equiv
1199091 minus 119883
1205901 2 +
1199092 minus 119883
1205902 2
To find the best value for χ find the maximum Prob or minimum χ2
Weighted Averages
This leads to 119909119882_119886119907119892 = 11990811199091+11990821199092
1199081+1199082=
119908119894 119909119894
119908119894 119908ℎ119890119903119890 119908119894 = 1
120590119894 2
Best Estimate of χ is from maximum probibility or minimum summation
Minimize Sum Solve for best estimate of χ Solve for derivative wrst χ set to 0
Note If w1=w2 we recover the standard result Xwavg= (12) (x1+x2)
Finally the width of a weighted average distribution is
120590119908119894119890119892 ℎ119905119890119889 119886119907119892 = 1
119908119894119894
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 51
Lecture 3 Slide 51
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Weighted Averages on Steroids
A very powerful method for combining data from different sources
with different methods and uncertainties (or indeed data of
related measured and calculated quantities) is Kalman filtering
The Kalman filter also known as
linear quadratic estimation (LQE)
is an algorithm that uses a series
of measurements observed over
time containing noise (random
variations) and other
inaccuracies and produces
estimates of unknown variables
that tend to be more precise than
those based on a single
measurement alone
The Kalman filter keeps track of the estimated state of the system and the variance or uncertainty of the
estimate The estimate is updated using a state transition model and measurements xk|k-1 denotes the
estimate of the systems state at time step k before the kth measurement yk has been taken into account Pk|k-1
is the corresponding uncertainty --Wikipedia 2013
Ludger Scherliess of USU Physics is a world expert at using Kalman filtering for the assimilation
of satellite and ground-based data and the USU GAMES model to predict space weather
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 52
Lecture 3 Slide 52
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Rejecting Data
Chauvenetrsquos Criterion
References Taylor Ch 6
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 53
Lecture 3 Slide 53
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly
ldquounreasonablerdquo data point from a set of randomly distributed
measurements
bull Never
bull Whenever it makes things look better
bull Chauvenetrsquos criterion provides a (quantitative) compromise
Rejecting Data
What is a good criteria for
rejecting data
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 54
Lecture 3 Slide 54
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly
ldquounreasonablerdquo data point from a set of randomly distributed
measurements
bull Never
Rejecting Data
Zallenrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 55
Lecture 3 Slide 55
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly
ldquounreasonablerdquo data point from a set of randomly distributed
measurements
bull Whenever it makes things look better
Rejecting Data
Disneyrsquos Criterion
Disneyrsquos First Law
Wishing will make it so
Disneyrsquos Second Law
Dreams are more colorful than reality
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 56
Lecture 3 Slide 56
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Data may be rejected if the expected number of measurements at
least as deviant as the suspect measurement is less than 50
Consider a set of N measurements of a single quantity
x1x2 helliphellipxN
Calculate ltxgt and σx and then determine the fractional deviations from the mean of all
the points
For the suspect point(s) xsuspect find the probability of such a point occurring in N
measurements
n = (expected number as deviant as xsuspect)
= N Prob(outside xsuspectσx)
Rejecting Data
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 57
Lecture 3 Slide 57
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Function of Gaussian Distribution
Area under curve
(probability that
ndashtσltxlt+tσ) is given
in Table at right
Probable Error (probability that
ndash067σltxlt+067σ) is 50
Error Function (probability that ndashtσltxlt+tσ )
Error Function Tabulated values-see App A
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 58
Lecture 3 Slide 58
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 59
Lecture 3 Slide 59
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashExample 1
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 60
Lecture 3 Slide 60
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Details (1)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 61
Lecture 3 Slide 61
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashDetails (2)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 62
Lecture 3 Slide 62
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos
Criterionmdash
Details (3)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 63
Lecture 3 Slide 63
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos
Criterionmdash
Example 2
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 64
Lecture 3 Slide 64
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Summary of Probability Theory
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 65
Lecture 3 Slide 65
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of
Probability
Theory-I
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 66
Lecture 3 Slide 66
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of
Probability
Theory-II
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 67
Lecture 3 Slide 67
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-III
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 68
Lecture 3 Slide 68
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-IV
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 37
Lecture 3 Slide 37
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Analysis and Gaussian Distribution
Adding a Constant Multiplying by a Constant
XX+A XBX and σB σ
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 38
Lecture 3 Slide 38
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Sum of Two Variables
Consider the derived quantity
Z=X + Y
(with X=0 and Y=0)
Error Propagation Addition
Error in Z
Multiple two probabilities 119875119903119900119887 119909 119910 = 119875119903119900119887 119909 ∙ 119875119903119900119887 119909 prop 119890119909119901 minus
1
2
1199092
1205901199092+
1199102
1205901199102
prop 119890119909119901 minus1
2
(119909+119910)2
1205901199092+120590119910
2 minus1199112
prop 119890119909119901 minus1
2
(119909+119910)2
1205901199092+120590119910
2 119890119909119901 minus
1199112
2
ICBST (Eq 553)
X+YZ and σx2 + σy
2 σz
2
(addition in quadrature for random independent variables)
Integrates to radic2π
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 39
Lecture 3 Slide 39
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Error Propagation
General formula for error propagation see [Taylor Secs 35 and 39]
Uncertainty as a function of one variable [Taylor Sec 35]
1 Consider a graphical method of estimating error
a) Consider an arbitaray function q(x)
b) Plot q(x) vs x
c) On the graph label
(1) qbest = q(xbest)
(2) qhi = q(xbest + δx)
(3) qlow = q(xbest- δx)
d) Making a linear approximation
x
qqxslopeqq
x
qqxslopeqq
bestbestlow
bestbesthi
e) Therefore
xx
Note the absolute value
How do we determine the error of a derived quantity Z(XYhellip) from errors in XYhellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 40
Lecture 3 Slide 40
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Error Propagation
General formula for uncertainty of a function of one variable
xx
[Taylor Eq 323]
Can you now derive for specific rules of error propagation
1 Addition and Subtraction [Taylor p 49]
2 Multiplication and Division [Taylor p 51]
3 Multiplication by a constant (exact number) [Taylor p 54]
4 Exponentiation (powers) [Taylor p 56]
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 41
Lecture 3 Slide 41
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Multiple Variables
Uncertainty of a function of multiple variables [Taylor Sec 311]
1 It can easily (no really) be shown that (see Taylor Sec 311) for a
function of several variables
)(
z
z
qy
y
qx
x
qzyxq
[Taylor Eq 347]
2 More correctly it can be shown that (see Taylor Sec 311) for a
function of several variables
)(
z
z
qy
y
qx
x
qzyxq
[Taylor Eq 347]
where the equals sign represents an upper bound as discussed above
3 For a function of several independent and random variables
)(
222
z
z
qy
y
qx
x
qzyxq
[Taylor Eq 348]
Again the proof is left for Ch 5
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 42
Lecture 3 Slide 42
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Product of Two Variables
Consider the arbitrary derived quantity
q(xy) of two independent random variables x and y
Expand q(xy) in a Taylor series about the expected values of x and y
(ie at points near X and Y)
Error Propagation General Case
119902 119909 119910 = 119902 119883 119884 + 120597119902
120597119909
119883 119909 minus 119883 +
120597119902
120597119910
119884
(119910 minus 119884)
Fixed shifts peak of distribution
Fixed Distribution centered at X with width σX
120575119902 119909 119910 = 120590119902 = 119902 119883 119884 + 120597119902
120597119909
119883120590119909
2
+ 120597119902
120597119910
119884
120590119910
2
0
How do we determine the error of a derived quantity Z(XYhellip) from errors in XYhellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 43
Lecture 3 Slide 43
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
SDOM of Gaussian Distribution
Standard Deviation of the Mean
The SDOM decreases as the square root of the number of measurements
That is the relative width σẊ of the distribution gets narrower as more
measurements are made
Each measurement
has similar σXi= σẊ
and similar partial
derivatives
Thushellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 44
Lecture 3 Slide 44
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Two Key Theorems from Probability
Central Limit Theorem
For random independent measurements (each with a well-define
expectation value and well-defined variance) the arithmetic mean
(average) will be approximately normally distributed
Principle of Maximum Likelihood
Given the N observed measurements x1 x2hellipxN the best estimates
for Ẋ and σ are those values for which the observed x1 x2hellipxN are
most likely
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 45
Lecture 3 Slide 45
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Mean of Gaussian Distribution as ldquoBest Estimaterdquo
Principle of Maximum Likelihood
Best Estimate of X is from maximum probability or minimum summation
Consider a data set
Each randomly distributed with
x1 x2 x3 hellipxN
To find the most likely value of the mean (the best estimate of ẋ)
find X that yields the highest probability for the data set
The combined probability for the full data set is the product
Minimize
Sum
Solve for
derivative wrst
X set to 0
Best
estimate
of X
prop1
120590119890minus(1199091minus119883)2 2120590 times
1
120590119890minus(1199092minus119883)2 2120590 times hellip times
1
120590119890minus(119909119873minus119883)2 2120590 =
1
120590119873119890 minus(119909119894minus119883)2 2120590
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 46
Lecture 3 Slide 46
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Uncertainty of ldquoBest Estimatesrdquo of Gaussian Distribution Principle of Maximum Likelihood
Best Estimate of X is from maximum probability or minimum summation
Consider a data set x1 x2 x3 hellipxN
To find the most likely value of the standard deviation (the best estimate of
the width of the x distribution) find σx that yields the highest probability for
the data set
The combined probability for the full data set is the product
Minimize
Sum
Solve for
derivative
wrst X set to 0
Best
estimate
of X
Best Estimate of σ is from maximum probability or minimum summation
Minimize
Sum
Best
estimate
of σ
Solve for
derivative
wrst σ set to 0
See
Prob 526
prop1
120590119890minus(1199091minus119883)2 2120590 times
1
120590119890minus(1199092minus119883)2 2120590 times hellip times
1
120590119890minus(119909119873minus119883)2 2120590 =
1
120590119873119890 minus(119909119894minus119883)2 2120590
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 47
Lecture 3 Slide 47
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Combining Data Sets
Weighted Averages
References Taylor Ch 7
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 48
Lecture 3 Slide 48
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Weighted Averages
Question How can we properly combine two or more separate
independent measurements of the same randomly distributed
quantity to determine a best combined value with uncertainty
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 49
Lecture 3 Slide 49
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider two measurements of the same quantity described by a random Gaussian distribution
ltx1gt x1 and ltx2gt x2
The probability of measuring two such measurements is
119875119903119900119887119909 1199091 1199092 = 119875119903119900119887119909 1199091 119875119903119900119887119909 1199092
= 1
12059011205902119890minus12059422 119908ℎ119890119903119890 1205942 equiv
1199091 minus 119883
1205901 2 +
1199092 minus 119883
1205902 2
To find the best value for χ find the maximum Prob or minimum χ2
Weighted Averages
Assume negligible systematic errors
Note χ2 or Chi squared is the sum of the squares of the deviations from the mean divided by
the corresponding uncertainty
Such methods are called ldquoMethods of Least Squaresrdquo They follow directly from the Principle of
Maximum Likelihood
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 50
Lecture 3 Slide 50
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
The probability of measuring two such measurements is
119875119903119900119887119909 1199091 1199092 = 119875119903119900119887119909 1199091 119875119903119900119887119909 1199092
= 1
12059011205902119890minus12059422 119908ℎ119890119903119890 1205942 equiv
1199091 minus 119883
1205901 2 +
1199092 minus 119883
1205902 2
To find the best value for χ find the maximum Prob or minimum χ2
Weighted Averages
This leads to 119909119882_119886119907119892 = 11990811199091+11990821199092
1199081+1199082=
119908119894 119909119894
119908119894 119908ℎ119890119903119890 119908119894 = 1
120590119894 2
Best Estimate of χ is from maximum probibility or minimum summation
Minimize Sum Solve for best estimate of χ Solve for derivative wrst χ set to 0
Note If w1=w2 we recover the standard result Xwavg= (12) (x1+x2)
Finally the width of a weighted average distribution is
120590119908119894119890119892 ℎ119905119890119889 119886119907119892 = 1
119908119894119894
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 51
Lecture 3 Slide 51
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Weighted Averages on Steroids
A very powerful method for combining data from different sources
with different methods and uncertainties (or indeed data of
related measured and calculated quantities) is Kalman filtering
The Kalman filter also known as
linear quadratic estimation (LQE)
is an algorithm that uses a series
of measurements observed over
time containing noise (random
variations) and other
inaccuracies and produces
estimates of unknown variables
that tend to be more precise than
those based on a single
measurement alone
The Kalman filter keeps track of the estimated state of the system and the variance or uncertainty of the
estimate The estimate is updated using a state transition model and measurements xk|k-1 denotes the
estimate of the systems state at time step k before the kth measurement yk has been taken into account Pk|k-1
is the corresponding uncertainty --Wikipedia 2013
Ludger Scherliess of USU Physics is a world expert at using Kalman filtering for the assimilation
of satellite and ground-based data and the USU GAMES model to predict space weather
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 52
Lecture 3 Slide 52
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Rejecting Data
Chauvenetrsquos Criterion
References Taylor Ch 6
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 53
Lecture 3 Slide 53
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly
ldquounreasonablerdquo data point from a set of randomly distributed
measurements
bull Never
bull Whenever it makes things look better
bull Chauvenetrsquos criterion provides a (quantitative) compromise
Rejecting Data
What is a good criteria for
rejecting data
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 54
Lecture 3 Slide 54
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly
ldquounreasonablerdquo data point from a set of randomly distributed
measurements
bull Never
Rejecting Data
Zallenrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 55
Lecture 3 Slide 55
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly
ldquounreasonablerdquo data point from a set of randomly distributed
measurements
bull Whenever it makes things look better
Rejecting Data
Disneyrsquos Criterion
Disneyrsquos First Law
Wishing will make it so
Disneyrsquos Second Law
Dreams are more colorful than reality
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 56
Lecture 3 Slide 56
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Data may be rejected if the expected number of measurements at
least as deviant as the suspect measurement is less than 50
Consider a set of N measurements of a single quantity
x1x2 helliphellipxN
Calculate ltxgt and σx and then determine the fractional deviations from the mean of all
the points
For the suspect point(s) xsuspect find the probability of such a point occurring in N
measurements
n = (expected number as deviant as xsuspect)
= N Prob(outside xsuspectσx)
Rejecting Data
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 57
Lecture 3 Slide 57
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Function of Gaussian Distribution
Area under curve
(probability that
ndashtσltxlt+tσ) is given
in Table at right
Probable Error (probability that
ndash067σltxlt+067σ) is 50
Error Function (probability that ndashtσltxlt+tσ )
Error Function Tabulated values-see App A
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 58
Lecture 3 Slide 58
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 59
Lecture 3 Slide 59
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashExample 1
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 60
Lecture 3 Slide 60
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Details (1)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 61
Lecture 3 Slide 61
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashDetails (2)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 62
Lecture 3 Slide 62
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos
Criterionmdash
Details (3)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 63
Lecture 3 Slide 63
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos
Criterionmdash
Example 2
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 64
Lecture 3 Slide 64
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Summary of Probability Theory
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 65
Lecture 3 Slide 65
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of
Probability
Theory-I
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 66
Lecture 3 Slide 66
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of
Probability
Theory-II
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 67
Lecture 3 Slide 67
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-III
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 68
Lecture 3 Slide 68
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-IV
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 38
Lecture 3 Slide 38
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Sum of Two Variables
Consider the derived quantity
Z=X + Y
(with X=0 and Y=0)
Error Propagation Addition
Error in Z
Multiple two probabilities 119875119903119900119887 119909 119910 = 119875119903119900119887 119909 ∙ 119875119903119900119887 119909 prop 119890119909119901 minus
1
2
1199092
1205901199092+
1199102
1205901199102
prop 119890119909119901 minus1
2
(119909+119910)2
1205901199092+120590119910
2 minus1199112
prop 119890119909119901 minus1
2
(119909+119910)2
1205901199092+120590119910
2 119890119909119901 minus
1199112
2
ICBST (Eq 553)
X+YZ and σx2 + σy
2 σz
2
(addition in quadrature for random independent variables)
Integrates to radic2π
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 39
Lecture 3 Slide 39
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Error Propagation
General formula for error propagation see [Taylor Secs 35 and 39]
Uncertainty as a function of one variable [Taylor Sec 35]
1 Consider a graphical method of estimating error
a) Consider an arbitaray function q(x)
b) Plot q(x) vs x
c) On the graph label
(1) qbest = q(xbest)
(2) qhi = q(xbest + δx)
(3) qlow = q(xbest- δx)
d) Making a linear approximation
x
qqxslopeqq
x
qqxslopeqq
bestbestlow
bestbesthi
e) Therefore
xx
Note the absolute value
How do we determine the error of a derived quantity Z(XYhellip) from errors in XYhellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 40
Lecture 3 Slide 40
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Error Propagation
General formula for uncertainty of a function of one variable
xx
[Taylor Eq 323]
Can you now derive for specific rules of error propagation
1 Addition and Subtraction [Taylor p 49]
2 Multiplication and Division [Taylor p 51]
3 Multiplication by a constant (exact number) [Taylor p 54]
4 Exponentiation (powers) [Taylor p 56]
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 41
Lecture 3 Slide 41
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Multiple Variables
Uncertainty of a function of multiple variables [Taylor Sec 311]
1 It can easily (no really) be shown that (see Taylor Sec 311) for a
function of several variables
)(
z
z
qy
y
qx
x
qzyxq
[Taylor Eq 347]
2 More correctly it can be shown that (see Taylor Sec 311) for a
function of several variables
)(
z
z
qy
y
qx
x
qzyxq
[Taylor Eq 347]
where the equals sign represents an upper bound as discussed above
3 For a function of several independent and random variables
)(
222
z
z
qy
y
qx
x
qzyxq
[Taylor Eq 348]
Again the proof is left for Ch 5
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 42
Lecture 3 Slide 42
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Product of Two Variables
Consider the arbitrary derived quantity
q(xy) of two independent random variables x and y
Expand q(xy) in a Taylor series about the expected values of x and y
(ie at points near X and Y)
Error Propagation General Case
119902 119909 119910 = 119902 119883 119884 + 120597119902
120597119909
119883 119909 minus 119883 +
120597119902
120597119910
119884
(119910 minus 119884)
Fixed shifts peak of distribution
Fixed Distribution centered at X with width σX
120575119902 119909 119910 = 120590119902 = 119902 119883 119884 + 120597119902
120597119909
119883120590119909
2
+ 120597119902
120597119910
119884
120590119910
2
0
How do we determine the error of a derived quantity Z(XYhellip) from errors in XYhellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 43
Lecture 3 Slide 43
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
SDOM of Gaussian Distribution
Standard Deviation of the Mean
The SDOM decreases as the square root of the number of measurements
That is the relative width σẊ of the distribution gets narrower as more
measurements are made
Each measurement
has similar σXi= σẊ
and similar partial
derivatives
Thushellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 44
Lecture 3 Slide 44
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Two Key Theorems from Probability
Central Limit Theorem
For random independent measurements (each with a well-define
expectation value and well-defined variance) the arithmetic mean
(average) will be approximately normally distributed
Principle of Maximum Likelihood
Given the N observed measurements x1 x2hellipxN the best estimates
for Ẋ and σ are those values for which the observed x1 x2hellipxN are
most likely
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 45
Lecture 3 Slide 45
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Mean of Gaussian Distribution as ldquoBest Estimaterdquo
Principle of Maximum Likelihood
Best Estimate of X is from maximum probability or minimum summation
Consider a data set
Each randomly distributed with
x1 x2 x3 hellipxN
To find the most likely value of the mean (the best estimate of ẋ)
find X that yields the highest probability for the data set
The combined probability for the full data set is the product
Minimize
Sum
Solve for
derivative wrst
X set to 0
Best
estimate
of X
prop1
120590119890minus(1199091minus119883)2 2120590 times
1
120590119890minus(1199092minus119883)2 2120590 times hellip times
1
120590119890minus(119909119873minus119883)2 2120590 =
1
120590119873119890 minus(119909119894minus119883)2 2120590
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 46
Lecture 3 Slide 46
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Uncertainty of ldquoBest Estimatesrdquo of Gaussian Distribution Principle of Maximum Likelihood
Best Estimate of X is from maximum probability or minimum summation
Consider a data set x1 x2 x3 hellipxN
To find the most likely value of the standard deviation (the best estimate of
the width of the x distribution) find σx that yields the highest probability for
the data set
The combined probability for the full data set is the product
Minimize
Sum
Solve for
derivative
wrst X set to 0
Best
estimate
of X
Best Estimate of σ is from maximum probability or minimum summation
Minimize
Sum
Best
estimate
of σ
Solve for
derivative
wrst σ set to 0
See
Prob 526
prop1
120590119890minus(1199091minus119883)2 2120590 times
1
120590119890minus(1199092minus119883)2 2120590 times hellip times
1
120590119890minus(119909119873minus119883)2 2120590 =
1
120590119873119890 minus(119909119894minus119883)2 2120590
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 47
Lecture 3 Slide 47
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Combining Data Sets
Weighted Averages
References Taylor Ch 7
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 48
Lecture 3 Slide 48
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Weighted Averages
Question How can we properly combine two or more separate
independent measurements of the same randomly distributed
quantity to determine a best combined value with uncertainty
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 49
Lecture 3 Slide 49
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider two measurements of the same quantity described by a random Gaussian distribution
ltx1gt x1 and ltx2gt x2
The probability of measuring two such measurements is
119875119903119900119887119909 1199091 1199092 = 119875119903119900119887119909 1199091 119875119903119900119887119909 1199092
= 1
12059011205902119890minus12059422 119908ℎ119890119903119890 1205942 equiv
1199091 minus 119883
1205901 2 +
1199092 minus 119883
1205902 2
To find the best value for χ find the maximum Prob or minimum χ2
Weighted Averages
Assume negligible systematic errors
Note χ2 or Chi squared is the sum of the squares of the deviations from the mean divided by
the corresponding uncertainty
Such methods are called ldquoMethods of Least Squaresrdquo They follow directly from the Principle of
Maximum Likelihood
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 50
Lecture 3 Slide 50
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
The probability of measuring two such measurements is
119875119903119900119887119909 1199091 1199092 = 119875119903119900119887119909 1199091 119875119903119900119887119909 1199092
= 1
12059011205902119890minus12059422 119908ℎ119890119903119890 1205942 equiv
1199091 minus 119883
1205901 2 +
1199092 minus 119883
1205902 2
To find the best value for χ find the maximum Prob or minimum χ2
Weighted Averages
This leads to 119909119882_119886119907119892 = 11990811199091+11990821199092
1199081+1199082=
119908119894 119909119894
119908119894 119908ℎ119890119903119890 119908119894 = 1
120590119894 2
Best Estimate of χ is from maximum probibility or minimum summation
Minimize Sum Solve for best estimate of χ Solve for derivative wrst χ set to 0
Note If w1=w2 we recover the standard result Xwavg= (12) (x1+x2)
Finally the width of a weighted average distribution is
120590119908119894119890119892 ℎ119905119890119889 119886119907119892 = 1
119908119894119894
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 51
Lecture 3 Slide 51
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Weighted Averages on Steroids
A very powerful method for combining data from different sources
with different methods and uncertainties (or indeed data of
related measured and calculated quantities) is Kalman filtering
The Kalman filter also known as
linear quadratic estimation (LQE)
is an algorithm that uses a series
of measurements observed over
time containing noise (random
variations) and other
inaccuracies and produces
estimates of unknown variables
that tend to be more precise than
those based on a single
measurement alone
The Kalman filter keeps track of the estimated state of the system and the variance or uncertainty of the
estimate The estimate is updated using a state transition model and measurements xk|k-1 denotes the
estimate of the systems state at time step k before the kth measurement yk has been taken into account Pk|k-1
is the corresponding uncertainty --Wikipedia 2013
Ludger Scherliess of USU Physics is a world expert at using Kalman filtering for the assimilation
of satellite and ground-based data and the USU GAMES model to predict space weather
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 52
Lecture 3 Slide 52
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Rejecting Data
Chauvenetrsquos Criterion
References Taylor Ch 6
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 53
Lecture 3 Slide 53
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly
ldquounreasonablerdquo data point from a set of randomly distributed
measurements
bull Never
bull Whenever it makes things look better
bull Chauvenetrsquos criterion provides a (quantitative) compromise
Rejecting Data
What is a good criteria for
rejecting data
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 54
Lecture 3 Slide 54
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly
ldquounreasonablerdquo data point from a set of randomly distributed
measurements
bull Never
Rejecting Data
Zallenrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 55
Lecture 3 Slide 55
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly
ldquounreasonablerdquo data point from a set of randomly distributed
measurements
bull Whenever it makes things look better
Rejecting Data
Disneyrsquos Criterion
Disneyrsquos First Law
Wishing will make it so
Disneyrsquos Second Law
Dreams are more colorful than reality
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 56
Lecture 3 Slide 56
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Data may be rejected if the expected number of measurements at
least as deviant as the suspect measurement is less than 50
Consider a set of N measurements of a single quantity
x1x2 helliphellipxN
Calculate ltxgt and σx and then determine the fractional deviations from the mean of all
the points
For the suspect point(s) xsuspect find the probability of such a point occurring in N
measurements
n = (expected number as deviant as xsuspect)
= N Prob(outside xsuspectσx)
Rejecting Data
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 57
Lecture 3 Slide 57
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Function of Gaussian Distribution
Area under curve
(probability that
ndashtσltxlt+tσ) is given
in Table at right
Probable Error (probability that
ndash067σltxlt+067σ) is 50
Error Function (probability that ndashtσltxlt+tσ )
Error Function Tabulated values-see App A
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 58
Lecture 3 Slide 58
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 59
Lecture 3 Slide 59
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashExample 1
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 60
Lecture 3 Slide 60
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Details (1)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 61
Lecture 3 Slide 61
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashDetails (2)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 62
Lecture 3 Slide 62
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos
Criterionmdash
Details (3)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 63
Lecture 3 Slide 63
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos
Criterionmdash
Example 2
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 64
Lecture 3 Slide 64
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Summary of Probability Theory
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 65
Lecture 3 Slide 65
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of
Probability
Theory-I
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 66
Lecture 3 Slide 66
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of
Probability
Theory-II
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 67
Lecture 3 Slide 67
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-III
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 68
Lecture 3 Slide 68
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-IV
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 39
Lecture 3 Slide 39
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Error Propagation
General formula for error propagation see [Taylor Secs 35 and 39]
Uncertainty as a function of one variable [Taylor Sec 35]
1 Consider a graphical method of estimating error
a) Consider an arbitaray function q(x)
b) Plot q(x) vs x
c) On the graph label
(1) qbest = q(xbest)
(2) qhi = q(xbest + δx)
(3) qlow = q(xbest- δx)
d) Making a linear approximation
x
qqxslopeqq
x
qqxslopeqq
bestbestlow
bestbesthi
e) Therefore
xx
Note the absolute value
How do we determine the error of a derived quantity Z(XYhellip) from errors in XYhellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 40
Lecture 3 Slide 40
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Error Propagation
General formula for uncertainty of a function of one variable
xx
[Taylor Eq 323]
Can you now derive for specific rules of error propagation
1 Addition and Subtraction [Taylor p 49]
2 Multiplication and Division [Taylor p 51]
3 Multiplication by a constant (exact number) [Taylor p 54]
4 Exponentiation (powers) [Taylor p 56]
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 41
Lecture 3 Slide 41
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Multiple Variables
Uncertainty of a function of multiple variables [Taylor Sec 311]
1 It can easily (no really) be shown that (see Taylor Sec 311) for a
function of several variables
)(
z
z
qy
y
qx
x
qzyxq
[Taylor Eq 347]
2 More correctly it can be shown that (see Taylor Sec 311) for a
function of several variables
)(
z
z
qy
y
qx
x
qzyxq
[Taylor Eq 347]
where the equals sign represents an upper bound as discussed above
3 For a function of several independent and random variables
)(
222
z
z
qy
y
qx
x
qzyxq
[Taylor Eq 348]
Again the proof is left for Ch 5
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 42
Lecture 3 Slide 42
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Product of Two Variables
Consider the arbitrary derived quantity
q(xy) of two independent random variables x and y
Expand q(xy) in a Taylor series about the expected values of x and y
(ie at points near X and Y)
Error Propagation General Case
119902 119909 119910 = 119902 119883 119884 + 120597119902
120597119909
119883 119909 minus 119883 +
120597119902
120597119910
119884
(119910 minus 119884)
Fixed shifts peak of distribution
Fixed Distribution centered at X with width σX
120575119902 119909 119910 = 120590119902 = 119902 119883 119884 + 120597119902
120597119909
119883120590119909
2
+ 120597119902
120597119910
119884
120590119910
2
0
How do we determine the error of a derived quantity Z(XYhellip) from errors in XYhellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 43
Lecture 3 Slide 43
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
SDOM of Gaussian Distribution
Standard Deviation of the Mean
The SDOM decreases as the square root of the number of measurements
That is the relative width σẊ of the distribution gets narrower as more
measurements are made
Each measurement
has similar σXi= σẊ
and similar partial
derivatives
Thushellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 44
Lecture 3 Slide 44
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Two Key Theorems from Probability
Central Limit Theorem
For random independent measurements (each with a well-define
expectation value and well-defined variance) the arithmetic mean
(average) will be approximately normally distributed
Principle of Maximum Likelihood
Given the N observed measurements x1 x2hellipxN the best estimates
for Ẋ and σ are those values for which the observed x1 x2hellipxN are
most likely
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 45
Lecture 3 Slide 45
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Mean of Gaussian Distribution as ldquoBest Estimaterdquo
Principle of Maximum Likelihood
Best Estimate of X is from maximum probability or minimum summation
Consider a data set
Each randomly distributed with
x1 x2 x3 hellipxN
To find the most likely value of the mean (the best estimate of ẋ)
find X that yields the highest probability for the data set
The combined probability for the full data set is the product
Minimize
Sum
Solve for
derivative wrst
X set to 0
Best
estimate
of X
prop1
120590119890minus(1199091minus119883)2 2120590 times
1
120590119890minus(1199092minus119883)2 2120590 times hellip times
1
120590119890minus(119909119873minus119883)2 2120590 =
1
120590119873119890 minus(119909119894minus119883)2 2120590
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 46
Lecture 3 Slide 46
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Uncertainty of ldquoBest Estimatesrdquo of Gaussian Distribution Principle of Maximum Likelihood
Best Estimate of X is from maximum probability or minimum summation
Consider a data set x1 x2 x3 hellipxN
To find the most likely value of the standard deviation (the best estimate of
the width of the x distribution) find σx that yields the highest probability for
the data set
The combined probability for the full data set is the product
Minimize
Sum
Solve for
derivative
wrst X set to 0
Best
estimate
of X
Best Estimate of σ is from maximum probability or minimum summation
Minimize
Sum
Best
estimate
of σ
Solve for
derivative
wrst σ set to 0
See
Prob 526
prop1
120590119890minus(1199091minus119883)2 2120590 times
1
120590119890minus(1199092minus119883)2 2120590 times hellip times
1
120590119890minus(119909119873minus119883)2 2120590 =
1
120590119873119890 minus(119909119894minus119883)2 2120590
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 47
Lecture 3 Slide 47
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Combining Data Sets
Weighted Averages
References Taylor Ch 7
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 48
Lecture 3 Slide 48
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Weighted Averages
Question How can we properly combine two or more separate
independent measurements of the same randomly distributed
quantity to determine a best combined value with uncertainty
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 49
Lecture 3 Slide 49
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider two measurements of the same quantity described by a random Gaussian distribution
ltx1gt x1 and ltx2gt x2
The probability of measuring two such measurements is
119875119903119900119887119909 1199091 1199092 = 119875119903119900119887119909 1199091 119875119903119900119887119909 1199092
= 1
12059011205902119890minus12059422 119908ℎ119890119903119890 1205942 equiv
1199091 minus 119883
1205901 2 +
1199092 minus 119883
1205902 2
To find the best value for χ find the maximum Prob or minimum χ2
Weighted Averages
Assume negligible systematic errors
Note χ2 or Chi squared is the sum of the squares of the deviations from the mean divided by
the corresponding uncertainty
Such methods are called ldquoMethods of Least Squaresrdquo They follow directly from the Principle of
Maximum Likelihood
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 50
Lecture 3 Slide 50
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
The probability of measuring two such measurements is
119875119903119900119887119909 1199091 1199092 = 119875119903119900119887119909 1199091 119875119903119900119887119909 1199092
= 1
12059011205902119890minus12059422 119908ℎ119890119903119890 1205942 equiv
1199091 minus 119883
1205901 2 +
1199092 minus 119883
1205902 2
To find the best value for χ find the maximum Prob or minimum χ2
Weighted Averages
This leads to 119909119882_119886119907119892 = 11990811199091+11990821199092
1199081+1199082=
119908119894 119909119894
119908119894 119908ℎ119890119903119890 119908119894 = 1
120590119894 2
Best Estimate of χ is from maximum probibility or minimum summation
Minimize Sum Solve for best estimate of χ Solve for derivative wrst χ set to 0
Note If w1=w2 we recover the standard result Xwavg= (12) (x1+x2)
Finally the width of a weighted average distribution is
120590119908119894119890119892 ℎ119905119890119889 119886119907119892 = 1
119908119894119894
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 51
Lecture 3 Slide 51
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Weighted Averages on Steroids
A very powerful method for combining data from different sources
with different methods and uncertainties (or indeed data of
related measured and calculated quantities) is Kalman filtering
The Kalman filter also known as
linear quadratic estimation (LQE)
is an algorithm that uses a series
of measurements observed over
time containing noise (random
variations) and other
inaccuracies and produces
estimates of unknown variables
that tend to be more precise than
those based on a single
measurement alone
The Kalman filter keeps track of the estimated state of the system and the variance or uncertainty of the
estimate The estimate is updated using a state transition model and measurements xk|k-1 denotes the
estimate of the systems state at time step k before the kth measurement yk has been taken into account Pk|k-1
is the corresponding uncertainty --Wikipedia 2013
Ludger Scherliess of USU Physics is a world expert at using Kalman filtering for the assimilation
of satellite and ground-based data and the USU GAMES model to predict space weather
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 52
Lecture 3 Slide 52
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Rejecting Data
Chauvenetrsquos Criterion
References Taylor Ch 6
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 53
Lecture 3 Slide 53
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly
ldquounreasonablerdquo data point from a set of randomly distributed
measurements
bull Never
bull Whenever it makes things look better
bull Chauvenetrsquos criterion provides a (quantitative) compromise
Rejecting Data
What is a good criteria for
rejecting data
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 54
Lecture 3 Slide 54
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly
ldquounreasonablerdquo data point from a set of randomly distributed
measurements
bull Never
Rejecting Data
Zallenrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 55
Lecture 3 Slide 55
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly
ldquounreasonablerdquo data point from a set of randomly distributed
measurements
bull Whenever it makes things look better
Rejecting Data
Disneyrsquos Criterion
Disneyrsquos First Law
Wishing will make it so
Disneyrsquos Second Law
Dreams are more colorful than reality
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 56
Lecture 3 Slide 56
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Data may be rejected if the expected number of measurements at
least as deviant as the suspect measurement is less than 50
Consider a set of N measurements of a single quantity
x1x2 helliphellipxN
Calculate ltxgt and σx and then determine the fractional deviations from the mean of all
the points
For the suspect point(s) xsuspect find the probability of such a point occurring in N
measurements
n = (expected number as deviant as xsuspect)
= N Prob(outside xsuspectσx)
Rejecting Data
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 57
Lecture 3 Slide 57
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Function of Gaussian Distribution
Area under curve
(probability that
ndashtσltxlt+tσ) is given
in Table at right
Probable Error (probability that
ndash067σltxlt+067σ) is 50
Error Function (probability that ndashtσltxlt+tσ )
Error Function Tabulated values-see App A
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 58
Lecture 3 Slide 58
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 59
Lecture 3 Slide 59
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashExample 1
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 60
Lecture 3 Slide 60
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Details (1)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 61
Lecture 3 Slide 61
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashDetails (2)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 62
Lecture 3 Slide 62
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos
Criterionmdash
Details (3)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 63
Lecture 3 Slide 63
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos
Criterionmdash
Example 2
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 64
Lecture 3 Slide 64
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Summary of Probability Theory
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 65
Lecture 3 Slide 65
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of
Probability
Theory-I
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 66
Lecture 3 Slide 66
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of
Probability
Theory-II
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 67
Lecture 3 Slide 67
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-III
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 68
Lecture 3 Slide 68
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-IV
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 40
Lecture 3 Slide 40
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Error Propagation
General formula for uncertainty of a function of one variable
xx
[Taylor Eq 323]
Can you now derive for specific rules of error propagation
1 Addition and Subtraction [Taylor p 49]
2 Multiplication and Division [Taylor p 51]
3 Multiplication by a constant (exact number) [Taylor p 54]
4 Exponentiation (powers) [Taylor p 56]
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 41
Lecture 3 Slide 41
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Multiple Variables
Uncertainty of a function of multiple variables [Taylor Sec 311]
1 It can easily (no really) be shown that (see Taylor Sec 311) for a
function of several variables
)(
z
z
qy
y
qx
x
qzyxq
[Taylor Eq 347]
2 More correctly it can be shown that (see Taylor Sec 311) for a
function of several variables
)(
z
z
qy
y
qx
x
qzyxq
[Taylor Eq 347]
where the equals sign represents an upper bound as discussed above
3 For a function of several independent and random variables
)(
222
z
z
qy
y
qx
x
qzyxq
[Taylor Eq 348]
Again the proof is left for Ch 5
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 42
Lecture 3 Slide 42
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Product of Two Variables
Consider the arbitrary derived quantity
q(xy) of two independent random variables x and y
Expand q(xy) in a Taylor series about the expected values of x and y
(ie at points near X and Y)
Error Propagation General Case
119902 119909 119910 = 119902 119883 119884 + 120597119902
120597119909
119883 119909 minus 119883 +
120597119902
120597119910
119884
(119910 minus 119884)
Fixed shifts peak of distribution
Fixed Distribution centered at X with width σX
120575119902 119909 119910 = 120590119902 = 119902 119883 119884 + 120597119902
120597119909
119883120590119909
2
+ 120597119902
120597119910
119884
120590119910
2
0
How do we determine the error of a derived quantity Z(XYhellip) from errors in XYhellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 43
Lecture 3 Slide 43
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
SDOM of Gaussian Distribution
Standard Deviation of the Mean
The SDOM decreases as the square root of the number of measurements
That is the relative width σẊ of the distribution gets narrower as more
measurements are made
Each measurement
has similar σXi= σẊ
and similar partial
derivatives
Thushellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 44
Lecture 3 Slide 44
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Two Key Theorems from Probability
Central Limit Theorem
For random independent measurements (each with a well-define
expectation value and well-defined variance) the arithmetic mean
(average) will be approximately normally distributed
Principle of Maximum Likelihood
Given the N observed measurements x1 x2hellipxN the best estimates
for Ẋ and σ are those values for which the observed x1 x2hellipxN are
most likely
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 45
Lecture 3 Slide 45
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Mean of Gaussian Distribution as ldquoBest Estimaterdquo
Principle of Maximum Likelihood
Best Estimate of X is from maximum probability or minimum summation
Consider a data set
Each randomly distributed with
x1 x2 x3 hellipxN
To find the most likely value of the mean (the best estimate of ẋ)
find X that yields the highest probability for the data set
The combined probability for the full data set is the product
Minimize
Sum
Solve for
derivative wrst
X set to 0
Best
estimate
of X
prop1
120590119890minus(1199091minus119883)2 2120590 times
1
120590119890minus(1199092minus119883)2 2120590 times hellip times
1
120590119890minus(119909119873minus119883)2 2120590 =
1
120590119873119890 minus(119909119894minus119883)2 2120590
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 46
Lecture 3 Slide 46
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Uncertainty of ldquoBest Estimatesrdquo of Gaussian Distribution Principle of Maximum Likelihood
Best Estimate of X is from maximum probability or minimum summation
Consider a data set x1 x2 x3 hellipxN
To find the most likely value of the standard deviation (the best estimate of
the width of the x distribution) find σx that yields the highest probability for
the data set
The combined probability for the full data set is the product
Minimize
Sum
Solve for
derivative
wrst X set to 0
Best
estimate
of X
Best Estimate of σ is from maximum probability or minimum summation
Minimize
Sum
Best
estimate
of σ
Solve for
derivative
wrst σ set to 0
See
Prob 526
prop1
120590119890minus(1199091minus119883)2 2120590 times
1
120590119890minus(1199092minus119883)2 2120590 times hellip times
1
120590119890minus(119909119873minus119883)2 2120590 =
1
120590119873119890 minus(119909119894minus119883)2 2120590
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 47
Lecture 3 Slide 47
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Combining Data Sets
Weighted Averages
References Taylor Ch 7
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 48
Lecture 3 Slide 48
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Weighted Averages
Question How can we properly combine two or more separate
independent measurements of the same randomly distributed
quantity to determine a best combined value with uncertainty
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 49
Lecture 3 Slide 49
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider two measurements of the same quantity described by a random Gaussian distribution
ltx1gt x1 and ltx2gt x2
The probability of measuring two such measurements is
119875119903119900119887119909 1199091 1199092 = 119875119903119900119887119909 1199091 119875119903119900119887119909 1199092
= 1
12059011205902119890minus12059422 119908ℎ119890119903119890 1205942 equiv
1199091 minus 119883
1205901 2 +
1199092 minus 119883
1205902 2
To find the best value for χ find the maximum Prob or minimum χ2
Weighted Averages
Assume negligible systematic errors
Note χ2 or Chi squared is the sum of the squares of the deviations from the mean divided by
the corresponding uncertainty
Such methods are called ldquoMethods of Least Squaresrdquo They follow directly from the Principle of
Maximum Likelihood
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 50
Lecture 3 Slide 50
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
The probability of measuring two such measurements is
119875119903119900119887119909 1199091 1199092 = 119875119903119900119887119909 1199091 119875119903119900119887119909 1199092
= 1
12059011205902119890minus12059422 119908ℎ119890119903119890 1205942 equiv
1199091 minus 119883
1205901 2 +
1199092 minus 119883
1205902 2
To find the best value for χ find the maximum Prob or minimum χ2
Weighted Averages
This leads to 119909119882_119886119907119892 = 11990811199091+11990821199092
1199081+1199082=
119908119894 119909119894
119908119894 119908ℎ119890119903119890 119908119894 = 1
120590119894 2
Best Estimate of χ is from maximum probibility or minimum summation
Minimize Sum Solve for best estimate of χ Solve for derivative wrst χ set to 0
Note If w1=w2 we recover the standard result Xwavg= (12) (x1+x2)
Finally the width of a weighted average distribution is
120590119908119894119890119892 ℎ119905119890119889 119886119907119892 = 1
119908119894119894
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 51
Lecture 3 Slide 51
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Weighted Averages on Steroids
A very powerful method for combining data from different sources
with different methods and uncertainties (or indeed data of
related measured and calculated quantities) is Kalman filtering
The Kalman filter also known as
linear quadratic estimation (LQE)
is an algorithm that uses a series
of measurements observed over
time containing noise (random
variations) and other
inaccuracies and produces
estimates of unknown variables
that tend to be more precise than
those based on a single
measurement alone
The Kalman filter keeps track of the estimated state of the system and the variance or uncertainty of the
estimate The estimate is updated using a state transition model and measurements xk|k-1 denotes the
estimate of the systems state at time step k before the kth measurement yk has been taken into account Pk|k-1
is the corresponding uncertainty --Wikipedia 2013
Ludger Scherliess of USU Physics is a world expert at using Kalman filtering for the assimilation
of satellite and ground-based data and the USU GAMES model to predict space weather
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 52
Lecture 3 Slide 52
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Rejecting Data
Chauvenetrsquos Criterion
References Taylor Ch 6
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 53
Lecture 3 Slide 53
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly
ldquounreasonablerdquo data point from a set of randomly distributed
measurements
bull Never
bull Whenever it makes things look better
bull Chauvenetrsquos criterion provides a (quantitative) compromise
Rejecting Data
What is a good criteria for
rejecting data
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 54
Lecture 3 Slide 54
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly
ldquounreasonablerdquo data point from a set of randomly distributed
measurements
bull Never
Rejecting Data
Zallenrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 55
Lecture 3 Slide 55
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly
ldquounreasonablerdquo data point from a set of randomly distributed
measurements
bull Whenever it makes things look better
Rejecting Data
Disneyrsquos Criterion
Disneyrsquos First Law
Wishing will make it so
Disneyrsquos Second Law
Dreams are more colorful than reality
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 56
Lecture 3 Slide 56
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Data may be rejected if the expected number of measurements at
least as deviant as the suspect measurement is less than 50
Consider a set of N measurements of a single quantity
x1x2 helliphellipxN
Calculate ltxgt and σx and then determine the fractional deviations from the mean of all
the points
For the suspect point(s) xsuspect find the probability of such a point occurring in N
measurements
n = (expected number as deviant as xsuspect)
= N Prob(outside xsuspectσx)
Rejecting Data
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 57
Lecture 3 Slide 57
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Function of Gaussian Distribution
Area under curve
(probability that
ndashtσltxlt+tσ) is given
in Table at right
Probable Error (probability that
ndash067σltxlt+067σ) is 50
Error Function (probability that ndashtσltxlt+tσ )
Error Function Tabulated values-see App A
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 58
Lecture 3 Slide 58
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 59
Lecture 3 Slide 59
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashExample 1
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 60
Lecture 3 Slide 60
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Details (1)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 61
Lecture 3 Slide 61
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashDetails (2)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 62
Lecture 3 Slide 62
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos
Criterionmdash
Details (3)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 63
Lecture 3 Slide 63
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos
Criterionmdash
Example 2
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 64
Lecture 3 Slide 64
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Summary of Probability Theory
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 65
Lecture 3 Slide 65
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of
Probability
Theory-I
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 66
Lecture 3 Slide 66
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of
Probability
Theory-II
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 67
Lecture 3 Slide 67
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-III
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 68
Lecture 3 Slide 68
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-IV
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 41
Lecture 3 Slide 41
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Multiple Variables
Uncertainty of a function of multiple variables [Taylor Sec 311]
1 It can easily (no really) be shown that (see Taylor Sec 311) for a
function of several variables
)(
z
z
qy
y
qx
x
qzyxq
[Taylor Eq 347]
2 More correctly it can be shown that (see Taylor Sec 311) for a
function of several variables
)(
z
z
qy
y
qx
x
qzyxq
[Taylor Eq 347]
where the equals sign represents an upper bound as discussed above
3 For a function of several independent and random variables
)(
222
z
z
qy
y
qx
x
qzyxq
[Taylor Eq 348]
Again the proof is left for Ch 5
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 42
Lecture 3 Slide 42
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Product of Two Variables
Consider the arbitrary derived quantity
q(xy) of two independent random variables x and y
Expand q(xy) in a Taylor series about the expected values of x and y
(ie at points near X and Y)
Error Propagation General Case
119902 119909 119910 = 119902 119883 119884 + 120597119902
120597119909
119883 119909 minus 119883 +
120597119902
120597119910
119884
(119910 minus 119884)
Fixed shifts peak of distribution
Fixed Distribution centered at X with width σX
120575119902 119909 119910 = 120590119902 = 119902 119883 119884 + 120597119902
120597119909
119883120590119909
2
+ 120597119902
120597119910
119884
120590119910
2
0
How do we determine the error of a derived quantity Z(XYhellip) from errors in XYhellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 43
Lecture 3 Slide 43
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
SDOM of Gaussian Distribution
Standard Deviation of the Mean
The SDOM decreases as the square root of the number of measurements
That is the relative width σẊ of the distribution gets narrower as more
measurements are made
Each measurement
has similar σXi= σẊ
and similar partial
derivatives
Thushellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 44
Lecture 3 Slide 44
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Two Key Theorems from Probability
Central Limit Theorem
For random independent measurements (each with a well-define
expectation value and well-defined variance) the arithmetic mean
(average) will be approximately normally distributed
Principle of Maximum Likelihood
Given the N observed measurements x1 x2hellipxN the best estimates
for Ẋ and σ are those values for which the observed x1 x2hellipxN are
most likely
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 45
Lecture 3 Slide 45
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Mean of Gaussian Distribution as ldquoBest Estimaterdquo
Principle of Maximum Likelihood
Best Estimate of X is from maximum probability or minimum summation
Consider a data set
Each randomly distributed with
x1 x2 x3 hellipxN
To find the most likely value of the mean (the best estimate of ẋ)
find X that yields the highest probability for the data set
The combined probability for the full data set is the product
Minimize
Sum
Solve for
derivative wrst
X set to 0
Best
estimate
of X
prop1
120590119890minus(1199091minus119883)2 2120590 times
1
120590119890minus(1199092minus119883)2 2120590 times hellip times
1
120590119890minus(119909119873minus119883)2 2120590 =
1
120590119873119890 minus(119909119894minus119883)2 2120590
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 46
Lecture 3 Slide 46
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Uncertainty of ldquoBest Estimatesrdquo of Gaussian Distribution Principle of Maximum Likelihood
Best Estimate of X is from maximum probability or minimum summation
Consider a data set x1 x2 x3 hellipxN
To find the most likely value of the standard deviation (the best estimate of
the width of the x distribution) find σx that yields the highest probability for
the data set
The combined probability for the full data set is the product
Minimize
Sum
Solve for
derivative
wrst X set to 0
Best
estimate
of X
Best Estimate of σ is from maximum probability or minimum summation
Minimize
Sum
Best
estimate
of σ
Solve for
derivative
wrst σ set to 0
See
Prob 526
prop1
120590119890minus(1199091minus119883)2 2120590 times
1
120590119890minus(1199092minus119883)2 2120590 times hellip times
1
120590119890minus(119909119873minus119883)2 2120590 =
1
120590119873119890 minus(119909119894minus119883)2 2120590
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 47
Lecture 3 Slide 47
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Combining Data Sets
Weighted Averages
References Taylor Ch 7
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 48
Lecture 3 Slide 48
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Weighted Averages
Question How can we properly combine two or more separate
independent measurements of the same randomly distributed
quantity to determine a best combined value with uncertainty
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 49
Lecture 3 Slide 49
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider two measurements of the same quantity described by a random Gaussian distribution
ltx1gt x1 and ltx2gt x2
The probability of measuring two such measurements is
119875119903119900119887119909 1199091 1199092 = 119875119903119900119887119909 1199091 119875119903119900119887119909 1199092
= 1
12059011205902119890minus12059422 119908ℎ119890119903119890 1205942 equiv
1199091 minus 119883
1205901 2 +
1199092 minus 119883
1205902 2
To find the best value for χ find the maximum Prob or minimum χ2
Weighted Averages
Assume negligible systematic errors
Note χ2 or Chi squared is the sum of the squares of the deviations from the mean divided by
the corresponding uncertainty
Such methods are called ldquoMethods of Least Squaresrdquo They follow directly from the Principle of
Maximum Likelihood
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 50
Lecture 3 Slide 50
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
The probability of measuring two such measurements is
119875119903119900119887119909 1199091 1199092 = 119875119903119900119887119909 1199091 119875119903119900119887119909 1199092
= 1
12059011205902119890minus12059422 119908ℎ119890119903119890 1205942 equiv
1199091 minus 119883
1205901 2 +
1199092 minus 119883
1205902 2
To find the best value for χ find the maximum Prob or minimum χ2
Weighted Averages
This leads to 119909119882_119886119907119892 = 11990811199091+11990821199092
1199081+1199082=
119908119894 119909119894
119908119894 119908ℎ119890119903119890 119908119894 = 1
120590119894 2
Best Estimate of χ is from maximum probibility or minimum summation
Minimize Sum Solve for best estimate of χ Solve for derivative wrst χ set to 0
Note If w1=w2 we recover the standard result Xwavg= (12) (x1+x2)
Finally the width of a weighted average distribution is
120590119908119894119890119892 ℎ119905119890119889 119886119907119892 = 1
119908119894119894
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 51
Lecture 3 Slide 51
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Weighted Averages on Steroids
A very powerful method for combining data from different sources
with different methods and uncertainties (or indeed data of
related measured and calculated quantities) is Kalman filtering
The Kalman filter also known as
linear quadratic estimation (LQE)
is an algorithm that uses a series
of measurements observed over
time containing noise (random
variations) and other
inaccuracies and produces
estimates of unknown variables
that tend to be more precise than
those based on a single
measurement alone
The Kalman filter keeps track of the estimated state of the system and the variance or uncertainty of the
estimate The estimate is updated using a state transition model and measurements xk|k-1 denotes the
estimate of the systems state at time step k before the kth measurement yk has been taken into account Pk|k-1
is the corresponding uncertainty --Wikipedia 2013
Ludger Scherliess of USU Physics is a world expert at using Kalman filtering for the assimilation
of satellite and ground-based data and the USU GAMES model to predict space weather
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 52
Lecture 3 Slide 52
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Rejecting Data
Chauvenetrsquos Criterion
References Taylor Ch 6
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 53
Lecture 3 Slide 53
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly
ldquounreasonablerdquo data point from a set of randomly distributed
measurements
bull Never
bull Whenever it makes things look better
bull Chauvenetrsquos criterion provides a (quantitative) compromise
Rejecting Data
What is a good criteria for
rejecting data
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 54
Lecture 3 Slide 54
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly
ldquounreasonablerdquo data point from a set of randomly distributed
measurements
bull Never
Rejecting Data
Zallenrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 55
Lecture 3 Slide 55
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly
ldquounreasonablerdquo data point from a set of randomly distributed
measurements
bull Whenever it makes things look better
Rejecting Data
Disneyrsquos Criterion
Disneyrsquos First Law
Wishing will make it so
Disneyrsquos Second Law
Dreams are more colorful than reality
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 56
Lecture 3 Slide 56
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Data may be rejected if the expected number of measurements at
least as deviant as the suspect measurement is less than 50
Consider a set of N measurements of a single quantity
x1x2 helliphellipxN
Calculate ltxgt and σx and then determine the fractional deviations from the mean of all
the points
For the suspect point(s) xsuspect find the probability of such a point occurring in N
measurements
n = (expected number as deviant as xsuspect)
= N Prob(outside xsuspectσx)
Rejecting Data
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 57
Lecture 3 Slide 57
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Function of Gaussian Distribution
Area under curve
(probability that
ndashtσltxlt+tσ) is given
in Table at right
Probable Error (probability that
ndash067σltxlt+067σ) is 50
Error Function (probability that ndashtσltxlt+tσ )
Error Function Tabulated values-see App A
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 58
Lecture 3 Slide 58
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 59
Lecture 3 Slide 59
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashExample 1
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 60
Lecture 3 Slide 60
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Details (1)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 61
Lecture 3 Slide 61
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashDetails (2)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 62
Lecture 3 Slide 62
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos
Criterionmdash
Details (3)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 63
Lecture 3 Slide 63
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos
Criterionmdash
Example 2
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 64
Lecture 3 Slide 64
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Summary of Probability Theory
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 65
Lecture 3 Slide 65
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of
Probability
Theory-I
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 66
Lecture 3 Slide 66
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of
Probability
Theory-II
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 67
Lecture 3 Slide 67
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-III
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 68
Lecture 3 Slide 68
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-IV
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 42
Lecture 3 Slide 42
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Product of Two Variables
Consider the arbitrary derived quantity
q(xy) of two independent random variables x and y
Expand q(xy) in a Taylor series about the expected values of x and y
(ie at points near X and Y)
Error Propagation General Case
119902 119909 119910 = 119902 119883 119884 + 120597119902
120597119909
119883 119909 minus 119883 +
120597119902
120597119910
119884
(119910 minus 119884)
Fixed shifts peak of distribution
Fixed Distribution centered at X with width σX
120575119902 119909 119910 = 120590119902 = 119902 119883 119884 + 120597119902
120597119909
119883120590119909
2
+ 120597119902
120597119910
119884
120590119910
2
0
How do we determine the error of a derived quantity Z(XYhellip) from errors in XYhellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 43
Lecture 3 Slide 43
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
SDOM of Gaussian Distribution
Standard Deviation of the Mean
The SDOM decreases as the square root of the number of measurements
That is the relative width σẊ of the distribution gets narrower as more
measurements are made
Each measurement
has similar σXi= σẊ
and similar partial
derivatives
Thushellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 44
Lecture 3 Slide 44
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Two Key Theorems from Probability
Central Limit Theorem
For random independent measurements (each with a well-define
expectation value and well-defined variance) the arithmetic mean
(average) will be approximately normally distributed
Principle of Maximum Likelihood
Given the N observed measurements x1 x2hellipxN the best estimates
for Ẋ and σ are those values for which the observed x1 x2hellipxN are
most likely
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 45
Lecture 3 Slide 45
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Mean of Gaussian Distribution as ldquoBest Estimaterdquo
Principle of Maximum Likelihood
Best Estimate of X is from maximum probability or minimum summation
Consider a data set
Each randomly distributed with
x1 x2 x3 hellipxN
To find the most likely value of the mean (the best estimate of ẋ)
find X that yields the highest probability for the data set
The combined probability for the full data set is the product
Minimize
Sum
Solve for
derivative wrst
X set to 0
Best
estimate
of X
prop1
120590119890minus(1199091minus119883)2 2120590 times
1
120590119890minus(1199092minus119883)2 2120590 times hellip times
1
120590119890minus(119909119873minus119883)2 2120590 =
1
120590119873119890 minus(119909119894minus119883)2 2120590
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 46
Lecture 3 Slide 46
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Uncertainty of ldquoBest Estimatesrdquo of Gaussian Distribution Principle of Maximum Likelihood
Best Estimate of X is from maximum probability or minimum summation
Consider a data set x1 x2 x3 hellipxN
To find the most likely value of the standard deviation (the best estimate of
the width of the x distribution) find σx that yields the highest probability for
the data set
The combined probability for the full data set is the product
Minimize
Sum
Solve for
derivative
wrst X set to 0
Best
estimate
of X
Best Estimate of σ is from maximum probability or minimum summation
Minimize
Sum
Best
estimate
of σ
Solve for
derivative
wrst σ set to 0
See
Prob 526
prop1
120590119890minus(1199091minus119883)2 2120590 times
1
120590119890minus(1199092minus119883)2 2120590 times hellip times
1
120590119890minus(119909119873minus119883)2 2120590 =
1
120590119873119890 minus(119909119894minus119883)2 2120590
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 47
Lecture 3 Slide 47
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Combining Data Sets
Weighted Averages
References Taylor Ch 7
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 48
Lecture 3 Slide 48
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Weighted Averages
Question How can we properly combine two or more separate
independent measurements of the same randomly distributed
quantity to determine a best combined value with uncertainty
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 49
Lecture 3 Slide 49
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider two measurements of the same quantity described by a random Gaussian distribution
ltx1gt x1 and ltx2gt x2
The probability of measuring two such measurements is
119875119903119900119887119909 1199091 1199092 = 119875119903119900119887119909 1199091 119875119903119900119887119909 1199092
= 1
12059011205902119890minus12059422 119908ℎ119890119903119890 1205942 equiv
1199091 minus 119883
1205901 2 +
1199092 minus 119883
1205902 2
To find the best value for χ find the maximum Prob or minimum χ2
Weighted Averages
Assume negligible systematic errors
Note χ2 or Chi squared is the sum of the squares of the deviations from the mean divided by
the corresponding uncertainty
Such methods are called ldquoMethods of Least Squaresrdquo They follow directly from the Principle of
Maximum Likelihood
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 50
Lecture 3 Slide 50
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
The probability of measuring two such measurements is
119875119903119900119887119909 1199091 1199092 = 119875119903119900119887119909 1199091 119875119903119900119887119909 1199092
= 1
12059011205902119890minus12059422 119908ℎ119890119903119890 1205942 equiv
1199091 minus 119883
1205901 2 +
1199092 minus 119883
1205902 2
To find the best value for χ find the maximum Prob or minimum χ2
Weighted Averages
This leads to 119909119882_119886119907119892 = 11990811199091+11990821199092
1199081+1199082=
119908119894 119909119894
119908119894 119908ℎ119890119903119890 119908119894 = 1
120590119894 2
Best Estimate of χ is from maximum probibility or minimum summation
Minimize Sum Solve for best estimate of χ Solve for derivative wrst χ set to 0
Note If w1=w2 we recover the standard result Xwavg= (12) (x1+x2)
Finally the width of a weighted average distribution is
120590119908119894119890119892 ℎ119905119890119889 119886119907119892 = 1
119908119894119894
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 51
Lecture 3 Slide 51
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Weighted Averages on Steroids
A very powerful method for combining data from different sources
with different methods and uncertainties (or indeed data of
related measured and calculated quantities) is Kalman filtering
The Kalman filter also known as
linear quadratic estimation (LQE)
is an algorithm that uses a series
of measurements observed over
time containing noise (random
variations) and other
inaccuracies and produces
estimates of unknown variables
that tend to be more precise than
those based on a single
measurement alone
The Kalman filter keeps track of the estimated state of the system and the variance or uncertainty of the
estimate The estimate is updated using a state transition model and measurements xk|k-1 denotes the
estimate of the systems state at time step k before the kth measurement yk has been taken into account Pk|k-1
is the corresponding uncertainty --Wikipedia 2013
Ludger Scherliess of USU Physics is a world expert at using Kalman filtering for the assimilation
of satellite and ground-based data and the USU GAMES model to predict space weather
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 52
Lecture 3 Slide 52
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Rejecting Data
Chauvenetrsquos Criterion
References Taylor Ch 6
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 53
Lecture 3 Slide 53
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly
ldquounreasonablerdquo data point from a set of randomly distributed
measurements
bull Never
bull Whenever it makes things look better
bull Chauvenetrsquos criterion provides a (quantitative) compromise
Rejecting Data
What is a good criteria for
rejecting data
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 54
Lecture 3 Slide 54
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly
ldquounreasonablerdquo data point from a set of randomly distributed
measurements
bull Never
Rejecting Data
Zallenrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 55
Lecture 3 Slide 55
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly
ldquounreasonablerdquo data point from a set of randomly distributed
measurements
bull Whenever it makes things look better
Rejecting Data
Disneyrsquos Criterion
Disneyrsquos First Law
Wishing will make it so
Disneyrsquos Second Law
Dreams are more colorful than reality
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 56
Lecture 3 Slide 56
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Data may be rejected if the expected number of measurements at
least as deviant as the suspect measurement is less than 50
Consider a set of N measurements of a single quantity
x1x2 helliphellipxN
Calculate ltxgt and σx and then determine the fractional deviations from the mean of all
the points
For the suspect point(s) xsuspect find the probability of such a point occurring in N
measurements
n = (expected number as deviant as xsuspect)
= N Prob(outside xsuspectσx)
Rejecting Data
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 57
Lecture 3 Slide 57
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Function of Gaussian Distribution
Area under curve
(probability that
ndashtσltxlt+tσ) is given
in Table at right
Probable Error (probability that
ndash067σltxlt+067σ) is 50
Error Function (probability that ndashtσltxlt+tσ )
Error Function Tabulated values-see App A
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 58
Lecture 3 Slide 58
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 59
Lecture 3 Slide 59
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashExample 1
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 60
Lecture 3 Slide 60
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Details (1)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 61
Lecture 3 Slide 61
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashDetails (2)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 62
Lecture 3 Slide 62
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos
Criterionmdash
Details (3)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 63
Lecture 3 Slide 63
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos
Criterionmdash
Example 2
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 64
Lecture 3 Slide 64
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Summary of Probability Theory
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 65
Lecture 3 Slide 65
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of
Probability
Theory-I
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 66
Lecture 3 Slide 66
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of
Probability
Theory-II
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 67
Lecture 3 Slide 67
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-III
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 68
Lecture 3 Slide 68
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-IV
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 43
Lecture 3 Slide 43
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
SDOM of Gaussian Distribution
Standard Deviation of the Mean
The SDOM decreases as the square root of the number of measurements
That is the relative width σẊ of the distribution gets narrower as more
measurements are made
Each measurement
has similar σXi= σẊ
and similar partial
derivatives
Thushellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 44
Lecture 3 Slide 44
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Two Key Theorems from Probability
Central Limit Theorem
For random independent measurements (each with a well-define
expectation value and well-defined variance) the arithmetic mean
(average) will be approximately normally distributed
Principle of Maximum Likelihood
Given the N observed measurements x1 x2hellipxN the best estimates
for Ẋ and σ are those values for which the observed x1 x2hellipxN are
most likely
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 45
Lecture 3 Slide 45
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Mean of Gaussian Distribution as ldquoBest Estimaterdquo
Principle of Maximum Likelihood
Best Estimate of X is from maximum probability or minimum summation
Consider a data set
Each randomly distributed with
x1 x2 x3 hellipxN
To find the most likely value of the mean (the best estimate of ẋ)
find X that yields the highest probability for the data set
The combined probability for the full data set is the product
Minimize
Sum
Solve for
derivative wrst
X set to 0
Best
estimate
of X
prop1
120590119890minus(1199091minus119883)2 2120590 times
1
120590119890minus(1199092minus119883)2 2120590 times hellip times
1
120590119890minus(119909119873minus119883)2 2120590 =
1
120590119873119890 minus(119909119894minus119883)2 2120590
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 46
Lecture 3 Slide 46
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Uncertainty of ldquoBest Estimatesrdquo of Gaussian Distribution Principle of Maximum Likelihood
Best Estimate of X is from maximum probability or minimum summation
Consider a data set x1 x2 x3 hellipxN
To find the most likely value of the standard deviation (the best estimate of
the width of the x distribution) find σx that yields the highest probability for
the data set
The combined probability for the full data set is the product
Minimize
Sum
Solve for
derivative
wrst X set to 0
Best
estimate
of X
Best Estimate of σ is from maximum probability or minimum summation
Minimize
Sum
Best
estimate
of σ
Solve for
derivative
wrst σ set to 0
See
Prob 526
prop1
120590119890minus(1199091minus119883)2 2120590 times
1
120590119890minus(1199092minus119883)2 2120590 times hellip times
1
120590119890minus(119909119873minus119883)2 2120590 =
1
120590119873119890 minus(119909119894minus119883)2 2120590
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 47
Lecture 3 Slide 47
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Combining Data Sets
Weighted Averages
References Taylor Ch 7
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 48
Lecture 3 Slide 48
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Weighted Averages
Question How can we properly combine two or more separate
independent measurements of the same randomly distributed
quantity to determine a best combined value with uncertainty
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 49
Lecture 3 Slide 49
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider two measurements of the same quantity described by a random Gaussian distribution
ltx1gt x1 and ltx2gt x2
The probability of measuring two such measurements is
119875119903119900119887119909 1199091 1199092 = 119875119903119900119887119909 1199091 119875119903119900119887119909 1199092
= 1
12059011205902119890minus12059422 119908ℎ119890119903119890 1205942 equiv
1199091 minus 119883
1205901 2 +
1199092 minus 119883
1205902 2
To find the best value for χ find the maximum Prob or minimum χ2
Weighted Averages
Assume negligible systematic errors
Note χ2 or Chi squared is the sum of the squares of the deviations from the mean divided by
the corresponding uncertainty
Such methods are called ldquoMethods of Least Squaresrdquo They follow directly from the Principle of
Maximum Likelihood
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 50
Lecture 3 Slide 50
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
The probability of measuring two such measurements is
119875119903119900119887119909 1199091 1199092 = 119875119903119900119887119909 1199091 119875119903119900119887119909 1199092
= 1
12059011205902119890minus12059422 119908ℎ119890119903119890 1205942 equiv
1199091 minus 119883
1205901 2 +
1199092 minus 119883
1205902 2
To find the best value for χ find the maximum Prob or minimum χ2
Weighted Averages
This leads to 119909119882_119886119907119892 = 11990811199091+11990821199092
1199081+1199082=
119908119894 119909119894
119908119894 119908ℎ119890119903119890 119908119894 = 1
120590119894 2
Best Estimate of χ is from maximum probibility or minimum summation
Minimize Sum Solve for best estimate of χ Solve for derivative wrst χ set to 0
Note If w1=w2 we recover the standard result Xwavg= (12) (x1+x2)
Finally the width of a weighted average distribution is
120590119908119894119890119892 ℎ119905119890119889 119886119907119892 = 1
119908119894119894
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 51
Lecture 3 Slide 51
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Weighted Averages on Steroids
A very powerful method for combining data from different sources
with different methods and uncertainties (or indeed data of
related measured and calculated quantities) is Kalman filtering
The Kalman filter also known as
linear quadratic estimation (LQE)
is an algorithm that uses a series
of measurements observed over
time containing noise (random
variations) and other
inaccuracies and produces
estimates of unknown variables
that tend to be more precise than
those based on a single
measurement alone
The Kalman filter keeps track of the estimated state of the system and the variance or uncertainty of the
estimate The estimate is updated using a state transition model and measurements xk|k-1 denotes the
estimate of the systems state at time step k before the kth measurement yk has been taken into account Pk|k-1
is the corresponding uncertainty --Wikipedia 2013
Ludger Scherliess of USU Physics is a world expert at using Kalman filtering for the assimilation
of satellite and ground-based data and the USU GAMES model to predict space weather
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 52
Lecture 3 Slide 52
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Rejecting Data
Chauvenetrsquos Criterion
References Taylor Ch 6
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 53
Lecture 3 Slide 53
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly
ldquounreasonablerdquo data point from a set of randomly distributed
measurements
bull Never
bull Whenever it makes things look better
bull Chauvenetrsquos criterion provides a (quantitative) compromise
Rejecting Data
What is a good criteria for
rejecting data
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 54
Lecture 3 Slide 54
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly
ldquounreasonablerdquo data point from a set of randomly distributed
measurements
bull Never
Rejecting Data
Zallenrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 55
Lecture 3 Slide 55
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly
ldquounreasonablerdquo data point from a set of randomly distributed
measurements
bull Whenever it makes things look better
Rejecting Data
Disneyrsquos Criterion
Disneyrsquos First Law
Wishing will make it so
Disneyrsquos Second Law
Dreams are more colorful than reality
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 56
Lecture 3 Slide 56
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Data may be rejected if the expected number of measurements at
least as deviant as the suspect measurement is less than 50
Consider a set of N measurements of a single quantity
x1x2 helliphellipxN
Calculate ltxgt and σx and then determine the fractional deviations from the mean of all
the points
For the suspect point(s) xsuspect find the probability of such a point occurring in N
measurements
n = (expected number as deviant as xsuspect)
= N Prob(outside xsuspectσx)
Rejecting Data
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 57
Lecture 3 Slide 57
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Function of Gaussian Distribution
Area under curve
(probability that
ndashtσltxlt+tσ) is given
in Table at right
Probable Error (probability that
ndash067σltxlt+067σ) is 50
Error Function (probability that ndashtσltxlt+tσ )
Error Function Tabulated values-see App A
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 58
Lecture 3 Slide 58
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 59
Lecture 3 Slide 59
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashExample 1
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 60
Lecture 3 Slide 60
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Details (1)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 61
Lecture 3 Slide 61
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashDetails (2)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 62
Lecture 3 Slide 62
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos
Criterionmdash
Details (3)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 63
Lecture 3 Slide 63
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos
Criterionmdash
Example 2
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 64
Lecture 3 Slide 64
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Summary of Probability Theory
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 65
Lecture 3 Slide 65
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of
Probability
Theory-I
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 66
Lecture 3 Slide 66
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of
Probability
Theory-II
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 67
Lecture 3 Slide 67
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-III
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 68
Lecture 3 Slide 68
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-IV
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 44
Lecture 3 Slide 44
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Two Key Theorems from Probability
Central Limit Theorem
For random independent measurements (each with a well-define
expectation value and well-defined variance) the arithmetic mean
(average) will be approximately normally distributed
Principle of Maximum Likelihood
Given the N observed measurements x1 x2hellipxN the best estimates
for Ẋ and σ are those values for which the observed x1 x2hellipxN are
most likely
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 45
Lecture 3 Slide 45
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Mean of Gaussian Distribution as ldquoBest Estimaterdquo
Principle of Maximum Likelihood
Best Estimate of X is from maximum probability or minimum summation
Consider a data set
Each randomly distributed with
x1 x2 x3 hellipxN
To find the most likely value of the mean (the best estimate of ẋ)
find X that yields the highest probability for the data set
The combined probability for the full data set is the product
Minimize
Sum
Solve for
derivative wrst
X set to 0
Best
estimate
of X
prop1
120590119890minus(1199091minus119883)2 2120590 times
1
120590119890minus(1199092minus119883)2 2120590 times hellip times
1
120590119890minus(119909119873minus119883)2 2120590 =
1
120590119873119890 minus(119909119894minus119883)2 2120590
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 46
Lecture 3 Slide 46
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Uncertainty of ldquoBest Estimatesrdquo of Gaussian Distribution Principle of Maximum Likelihood
Best Estimate of X is from maximum probability or minimum summation
Consider a data set x1 x2 x3 hellipxN
To find the most likely value of the standard deviation (the best estimate of
the width of the x distribution) find σx that yields the highest probability for
the data set
The combined probability for the full data set is the product
Minimize
Sum
Solve for
derivative
wrst X set to 0
Best
estimate
of X
Best Estimate of σ is from maximum probability or minimum summation
Minimize
Sum
Best
estimate
of σ
Solve for
derivative
wrst σ set to 0
See
Prob 526
prop1
120590119890minus(1199091minus119883)2 2120590 times
1
120590119890minus(1199092minus119883)2 2120590 times hellip times
1
120590119890minus(119909119873minus119883)2 2120590 =
1
120590119873119890 minus(119909119894minus119883)2 2120590
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 47
Lecture 3 Slide 47
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Combining Data Sets
Weighted Averages
References Taylor Ch 7
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 48
Lecture 3 Slide 48
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Weighted Averages
Question How can we properly combine two or more separate
independent measurements of the same randomly distributed
quantity to determine a best combined value with uncertainty
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 49
Lecture 3 Slide 49
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider two measurements of the same quantity described by a random Gaussian distribution
ltx1gt x1 and ltx2gt x2
The probability of measuring two such measurements is
119875119903119900119887119909 1199091 1199092 = 119875119903119900119887119909 1199091 119875119903119900119887119909 1199092
= 1
12059011205902119890minus12059422 119908ℎ119890119903119890 1205942 equiv
1199091 minus 119883
1205901 2 +
1199092 minus 119883
1205902 2
To find the best value for χ find the maximum Prob or minimum χ2
Weighted Averages
Assume negligible systematic errors
Note χ2 or Chi squared is the sum of the squares of the deviations from the mean divided by
the corresponding uncertainty
Such methods are called ldquoMethods of Least Squaresrdquo They follow directly from the Principle of
Maximum Likelihood
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 50
Lecture 3 Slide 50
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
The probability of measuring two such measurements is
119875119903119900119887119909 1199091 1199092 = 119875119903119900119887119909 1199091 119875119903119900119887119909 1199092
= 1
12059011205902119890minus12059422 119908ℎ119890119903119890 1205942 equiv
1199091 minus 119883
1205901 2 +
1199092 minus 119883
1205902 2
To find the best value for χ find the maximum Prob or minimum χ2
Weighted Averages
This leads to 119909119882_119886119907119892 = 11990811199091+11990821199092
1199081+1199082=
119908119894 119909119894
119908119894 119908ℎ119890119903119890 119908119894 = 1
120590119894 2
Best Estimate of χ is from maximum probibility or minimum summation
Minimize Sum Solve for best estimate of χ Solve for derivative wrst χ set to 0
Note If w1=w2 we recover the standard result Xwavg= (12) (x1+x2)
Finally the width of a weighted average distribution is
120590119908119894119890119892 ℎ119905119890119889 119886119907119892 = 1
119908119894119894
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 51
Lecture 3 Slide 51
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Weighted Averages on Steroids
A very powerful method for combining data from different sources
with different methods and uncertainties (or indeed data of
related measured and calculated quantities) is Kalman filtering
The Kalman filter also known as
linear quadratic estimation (LQE)
is an algorithm that uses a series
of measurements observed over
time containing noise (random
variations) and other
inaccuracies and produces
estimates of unknown variables
that tend to be more precise than
those based on a single
measurement alone
The Kalman filter keeps track of the estimated state of the system and the variance or uncertainty of the
estimate The estimate is updated using a state transition model and measurements xk|k-1 denotes the
estimate of the systems state at time step k before the kth measurement yk has been taken into account Pk|k-1
is the corresponding uncertainty --Wikipedia 2013
Ludger Scherliess of USU Physics is a world expert at using Kalman filtering for the assimilation
of satellite and ground-based data and the USU GAMES model to predict space weather
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 52
Lecture 3 Slide 52
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Rejecting Data
Chauvenetrsquos Criterion
References Taylor Ch 6
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 53
Lecture 3 Slide 53
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly
ldquounreasonablerdquo data point from a set of randomly distributed
measurements
bull Never
bull Whenever it makes things look better
bull Chauvenetrsquos criterion provides a (quantitative) compromise
Rejecting Data
What is a good criteria for
rejecting data
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 54
Lecture 3 Slide 54
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly
ldquounreasonablerdquo data point from a set of randomly distributed
measurements
bull Never
Rejecting Data
Zallenrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 55
Lecture 3 Slide 55
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly
ldquounreasonablerdquo data point from a set of randomly distributed
measurements
bull Whenever it makes things look better
Rejecting Data
Disneyrsquos Criterion
Disneyrsquos First Law
Wishing will make it so
Disneyrsquos Second Law
Dreams are more colorful than reality
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 56
Lecture 3 Slide 56
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Data may be rejected if the expected number of measurements at
least as deviant as the suspect measurement is less than 50
Consider a set of N measurements of a single quantity
x1x2 helliphellipxN
Calculate ltxgt and σx and then determine the fractional deviations from the mean of all
the points
For the suspect point(s) xsuspect find the probability of such a point occurring in N
measurements
n = (expected number as deviant as xsuspect)
= N Prob(outside xsuspectσx)
Rejecting Data
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 57
Lecture 3 Slide 57
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Function of Gaussian Distribution
Area under curve
(probability that
ndashtσltxlt+tσ) is given
in Table at right
Probable Error (probability that
ndash067σltxlt+067σ) is 50
Error Function (probability that ndashtσltxlt+tσ )
Error Function Tabulated values-see App A
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 58
Lecture 3 Slide 58
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 59
Lecture 3 Slide 59
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashExample 1
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 60
Lecture 3 Slide 60
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Details (1)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 61
Lecture 3 Slide 61
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashDetails (2)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 62
Lecture 3 Slide 62
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos
Criterionmdash
Details (3)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 63
Lecture 3 Slide 63
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos
Criterionmdash
Example 2
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 64
Lecture 3 Slide 64
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Summary of Probability Theory
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 65
Lecture 3 Slide 65
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of
Probability
Theory-I
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 66
Lecture 3 Slide 66
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of
Probability
Theory-II
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 67
Lecture 3 Slide 67
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-III
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 68
Lecture 3 Slide 68
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-IV
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 45
Lecture 3 Slide 45
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Mean of Gaussian Distribution as ldquoBest Estimaterdquo
Principle of Maximum Likelihood
Best Estimate of X is from maximum probability or minimum summation
Consider a data set
Each randomly distributed with
x1 x2 x3 hellipxN
To find the most likely value of the mean (the best estimate of ẋ)
find X that yields the highest probability for the data set
The combined probability for the full data set is the product
Minimize
Sum
Solve for
derivative wrst
X set to 0
Best
estimate
of X
prop1
120590119890minus(1199091minus119883)2 2120590 times
1
120590119890minus(1199092minus119883)2 2120590 times hellip times
1
120590119890minus(119909119873minus119883)2 2120590 =
1
120590119873119890 minus(119909119894minus119883)2 2120590
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 46
Lecture 3 Slide 46
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Uncertainty of ldquoBest Estimatesrdquo of Gaussian Distribution Principle of Maximum Likelihood
Best Estimate of X is from maximum probability or minimum summation
Consider a data set x1 x2 x3 hellipxN
To find the most likely value of the standard deviation (the best estimate of
the width of the x distribution) find σx that yields the highest probability for
the data set
The combined probability for the full data set is the product
Minimize
Sum
Solve for
derivative
wrst X set to 0
Best
estimate
of X
Best Estimate of σ is from maximum probability or minimum summation
Minimize
Sum
Best
estimate
of σ
Solve for
derivative
wrst σ set to 0
See
Prob 526
prop1
120590119890minus(1199091minus119883)2 2120590 times
1
120590119890minus(1199092minus119883)2 2120590 times hellip times
1
120590119890minus(119909119873minus119883)2 2120590 =
1
120590119873119890 minus(119909119894minus119883)2 2120590
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 47
Lecture 3 Slide 47
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Combining Data Sets
Weighted Averages
References Taylor Ch 7
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 48
Lecture 3 Slide 48
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Weighted Averages
Question How can we properly combine two or more separate
independent measurements of the same randomly distributed
quantity to determine a best combined value with uncertainty
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 49
Lecture 3 Slide 49
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider two measurements of the same quantity described by a random Gaussian distribution
ltx1gt x1 and ltx2gt x2
The probability of measuring two such measurements is
119875119903119900119887119909 1199091 1199092 = 119875119903119900119887119909 1199091 119875119903119900119887119909 1199092
= 1
12059011205902119890minus12059422 119908ℎ119890119903119890 1205942 equiv
1199091 minus 119883
1205901 2 +
1199092 minus 119883
1205902 2
To find the best value for χ find the maximum Prob or minimum χ2
Weighted Averages
Assume negligible systematic errors
Note χ2 or Chi squared is the sum of the squares of the deviations from the mean divided by
the corresponding uncertainty
Such methods are called ldquoMethods of Least Squaresrdquo They follow directly from the Principle of
Maximum Likelihood
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 50
Lecture 3 Slide 50
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
The probability of measuring two such measurements is
119875119903119900119887119909 1199091 1199092 = 119875119903119900119887119909 1199091 119875119903119900119887119909 1199092
= 1
12059011205902119890minus12059422 119908ℎ119890119903119890 1205942 equiv
1199091 minus 119883
1205901 2 +
1199092 minus 119883
1205902 2
To find the best value for χ find the maximum Prob or minimum χ2
Weighted Averages
This leads to 119909119882_119886119907119892 = 11990811199091+11990821199092
1199081+1199082=
119908119894 119909119894
119908119894 119908ℎ119890119903119890 119908119894 = 1
120590119894 2
Best Estimate of χ is from maximum probibility or minimum summation
Minimize Sum Solve for best estimate of χ Solve for derivative wrst χ set to 0
Note If w1=w2 we recover the standard result Xwavg= (12) (x1+x2)
Finally the width of a weighted average distribution is
120590119908119894119890119892 ℎ119905119890119889 119886119907119892 = 1
119908119894119894
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 51
Lecture 3 Slide 51
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Weighted Averages on Steroids
A very powerful method for combining data from different sources
with different methods and uncertainties (or indeed data of
related measured and calculated quantities) is Kalman filtering
The Kalman filter also known as
linear quadratic estimation (LQE)
is an algorithm that uses a series
of measurements observed over
time containing noise (random
variations) and other
inaccuracies and produces
estimates of unknown variables
that tend to be more precise than
those based on a single
measurement alone
The Kalman filter keeps track of the estimated state of the system and the variance or uncertainty of the
estimate The estimate is updated using a state transition model and measurements xk|k-1 denotes the
estimate of the systems state at time step k before the kth measurement yk has been taken into account Pk|k-1
is the corresponding uncertainty --Wikipedia 2013
Ludger Scherliess of USU Physics is a world expert at using Kalman filtering for the assimilation
of satellite and ground-based data and the USU GAMES model to predict space weather
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 52
Lecture 3 Slide 52
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Rejecting Data
Chauvenetrsquos Criterion
References Taylor Ch 6
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 53
Lecture 3 Slide 53
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly
ldquounreasonablerdquo data point from a set of randomly distributed
measurements
bull Never
bull Whenever it makes things look better
bull Chauvenetrsquos criterion provides a (quantitative) compromise
Rejecting Data
What is a good criteria for
rejecting data
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 54
Lecture 3 Slide 54
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly
ldquounreasonablerdquo data point from a set of randomly distributed
measurements
bull Never
Rejecting Data
Zallenrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 55
Lecture 3 Slide 55
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly
ldquounreasonablerdquo data point from a set of randomly distributed
measurements
bull Whenever it makes things look better
Rejecting Data
Disneyrsquos Criterion
Disneyrsquos First Law
Wishing will make it so
Disneyrsquos Second Law
Dreams are more colorful than reality
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 56
Lecture 3 Slide 56
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Data may be rejected if the expected number of measurements at
least as deviant as the suspect measurement is less than 50
Consider a set of N measurements of a single quantity
x1x2 helliphellipxN
Calculate ltxgt and σx and then determine the fractional deviations from the mean of all
the points
For the suspect point(s) xsuspect find the probability of such a point occurring in N
measurements
n = (expected number as deviant as xsuspect)
= N Prob(outside xsuspectσx)
Rejecting Data
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 57
Lecture 3 Slide 57
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Function of Gaussian Distribution
Area under curve
(probability that
ndashtσltxlt+tσ) is given
in Table at right
Probable Error (probability that
ndash067σltxlt+067σ) is 50
Error Function (probability that ndashtσltxlt+tσ )
Error Function Tabulated values-see App A
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 58
Lecture 3 Slide 58
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 59
Lecture 3 Slide 59
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashExample 1
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 60
Lecture 3 Slide 60
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Details (1)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 61
Lecture 3 Slide 61
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashDetails (2)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 62
Lecture 3 Slide 62
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos
Criterionmdash
Details (3)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 63
Lecture 3 Slide 63
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos
Criterionmdash
Example 2
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 64
Lecture 3 Slide 64
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Summary of Probability Theory
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 65
Lecture 3 Slide 65
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of
Probability
Theory-I
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 66
Lecture 3 Slide 66
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of
Probability
Theory-II
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 67
Lecture 3 Slide 67
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-III
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 68
Lecture 3 Slide 68
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-IV
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 46
Lecture 3 Slide 46
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Uncertainty of ldquoBest Estimatesrdquo of Gaussian Distribution Principle of Maximum Likelihood
Best Estimate of X is from maximum probability or minimum summation
Consider a data set x1 x2 x3 hellipxN
To find the most likely value of the standard deviation (the best estimate of
the width of the x distribution) find σx that yields the highest probability for
the data set
The combined probability for the full data set is the product
Minimize
Sum
Solve for
derivative
wrst X set to 0
Best
estimate
of X
Best Estimate of σ is from maximum probability or minimum summation
Minimize
Sum
Best
estimate
of σ
Solve for
derivative
wrst σ set to 0
See
Prob 526
prop1
120590119890minus(1199091minus119883)2 2120590 times
1
120590119890minus(1199092minus119883)2 2120590 times hellip times
1
120590119890minus(119909119873minus119883)2 2120590 =
1
120590119873119890 minus(119909119894minus119883)2 2120590
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 47
Lecture 3 Slide 47
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Combining Data Sets
Weighted Averages
References Taylor Ch 7
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 48
Lecture 3 Slide 48
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Weighted Averages
Question How can we properly combine two or more separate
independent measurements of the same randomly distributed
quantity to determine a best combined value with uncertainty
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 49
Lecture 3 Slide 49
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider two measurements of the same quantity described by a random Gaussian distribution
ltx1gt x1 and ltx2gt x2
The probability of measuring two such measurements is
119875119903119900119887119909 1199091 1199092 = 119875119903119900119887119909 1199091 119875119903119900119887119909 1199092
= 1
12059011205902119890minus12059422 119908ℎ119890119903119890 1205942 equiv
1199091 minus 119883
1205901 2 +
1199092 minus 119883
1205902 2
To find the best value for χ find the maximum Prob or minimum χ2
Weighted Averages
Assume negligible systematic errors
Note χ2 or Chi squared is the sum of the squares of the deviations from the mean divided by
the corresponding uncertainty
Such methods are called ldquoMethods of Least Squaresrdquo They follow directly from the Principle of
Maximum Likelihood
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 50
Lecture 3 Slide 50
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
The probability of measuring two such measurements is
119875119903119900119887119909 1199091 1199092 = 119875119903119900119887119909 1199091 119875119903119900119887119909 1199092
= 1
12059011205902119890minus12059422 119908ℎ119890119903119890 1205942 equiv
1199091 minus 119883
1205901 2 +
1199092 minus 119883
1205902 2
To find the best value for χ find the maximum Prob or minimum χ2
Weighted Averages
This leads to 119909119882_119886119907119892 = 11990811199091+11990821199092
1199081+1199082=
119908119894 119909119894
119908119894 119908ℎ119890119903119890 119908119894 = 1
120590119894 2
Best Estimate of χ is from maximum probibility or minimum summation
Minimize Sum Solve for best estimate of χ Solve for derivative wrst χ set to 0
Note If w1=w2 we recover the standard result Xwavg= (12) (x1+x2)
Finally the width of a weighted average distribution is
120590119908119894119890119892 ℎ119905119890119889 119886119907119892 = 1
119908119894119894
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 51
Lecture 3 Slide 51
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Weighted Averages on Steroids
A very powerful method for combining data from different sources
with different methods and uncertainties (or indeed data of
related measured and calculated quantities) is Kalman filtering
The Kalman filter also known as
linear quadratic estimation (LQE)
is an algorithm that uses a series
of measurements observed over
time containing noise (random
variations) and other
inaccuracies and produces
estimates of unknown variables
that tend to be more precise than
those based on a single
measurement alone
The Kalman filter keeps track of the estimated state of the system and the variance or uncertainty of the
estimate The estimate is updated using a state transition model and measurements xk|k-1 denotes the
estimate of the systems state at time step k before the kth measurement yk has been taken into account Pk|k-1
is the corresponding uncertainty --Wikipedia 2013
Ludger Scherliess of USU Physics is a world expert at using Kalman filtering for the assimilation
of satellite and ground-based data and the USU GAMES model to predict space weather
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 52
Lecture 3 Slide 52
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Rejecting Data
Chauvenetrsquos Criterion
References Taylor Ch 6
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 53
Lecture 3 Slide 53
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly
ldquounreasonablerdquo data point from a set of randomly distributed
measurements
bull Never
bull Whenever it makes things look better
bull Chauvenetrsquos criterion provides a (quantitative) compromise
Rejecting Data
What is a good criteria for
rejecting data
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 54
Lecture 3 Slide 54
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly
ldquounreasonablerdquo data point from a set of randomly distributed
measurements
bull Never
Rejecting Data
Zallenrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 55
Lecture 3 Slide 55
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly
ldquounreasonablerdquo data point from a set of randomly distributed
measurements
bull Whenever it makes things look better
Rejecting Data
Disneyrsquos Criterion
Disneyrsquos First Law
Wishing will make it so
Disneyrsquos Second Law
Dreams are more colorful than reality
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 56
Lecture 3 Slide 56
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Data may be rejected if the expected number of measurements at
least as deviant as the suspect measurement is less than 50
Consider a set of N measurements of a single quantity
x1x2 helliphellipxN
Calculate ltxgt and σx and then determine the fractional deviations from the mean of all
the points
For the suspect point(s) xsuspect find the probability of such a point occurring in N
measurements
n = (expected number as deviant as xsuspect)
= N Prob(outside xsuspectσx)
Rejecting Data
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 57
Lecture 3 Slide 57
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Function of Gaussian Distribution
Area under curve
(probability that
ndashtσltxlt+tσ) is given
in Table at right
Probable Error (probability that
ndash067σltxlt+067σ) is 50
Error Function (probability that ndashtσltxlt+tσ )
Error Function Tabulated values-see App A
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 58
Lecture 3 Slide 58
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 59
Lecture 3 Slide 59
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashExample 1
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 60
Lecture 3 Slide 60
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Details (1)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 61
Lecture 3 Slide 61
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashDetails (2)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 62
Lecture 3 Slide 62
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos
Criterionmdash
Details (3)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 63
Lecture 3 Slide 63
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos
Criterionmdash
Example 2
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 64
Lecture 3 Slide 64
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Summary of Probability Theory
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 65
Lecture 3 Slide 65
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of
Probability
Theory-I
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 66
Lecture 3 Slide 66
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of
Probability
Theory-II
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 67
Lecture 3 Slide 67
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-III
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 68
Lecture 3 Slide 68
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-IV
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 47
Lecture 3 Slide 47
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Combining Data Sets
Weighted Averages
References Taylor Ch 7
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 48
Lecture 3 Slide 48
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Weighted Averages
Question How can we properly combine two or more separate
independent measurements of the same randomly distributed
quantity to determine a best combined value with uncertainty
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 49
Lecture 3 Slide 49
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider two measurements of the same quantity described by a random Gaussian distribution
ltx1gt x1 and ltx2gt x2
The probability of measuring two such measurements is
119875119903119900119887119909 1199091 1199092 = 119875119903119900119887119909 1199091 119875119903119900119887119909 1199092
= 1
12059011205902119890minus12059422 119908ℎ119890119903119890 1205942 equiv
1199091 minus 119883
1205901 2 +
1199092 minus 119883
1205902 2
To find the best value for χ find the maximum Prob or minimum χ2
Weighted Averages
Assume negligible systematic errors
Note χ2 or Chi squared is the sum of the squares of the deviations from the mean divided by
the corresponding uncertainty
Such methods are called ldquoMethods of Least Squaresrdquo They follow directly from the Principle of
Maximum Likelihood
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 50
Lecture 3 Slide 50
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
The probability of measuring two such measurements is
119875119903119900119887119909 1199091 1199092 = 119875119903119900119887119909 1199091 119875119903119900119887119909 1199092
= 1
12059011205902119890minus12059422 119908ℎ119890119903119890 1205942 equiv
1199091 minus 119883
1205901 2 +
1199092 minus 119883
1205902 2
To find the best value for χ find the maximum Prob or minimum χ2
Weighted Averages
This leads to 119909119882_119886119907119892 = 11990811199091+11990821199092
1199081+1199082=
119908119894 119909119894
119908119894 119908ℎ119890119903119890 119908119894 = 1
120590119894 2
Best Estimate of χ is from maximum probibility or minimum summation
Minimize Sum Solve for best estimate of χ Solve for derivative wrst χ set to 0
Note If w1=w2 we recover the standard result Xwavg= (12) (x1+x2)
Finally the width of a weighted average distribution is
120590119908119894119890119892 ℎ119905119890119889 119886119907119892 = 1
119908119894119894
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 51
Lecture 3 Slide 51
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Weighted Averages on Steroids
A very powerful method for combining data from different sources
with different methods and uncertainties (or indeed data of
related measured and calculated quantities) is Kalman filtering
The Kalman filter also known as
linear quadratic estimation (LQE)
is an algorithm that uses a series
of measurements observed over
time containing noise (random
variations) and other
inaccuracies and produces
estimates of unknown variables
that tend to be more precise than
those based on a single
measurement alone
The Kalman filter keeps track of the estimated state of the system and the variance or uncertainty of the
estimate The estimate is updated using a state transition model and measurements xk|k-1 denotes the
estimate of the systems state at time step k before the kth measurement yk has been taken into account Pk|k-1
is the corresponding uncertainty --Wikipedia 2013
Ludger Scherliess of USU Physics is a world expert at using Kalman filtering for the assimilation
of satellite and ground-based data and the USU GAMES model to predict space weather
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 52
Lecture 3 Slide 52
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Rejecting Data
Chauvenetrsquos Criterion
References Taylor Ch 6
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 53
Lecture 3 Slide 53
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly
ldquounreasonablerdquo data point from a set of randomly distributed
measurements
bull Never
bull Whenever it makes things look better
bull Chauvenetrsquos criterion provides a (quantitative) compromise
Rejecting Data
What is a good criteria for
rejecting data
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 54
Lecture 3 Slide 54
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly
ldquounreasonablerdquo data point from a set of randomly distributed
measurements
bull Never
Rejecting Data
Zallenrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 55
Lecture 3 Slide 55
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly
ldquounreasonablerdquo data point from a set of randomly distributed
measurements
bull Whenever it makes things look better
Rejecting Data
Disneyrsquos Criterion
Disneyrsquos First Law
Wishing will make it so
Disneyrsquos Second Law
Dreams are more colorful than reality
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 56
Lecture 3 Slide 56
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Data may be rejected if the expected number of measurements at
least as deviant as the suspect measurement is less than 50
Consider a set of N measurements of a single quantity
x1x2 helliphellipxN
Calculate ltxgt and σx and then determine the fractional deviations from the mean of all
the points
For the suspect point(s) xsuspect find the probability of such a point occurring in N
measurements
n = (expected number as deviant as xsuspect)
= N Prob(outside xsuspectσx)
Rejecting Data
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 57
Lecture 3 Slide 57
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Function of Gaussian Distribution
Area under curve
(probability that
ndashtσltxlt+tσ) is given
in Table at right
Probable Error (probability that
ndash067σltxlt+067σ) is 50
Error Function (probability that ndashtσltxlt+tσ )
Error Function Tabulated values-see App A
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 58
Lecture 3 Slide 58
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 59
Lecture 3 Slide 59
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashExample 1
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 60
Lecture 3 Slide 60
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Details (1)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 61
Lecture 3 Slide 61
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashDetails (2)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 62
Lecture 3 Slide 62
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos
Criterionmdash
Details (3)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 63
Lecture 3 Slide 63
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos
Criterionmdash
Example 2
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 64
Lecture 3 Slide 64
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Summary of Probability Theory
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 65
Lecture 3 Slide 65
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of
Probability
Theory-I
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 66
Lecture 3 Slide 66
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of
Probability
Theory-II
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 67
Lecture 3 Slide 67
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-III
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 68
Lecture 3 Slide 68
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-IV
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 48
Lecture 3 Slide 48
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Weighted Averages
Question How can we properly combine two or more separate
independent measurements of the same randomly distributed
quantity to determine a best combined value with uncertainty
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 49
Lecture 3 Slide 49
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider two measurements of the same quantity described by a random Gaussian distribution
ltx1gt x1 and ltx2gt x2
The probability of measuring two such measurements is
119875119903119900119887119909 1199091 1199092 = 119875119903119900119887119909 1199091 119875119903119900119887119909 1199092
= 1
12059011205902119890minus12059422 119908ℎ119890119903119890 1205942 equiv
1199091 minus 119883
1205901 2 +
1199092 minus 119883
1205902 2
To find the best value for χ find the maximum Prob or minimum χ2
Weighted Averages
Assume negligible systematic errors
Note χ2 or Chi squared is the sum of the squares of the deviations from the mean divided by
the corresponding uncertainty
Such methods are called ldquoMethods of Least Squaresrdquo They follow directly from the Principle of
Maximum Likelihood
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 50
Lecture 3 Slide 50
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
The probability of measuring two such measurements is
119875119903119900119887119909 1199091 1199092 = 119875119903119900119887119909 1199091 119875119903119900119887119909 1199092
= 1
12059011205902119890minus12059422 119908ℎ119890119903119890 1205942 equiv
1199091 minus 119883
1205901 2 +
1199092 minus 119883
1205902 2
To find the best value for χ find the maximum Prob or minimum χ2
Weighted Averages
This leads to 119909119882_119886119907119892 = 11990811199091+11990821199092
1199081+1199082=
119908119894 119909119894
119908119894 119908ℎ119890119903119890 119908119894 = 1
120590119894 2
Best Estimate of χ is from maximum probibility or minimum summation
Minimize Sum Solve for best estimate of χ Solve for derivative wrst χ set to 0
Note If w1=w2 we recover the standard result Xwavg= (12) (x1+x2)
Finally the width of a weighted average distribution is
120590119908119894119890119892 ℎ119905119890119889 119886119907119892 = 1
119908119894119894
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 51
Lecture 3 Slide 51
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Weighted Averages on Steroids
A very powerful method for combining data from different sources
with different methods and uncertainties (or indeed data of
related measured and calculated quantities) is Kalman filtering
The Kalman filter also known as
linear quadratic estimation (LQE)
is an algorithm that uses a series
of measurements observed over
time containing noise (random
variations) and other
inaccuracies and produces
estimates of unknown variables
that tend to be more precise than
those based on a single
measurement alone
The Kalman filter keeps track of the estimated state of the system and the variance or uncertainty of the
estimate The estimate is updated using a state transition model and measurements xk|k-1 denotes the
estimate of the systems state at time step k before the kth measurement yk has been taken into account Pk|k-1
is the corresponding uncertainty --Wikipedia 2013
Ludger Scherliess of USU Physics is a world expert at using Kalman filtering for the assimilation
of satellite and ground-based data and the USU GAMES model to predict space weather
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 52
Lecture 3 Slide 52
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Rejecting Data
Chauvenetrsquos Criterion
References Taylor Ch 6
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 53
Lecture 3 Slide 53
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly
ldquounreasonablerdquo data point from a set of randomly distributed
measurements
bull Never
bull Whenever it makes things look better
bull Chauvenetrsquos criterion provides a (quantitative) compromise
Rejecting Data
What is a good criteria for
rejecting data
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 54
Lecture 3 Slide 54
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly
ldquounreasonablerdquo data point from a set of randomly distributed
measurements
bull Never
Rejecting Data
Zallenrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 55
Lecture 3 Slide 55
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly
ldquounreasonablerdquo data point from a set of randomly distributed
measurements
bull Whenever it makes things look better
Rejecting Data
Disneyrsquos Criterion
Disneyrsquos First Law
Wishing will make it so
Disneyrsquos Second Law
Dreams are more colorful than reality
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 56
Lecture 3 Slide 56
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Data may be rejected if the expected number of measurements at
least as deviant as the suspect measurement is less than 50
Consider a set of N measurements of a single quantity
x1x2 helliphellipxN
Calculate ltxgt and σx and then determine the fractional deviations from the mean of all
the points
For the suspect point(s) xsuspect find the probability of such a point occurring in N
measurements
n = (expected number as deviant as xsuspect)
= N Prob(outside xsuspectσx)
Rejecting Data
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 57
Lecture 3 Slide 57
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Function of Gaussian Distribution
Area under curve
(probability that
ndashtσltxlt+tσ) is given
in Table at right
Probable Error (probability that
ndash067σltxlt+067σ) is 50
Error Function (probability that ndashtσltxlt+tσ )
Error Function Tabulated values-see App A
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 58
Lecture 3 Slide 58
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 59
Lecture 3 Slide 59
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashExample 1
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 60
Lecture 3 Slide 60
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Details (1)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 61
Lecture 3 Slide 61
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashDetails (2)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 62
Lecture 3 Slide 62
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos
Criterionmdash
Details (3)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 63
Lecture 3 Slide 63
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos
Criterionmdash
Example 2
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 64
Lecture 3 Slide 64
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Summary of Probability Theory
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 65
Lecture 3 Slide 65
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of
Probability
Theory-I
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 66
Lecture 3 Slide 66
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of
Probability
Theory-II
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 67
Lecture 3 Slide 67
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-III
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 68
Lecture 3 Slide 68
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-IV
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 49
Lecture 3 Slide 49
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider two measurements of the same quantity described by a random Gaussian distribution
ltx1gt x1 and ltx2gt x2
The probability of measuring two such measurements is
119875119903119900119887119909 1199091 1199092 = 119875119903119900119887119909 1199091 119875119903119900119887119909 1199092
= 1
12059011205902119890minus12059422 119908ℎ119890119903119890 1205942 equiv
1199091 minus 119883
1205901 2 +
1199092 minus 119883
1205902 2
To find the best value for χ find the maximum Prob or minimum χ2
Weighted Averages
Assume negligible systematic errors
Note χ2 or Chi squared is the sum of the squares of the deviations from the mean divided by
the corresponding uncertainty
Such methods are called ldquoMethods of Least Squaresrdquo They follow directly from the Principle of
Maximum Likelihood
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 50
Lecture 3 Slide 50
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
The probability of measuring two such measurements is
119875119903119900119887119909 1199091 1199092 = 119875119903119900119887119909 1199091 119875119903119900119887119909 1199092
= 1
12059011205902119890minus12059422 119908ℎ119890119903119890 1205942 equiv
1199091 minus 119883
1205901 2 +
1199092 minus 119883
1205902 2
To find the best value for χ find the maximum Prob or minimum χ2
Weighted Averages
This leads to 119909119882_119886119907119892 = 11990811199091+11990821199092
1199081+1199082=
119908119894 119909119894
119908119894 119908ℎ119890119903119890 119908119894 = 1
120590119894 2
Best Estimate of χ is from maximum probibility or minimum summation
Minimize Sum Solve for best estimate of χ Solve for derivative wrst χ set to 0
Note If w1=w2 we recover the standard result Xwavg= (12) (x1+x2)
Finally the width of a weighted average distribution is
120590119908119894119890119892 ℎ119905119890119889 119886119907119892 = 1
119908119894119894
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 51
Lecture 3 Slide 51
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Weighted Averages on Steroids
A very powerful method for combining data from different sources
with different methods and uncertainties (or indeed data of
related measured and calculated quantities) is Kalman filtering
The Kalman filter also known as
linear quadratic estimation (LQE)
is an algorithm that uses a series
of measurements observed over
time containing noise (random
variations) and other
inaccuracies and produces
estimates of unknown variables
that tend to be more precise than
those based on a single
measurement alone
The Kalman filter keeps track of the estimated state of the system and the variance or uncertainty of the
estimate The estimate is updated using a state transition model and measurements xk|k-1 denotes the
estimate of the systems state at time step k before the kth measurement yk has been taken into account Pk|k-1
is the corresponding uncertainty --Wikipedia 2013
Ludger Scherliess of USU Physics is a world expert at using Kalman filtering for the assimilation
of satellite and ground-based data and the USU GAMES model to predict space weather
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 52
Lecture 3 Slide 52
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Rejecting Data
Chauvenetrsquos Criterion
References Taylor Ch 6
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 53
Lecture 3 Slide 53
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly
ldquounreasonablerdquo data point from a set of randomly distributed
measurements
bull Never
bull Whenever it makes things look better
bull Chauvenetrsquos criterion provides a (quantitative) compromise
Rejecting Data
What is a good criteria for
rejecting data
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 54
Lecture 3 Slide 54
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly
ldquounreasonablerdquo data point from a set of randomly distributed
measurements
bull Never
Rejecting Data
Zallenrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 55
Lecture 3 Slide 55
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly
ldquounreasonablerdquo data point from a set of randomly distributed
measurements
bull Whenever it makes things look better
Rejecting Data
Disneyrsquos Criterion
Disneyrsquos First Law
Wishing will make it so
Disneyrsquos Second Law
Dreams are more colorful than reality
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 56
Lecture 3 Slide 56
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Data may be rejected if the expected number of measurements at
least as deviant as the suspect measurement is less than 50
Consider a set of N measurements of a single quantity
x1x2 helliphellipxN
Calculate ltxgt and σx and then determine the fractional deviations from the mean of all
the points
For the suspect point(s) xsuspect find the probability of such a point occurring in N
measurements
n = (expected number as deviant as xsuspect)
= N Prob(outside xsuspectσx)
Rejecting Data
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 57
Lecture 3 Slide 57
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Function of Gaussian Distribution
Area under curve
(probability that
ndashtσltxlt+tσ) is given
in Table at right
Probable Error (probability that
ndash067σltxlt+067σ) is 50
Error Function (probability that ndashtσltxlt+tσ )
Error Function Tabulated values-see App A
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 58
Lecture 3 Slide 58
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 59
Lecture 3 Slide 59
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashExample 1
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 60
Lecture 3 Slide 60
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Details (1)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 61
Lecture 3 Slide 61
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashDetails (2)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 62
Lecture 3 Slide 62
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos
Criterionmdash
Details (3)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 63
Lecture 3 Slide 63
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos
Criterionmdash
Example 2
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 64
Lecture 3 Slide 64
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Summary of Probability Theory
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 65
Lecture 3 Slide 65
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of
Probability
Theory-I
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 66
Lecture 3 Slide 66
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of
Probability
Theory-II
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 67
Lecture 3 Slide 67
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-III
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 68
Lecture 3 Slide 68
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-IV
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 50
Lecture 3 Slide 50
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
The probability of measuring two such measurements is
119875119903119900119887119909 1199091 1199092 = 119875119903119900119887119909 1199091 119875119903119900119887119909 1199092
= 1
12059011205902119890minus12059422 119908ℎ119890119903119890 1205942 equiv
1199091 minus 119883
1205901 2 +
1199092 minus 119883
1205902 2
To find the best value for χ find the maximum Prob or minimum χ2
Weighted Averages
This leads to 119909119882_119886119907119892 = 11990811199091+11990821199092
1199081+1199082=
119908119894 119909119894
119908119894 119908ℎ119890119903119890 119908119894 = 1
120590119894 2
Best Estimate of χ is from maximum probibility or minimum summation
Minimize Sum Solve for best estimate of χ Solve for derivative wrst χ set to 0
Note If w1=w2 we recover the standard result Xwavg= (12) (x1+x2)
Finally the width of a weighted average distribution is
120590119908119894119890119892 ℎ119905119890119889 119886119907119892 = 1
119908119894119894
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 51
Lecture 3 Slide 51
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Weighted Averages on Steroids
A very powerful method for combining data from different sources
with different methods and uncertainties (or indeed data of
related measured and calculated quantities) is Kalman filtering
The Kalman filter also known as
linear quadratic estimation (LQE)
is an algorithm that uses a series
of measurements observed over
time containing noise (random
variations) and other
inaccuracies and produces
estimates of unknown variables
that tend to be more precise than
those based on a single
measurement alone
The Kalman filter keeps track of the estimated state of the system and the variance or uncertainty of the
estimate The estimate is updated using a state transition model and measurements xk|k-1 denotes the
estimate of the systems state at time step k before the kth measurement yk has been taken into account Pk|k-1
is the corresponding uncertainty --Wikipedia 2013
Ludger Scherliess of USU Physics is a world expert at using Kalman filtering for the assimilation
of satellite and ground-based data and the USU GAMES model to predict space weather
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 52
Lecture 3 Slide 52
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Rejecting Data
Chauvenetrsquos Criterion
References Taylor Ch 6
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 53
Lecture 3 Slide 53
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly
ldquounreasonablerdquo data point from a set of randomly distributed
measurements
bull Never
bull Whenever it makes things look better
bull Chauvenetrsquos criterion provides a (quantitative) compromise
Rejecting Data
What is a good criteria for
rejecting data
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 54
Lecture 3 Slide 54
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly
ldquounreasonablerdquo data point from a set of randomly distributed
measurements
bull Never
Rejecting Data
Zallenrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 55
Lecture 3 Slide 55
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly
ldquounreasonablerdquo data point from a set of randomly distributed
measurements
bull Whenever it makes things look better
Rejecting Data
Disneyrsquos Criterion
Disneyrsquos First Law
Wishing will make it so
Disneyrsquos Second Law
Dreams are more colorful than reality
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 56
Lecture 3 Slide 56
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Data may be rejected if the expected number of measurements at
least as deviant as the suspect measurement is less than 50
Consider a set of N measurements of a single quantity
x1x2 helliphellipxN
Calculate ltxgt and σx and then determine the fractional deviations from the mean of all
the points
For the suspect point(s) xsuspect find the probability of such a point occurring in N
measurements
n = (expected number as deviant as xsuspect)
= N Prob(outside xsuspectσx)
Rejecting Data
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 57
Lecture 3 Slide 57
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Function of Gaussian Distribution
Area under curve
(probability that
ndashtσltxlt+tσ) is given
in Table at right
Probable Error (probability that
ndash067σltxlt+067σ) is 50
Error Function (probability that ndashtσltxlt+tσ )
Error Function Tabulated values-see App A
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 58
Lecture 3 Slide 58
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 59
Lecture 3 Slide 59
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashExample 1
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 60
Lecture 3 Slide 60
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Details (1)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 61
Lecture 3 Slide 61
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashDetails (2)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 62
Lecture 3 Slide 62
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos
Criterionmdash
Details (3)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 63
Lecture 3 Slide 63
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos
Criterionmdash
Example 2
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 64
Lecture 3 Slide 64
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Summary of Probability Theory
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 65
Lecture 3 Slide 65
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of
Probability
Theory-I
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 66
Lecture 3 Slide 66
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of
Probability
Theory-II
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 67
Lecture 3 Slide 67
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-III
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 68
Lecture 3 Slide 68
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-IV
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 51
Lecture 3 Slide 51
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Weighted Averages on Steroids
A very powerful method for combining data from different sources
with different methods and uncertainties (or indeed data of
related measured and calculated quantities) is Kalman filtering
The Kalman filter also known as
linear quadratic estimation (LQE)
is an algorithm that uses a series
of measurements observed over
time containing noise (random
variations) and other
inaccuracies and produces
estimates of unknown variables
that tend to be more precise than
those based on a single
measurement alone
The Kalman filter keeps track of the estimated state of the system and the variance or uncertainty of the
estimate The estimate is updated using a state transition model and measurements xk|k-1 denotes the
estimate of the systems state at time step k before the kth measurement yk has been taken into account Pk|k-1
is the corresponding uncertainty --Wikipedia 2013
Ludger Scherliess of USU Physics is a world expert at using Kalman filtering for the assimilation
of satellite and ground-based data and the USU GAMES model to predict space weather
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 52
Lecture 3 Slide 52
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Rejecting Data
Chauvenetrsquos Criterion
References Taylor Ch 6
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 53
Lecture 3 Slide 53
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly
ldquounreasonablerdquo data point from a set of randomly distributed
measurements
bull Never
bull Whenever it makes things look better
bull Chauvenetrsquos criterion provides a (quantitative) compromise
Rejecting Data
What is a good criteria for
rejecting data
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 54
Lecture 3 Slide 54
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly
ldquounreasonablerdquo data point from a set of randomly distributed
measurements
bull Never
Rejecting Data
Zallenrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 55
Lecture 3 Slide 55
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly
ldquounreasonablerdquo data point from a set of randomly distributed
measurements
bull Whenever it makes things look better
Rejecting Data
Disneyrsquos Criterion
Disneyrsquos First Law
Wishing will make it so
Disneyrsquos Second Law
Dreams are more colorful than reality
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 56
Lecture 3 Slide 56
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Data may be rejected if the expected number of measurements at
least as deviant as the suspect measurement is less than 50
Consider a set of N measurements of a single quantity
x1x2 helliphellipxN
Calculate ltxgt and σx and then determine the fractional deviations from the mean of all
the points
For the suspect point(s) xsuspect find the probability of such a point occurring in N
measurements
n = (expected number as deviant as xsuspect)
= N Prob(outside xsuspectσx)
Rejecting Data
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 57
Lecture 3 Slide 57
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Function of Gaussian Distribution
Area under curve
(probability that
ndashtσltxlt+tσ) is given
in Table at right
Probable Error (probability that
ndash067σltxlt+067σ) is 50
Error Function (probability that ndashtσltxlt+tσ )
Error Function Tabulated values-see App A
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 58
Lecture 3 Slide 58
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 59
Lecture 3 Slide 59
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashExample 1
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 60
Lecture 3 Slide 60
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Details (1)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 61
Lecture 3 Slide 61
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashDetails (2)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 62
Lecture 3 Slide 62
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos
Criterionmdash
Details (3)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 63
Lecture 3 Slide 63
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos
Criterionmdash
Example 2
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 64
Lecture 3 Slide 64
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Summary of Probability Theory
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 65
Lecture 3 Slide 65
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of
Probability
Theory-I
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 66
Lecture 3 Slide 66
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of
Probability
Theory-II
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 67
Lecture 3 Slide 67
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-III
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 68
Lecture 3 Slide 68
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-IV
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 52
Lecture 3 Slide 52
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Rejecting Data
Chauvenetrsquos Criterion
References Taylor Ch 6
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 53
Lecture 3 Slide 53
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly
ldquounreasonablerdquo data point from a set of randomly distributed
measurements
bull Never
bull Whenever it makes things look better
bull Chauvenetrsquos criterion provides a (quantitative) compromise
Rejecting Data
What is a good criteria for
rejecting data
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 54
Lecture 3 Slide 54
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly
ldquounreasonablerdquo data point from a set of randomly distributed
measurements
bull Never
Rejecting Data
Zallenrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 55
Lecture 3 Slide 55
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly
ldquounreasonablerdquo data point from a set of randomly distributed
measurements
bull Whenever it makes things look better
Rejecting Data
Disneyrsquos Criterion
Disneyrsquos First Law
Wishing will make it so
Disneyrsquos Second Law
Dreams are more colorful than reality
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 56
Lecture 3 Slide 56
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Data may be rejected if the expected number of measurements at
least as deviant as the suspect measurement is less than 50
Consider a set of N measurements of a single quantity
x1x2 helliphellipxN
Calculate ltxgt and σx and then determine the fractional deviations from the mean of all
the points
For the suspect point(s) xsuspect find the probability of such a point occurring in N
measurements
n = (expected number as deviant as xsuspect)
= N Prob(outside xsuspectσx)
Rejecting Data
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 57
Lecture 3 Slide 57
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Function of Gaussian Distribution
Area under curve
(probability that
ndashtσltxlt+tσ) is given
in Table at right
Probable Error (probability that
ndash067σltxlt+067σ) is 50
Error Function (probability that ndashtσltxlt+tσ )
Error Function Tabulated values-see App A
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 58
Lecture 3 Slide 58
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 59
Lecture 3 Slide 59
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashExample 1
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 60
Lecture 3 Slide 60
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Details (1)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 61
Lecture 3 Slide 61
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashDetails (2)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 62
Lecture 3 Slide 62
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos
Criterionmdash
Details (3)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 63
Lecture 3 Slide 63
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos
Criterionmdash
Example 2
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 64
Lecture 3 Slide 64
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Summary of Probability Theory
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 65
Lecture 3 Slide 65
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of
Probability
Theory-I
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 66
Lecture 3 Slide 66
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of
Probability
Theory-II
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 67
Lecture 3 Slide 67
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-III
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 68
Lecture 3 Slide 68
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-IV
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 53
Lecture 3 Slide 53
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly
ldquounreasonablerdquo data point from a set of randomly distributed
measurements
bull Never
bull Whenever it makes things look better
bull Chauvenetrsquos criterion provides a (quantitative) compromise
Rejecting Data
What is a good criteria for
rejecting data
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 54
Lecture 3 Slide 54
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly
ldquounreasonablerdquo data point from a set of randomly distributed
measurements
bull Never
Rejecting Data
Zallenrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 55
Lecture 3 Slide 55
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly
ldquounreasonablerdquo data point from a set of randomly distributed
measurements
bull Whenever it makes things look better
Rejecting Data
Disneyrsquos Criterion
Disneyrsquos First Law
Wishing will make it so
Disneyrsquos Second Law
Dreams are more colorful than reality
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 56
Lecture 3 Slide 56
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Data may be rejected if the expected number of measurements at
least as deviant as the suspect measurement is less than 50
Consider a set of N measurements of a single quantity
x1x2 helliphellipxN
Calculate ltxgt and σx and then determine the fractional deviations from the mean of all
the points
For the suspect point(s) xsuspect find the probability of such a point occurring in N
measurements
n = (expected number as deviant as xsuspect)
= N Prob(outside xsuspectσx)
Rejecting Data
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 57
Lecture 3 Slide 57
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Function of Gaussian Distribution
Area under curve
(probability that
ndashtσltxlt+tσ) is given
in Table at right
Probable Error (probability that
ndash067σltxlt+067σ) is 50
Error Function (probability that ndashtσltxlt+tσ )
Error Function Tabulated values-see App A
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 58
Lecture 3 Slide 58
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 59
Lecture 3 Slide 59
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashExample 1
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 60
Lecture 3 Slide 60
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Details (1)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 61
Lecture 3 Slide 61
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashDetails (2)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 62
Lecture 3 Slide 62
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos
Criterionmdash
Details (3)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 63
Lecture 3 Slide 63
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos
Criterionmdash
Example 2
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 64
Lecture 3 Slide 64
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Summary of Probability Theory
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 65
Lecture 3 Slide 65
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of
Probability
Theory-I
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 66
Lecture 3 Slide 66
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of
Probability
Theory-II
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 67
Lecture 3 Slide 67
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-III
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 68
Lecture 3 Slide 68
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-IV
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 54
Lecture 3 Slide 54
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly
ldquounreasonablerdquo data point from a set of randomly distributed
measurements
bull Never
Rejecting Data
Zallenrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 55
Lecture 3 Slide 55
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly
ldquounreasonablerdquo data point from a set of randomly distributed
measurements
bull Whenever it makes things look better
Rejecting Data
Disneyrsquos Criterion
Disneyrsquos First Law
Wishing will make it so
Disneyrsquos Second Law
Dreams are more colorful than reality
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 56
Lecture 3 Slide 56
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Data may be rejected if the expected number of measurements at
least as deviant as the suspect measurement is less than 50
Consider a set of N measurements of a single quantity
x1x2 helliphellipxN
Calculate ltxgt and σx and then determine the fractional deviations from the mean of all
the points
For the suspect point(s) xsuspect find the probability of such a point occurring in N
measurements
n = (expected number as deviant as xsuspect)
= N Prob(outside xsuspectσx)
Rejecting Data
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 57
Lecture 3 Slide 57
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Function of Gaussian Distribution
Area under curve
(probability that
ndashtσltxlt+tσ) is given
in Table at right
Probable Error (probability that
ndash067σltxlt+067σ) is 50
Error Function (probability that ndashtσltxlt+tσ )
Error Function Tabulated values-see App A
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 58
Lecture 3 Slide 58
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 59
Lecture 3 Slide 59
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashExample 1
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 60
Lecture 3 Slide 60
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Details (1)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 61
Lecture 3 Slide 61
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashDetails (2)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 62
Lecture 3 Slide 62
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos
Criterionmdash
Details (3)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 63
Lecture 3 Slide 63
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos
Criterionmdash
Example 2
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 64
Lecture 3 Slide 64
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Summary of Probability Theory
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 65
Lecture 3 Slide 65
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of
Probability
Theory-I
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 66
Lecture 3 Slide 66
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of
Probability
Theory-II
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 67
Lecture 3 Slide 67
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-III
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 68
Lecture 3 Slide 68
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-IV
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 55
Lecture 3 Slide 55
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly
ldquounreasonablerdquo data point from a set of randomly distributed
measurements
bull Whenever it makes things look better
Rejecting Data
Disneyrsquos Criterion
Disneyrsquos First Law
Wishing will make it so
Disneyrsquos Second Law
Dreams are more colorful than reality
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 56
Lecture 3 Slide 56
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Data may be rejected if the expected number of measurements at
least as deviant as the suspect measurement is less than 50
Consider a set of N measurements of a single quantity
x1x2 helliphellipxN
Calculate ltxgt and σx and then determine the fractional deviations from the mean of all
the points
For the suspect point(s) xsuspect find the probability of such a point occurring in N
measurements
n = (expected number as deviant as xsuspect)
= N Prob(outside xsuspectσx)
Rejecting Data
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 57
Lecture 3 Slide 57
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Function of Gaussian Distribution
Area under curve
(probability that
ndashtσltxlt+tσ) is given
in Table at right
Probable Error (probability that
ndash067σltxlt+067σ) is 50
Error Function (probability that ndashtσltxlt+tσ )
Error Function Tabulated values-see App A
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 58
Lecture 3 Slide 58
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 59
Lecture 3 Slide 59
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashExample 1
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 60
Lecture 3 Slide 60
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Details (1)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 61
Lecture 3 Slide 61
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashDetails (2)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 62
Lecture 3 Slide 62
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos
Criterionmdash
Details (3)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 63
Lecture 3 Slide 63
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos
Criterionmdash
Example 2
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 64
Lecture 3 Slide 64
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Summary of Probability Theory
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 65
Lecture 3 Slide 65
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of
Probability
Theory-I
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 66
Lecture 3 Slide 66
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of
Probability
Theory-II
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 67
Lecture 3 Slide 67
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-III
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 68
Lecture 3 Slide 68
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-IV
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 56
Lecture 3 Slide 56
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Data may be rejected if the expected number of measurements at
least as deviant as the suspect measurement is less than 50
Consider a set of N measurements of a single quantity
x1x2 helliphellipxN
Calculate ltxgt and σx and then determine the fractional deviations from the mean of all
the points
For the suspect point(s) xsuspect find the probability of such a point occurring in N
measurements
n = (expected number as deviant as xsuspect)
= N Prob(outside xsuspectσx)
Rejecting Data
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 57
Lecture 3 Slide 57
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Function of Gaussian Distribution
Area under curve
(probability that
ndashtσltxlt+tσ) is given
in Table at right
Probable Error (probability that
ndash067σltxlt+067σ) is 50
Error Function (probability that ndashtσltxlt+tσ )
Error Function Tabulated values-see App A
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 58
Lecture 3 Slide 58
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 59
Lecture 3 Slide 59
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashExample 1
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 60
Lecture 3 Slide 60
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Details (1)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 61
Lecture 3 Slide 61
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashDetails (2)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 62
Lecture 3 Slide 62
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos
Criterionmdash
Details (3)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 63
Lecture 3 Slide 63
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos
Criterionmdash
Example 2
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 64
Lecture 3 Slide 64
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Summary of Probability Theory
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 65
Lecture 3 Slide 65
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of
Probability
Theory-I
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 66
Lecture 3 Slide 66
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of
Probability
Theory-II
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 67
Lecture 3 Slide 67
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-III
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 68
Lecture 3 Slide 68
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-IV
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 57
Lecture 3 Slide 57
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Function of Gaussian Distribution
Area under curve
(probability that
ndashtσltxlt+tσ) is given
in Table at right
Probable Error (probability that
ndash067σltxlt+067σ) is 50
Error Function (probability that ndashtσltxlt+tσ )
Error Function Tabulated values-see App A
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 58
Lecture 3 Slide 58
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 59
Lecture 3 Slide 59
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashExample 1
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 60
Lecture 3 Slide 60
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Details (1)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 61
Lecture 3 Slide 61
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashDetails (2)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 62
Lecture 3 Slide 62
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos
Criterionmdash
Details (3)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 63
Lecture 3 Slide 63
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos
Criterionmdash
Example 2
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 64
Lecture 3 Slide 64
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Summary of Probability Theory
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 65
Lecture 3 Slide 65
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of
Probability
Theory-I
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 66
Lecture 3 Slide 66
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of
Probability
Theory-II
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 67
Lecture 3 Slide 67
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-III
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 68
Lecture 3 Slide 68
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-IV
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 58
Lecture 3 Slide 58
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 59
Lecture 3 Slide 59
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashExample 1
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 60
Lecture 3 Slide 60
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Details (1)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 61
Lecture 3 Slide 61
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashDetails (2)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 62
Lecture 3 Slide 62
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos
Criterionmdash
Details (3)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 63
Lecture 3 Slide 63
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos
Criterionmdash
Example 2
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 64
Lecture 3 Slide 64
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Summary of Probability Theory
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 65
Lecture 3 Slide 65
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of
Probability
Theory-I
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 66
Lecture 3 Slide 66
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of
Probability
Theory-II
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 67
Lecture 3 Slide 67
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-III
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 68
Lecture 3 Slide 68
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-IV
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 59
Lecture 3 Slide 59
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashExample 1
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 60
Lecture 3 Slide 60
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Details (1)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 61
Lecture 3 Slide 61
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashDetails (2)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 62
Lecture 3 Slide 62
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos
Criterionmdash
Details (3)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 63
Lecture 3 Slide 63
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos
Criterionmdash
Example 2
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 64
Lecture 3 Slide 64
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Summary of Probability Theory
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 65
Lecture 3 Slide 65
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of
Probability
Theory-I
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 66
Lecture 3 Slide 66
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of
Probability
Theory-II
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 67
Lecture 3 Slide 67
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-III
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 68
Lecture 3 Slide 68
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-IV
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 60
Lecture 3 Slide 60
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Details (1)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 61
Lecture 3 Slide 61
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashDetails (2)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 62
Lecture 3 Slide 62
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos
Criterionmdash
Details (3)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 63
Lecture 3 Slide 63
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos
Criterionmdash
Example 2
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 64
Lecture 3 Slide 64
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Summary of Probability Theory
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 65
Lecture 3 Slide 65
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of
Probability
Theory-I
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 66
Lecture 3 Slide 66
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of
Probability
Theory-II
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 67
Lecture 3 Slide 67
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-III
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 68
Lecture 3 Slide 68
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-IV
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 61
Lecture 3 Slide 61
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashDetails (2)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 62
Lecture 3 Slide 62
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos
Criterionmdash
Details (3)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 63
Lecture 3 Slide 63
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos
Criterionmdash
Example 2
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 64
Lecture 3 Slide 64
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Summary of Probability Theory
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 65
Lecture 3 Slide 65
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of
Probability
Theory-I
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 66
Lecture 3 Slide 66
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of
Probability
Theory-II
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 67
Lecture 3 Slide 67
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-III
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 68
Lecture 3 Slide 68
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-IV
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 62
Lecture 3 Slide 62
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos
Criterionmdash
Details (3)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 63
Lecture 3 Slide 63
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos
Criterionmdash
Example 2
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 64
Lecture 3 Slide 64
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Summary of Probability Theory
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 65
Lecture 3 Slide 65
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of
Probability
Theory-I
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 66
Lecture 3 Slide 66
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of
Probability
Theory-II
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 67
Lecture 3 Slide 67
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-III
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 68
Lecture 3 Slide 68
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-IV
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 63
Lecture 3 Slide 63
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos
Criterionmdash
Example 2
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 64
Lecture 3 Slide 64
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Summary of Probability Theory
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 65
Lecture 3 Slide 65
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of
Probability
Theory-I
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 66
Lecture 3 Slide 66
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of
Probability
Theory-II
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 67
Lecture 3 Slide 67
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-III
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 68
Lecture 3 Slide 68
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-IV
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 64
Lecture 3 Slide 64
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Summary of Probability Theory
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 65
Lecture 3 Slide 65
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of
Probability
Theory-I
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 66
Lecture 3 Slide 66
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of
Probability
Theory-II
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 67
Lecture 3 Slide 67
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-III
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 68
Lecture 3 Slide 68
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-IV
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 65
Lecture 3 Slide 65
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of
Probability
Theory-I
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 66
Lecture 3 Slide 66
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of
Probability
Theory-II
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 67
Lecture 3 Slide 67
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-III
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 68
Lecture 3 Slide 68
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-IV
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 66
Lecture 3 Slide 66
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of
Probability
Theory-II
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 67
Lecture 3 Slide 67
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-III
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 68
Lecture 3 Slide 68
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-IV
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 67
Lecture 3 Slide 67
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-III
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 68
Lecture 3 Slide 68
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-IV