continuous distributions the uniform distribution from a to b
TRANSCRIPT
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0
0.1
0.2
0.3
0.4
0 5 10 15
1
b a
a b
f x
x0
0.1
0.2
0.3
0.4
0 5 10 15
1
b a
a b
f x
x
1
b a
a b
f x
x
Continuous Distributions
The Uniform distribution from a to b
1
0 otherwise
a x bf x b a
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The Normal distribution (mean , standard deviation )
2
221
2
x
f x e
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0
0.1
0.2
-2 0 2 4 6 8 10
The Exponential distribution
0
0 0
xe xf x
x
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Weibull distribution with parameters and.
Thus 1x
F x e
1and 0x
f x F x x e x
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The Weibull density, f(x)
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0 1 2 3 4 5
( = 0.5, = 2)
( = 0.7, = 2)
( = 0.9, = 2)
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The Gamma distribution
Let the continuous random variable X have density function:
1 0
0 0
xx e xf x
x
Then X is said to have a Gamma distribution with parameters and .
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Expectation
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Let X denote a discrete random variable with probability function p(x) (probability density function f(x) if X is continuous) then the expected value of X, E(X) is defined to be:
i ix i
E X xp x x p x
E X xf x dx
and if X is continuous with probability density function f(x)
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0
0.1
0.2
0.3
0.4
0 1 2 3 4 5 6 7
Interpretation of E(X)
1. The expected value of X, E(X), is the centre of gravity of the probability distribution of X.
2. The expected value of X, E(X), is the long-run average value of X. (shown later –Law of Large Numbers)
E(X)
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Example:
The Uniform distribution
Suppose X has a uniform distribution from a to b.
Then:
1
0 ,b a a x b
f xx a x b
The expected value of X is:
1b
b aa
E X xf x dx x dx
2 2 2
1
2 2 2
b
b a
a
x b a a b
b a
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Example:
The Normal distribution
Suppose X has a Normal distribution with parameters and .
Then: 2
221
2
x
f x e
The expected value of X is:
2
221
2
x
E X xf x dx x e dx
xz
Make the substitution:1
and dz dx x z
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Hence
Now
2
21
2
z
E X z e dz
2 2
2 21
2 2
z z
e dz ze dz
2 2
2 21
1 and 02
z z
e dz ze dz
Thus E X
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Example:
The Gamma distribution
Suppose X has a Gamma distribution with parameters and .
Then: 1 0
0 0
xx e xf x
x
Note:
This is a very useful formula when working with the Gamma distribution.
1
0
1 if 0,xf x dx x e dx
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The expected value of X is:
1
0
xE X xf x dx x x e dx
This is now equal to 1. 0
xx e dx
1
10
1
1xx e dx
1
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Thus if X has a Gamma ( ,) distribution then the expected value of X is:
E X
Special Cases: ( ,) distribution then the expected value of X is:
1. Exponential () distribution: = 1, arbitrary
1E X
2. Chi-square () distribution: = /2, = ½.
21
2
E X
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The Gamma distribution
0
0.05
0.1
0.15
0.2
0.25
0.3
0 2 4 6 8 10
E X
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0
0.05
0.1
0.15
0.2
0.25
0 5 10 15 20 25
The Exponential distribution
1E X
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0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0 5 10 15 20 25
The Chi-square (2) distribution
E X
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Expectation of functions of Random Variables
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DefinitionLet X denote a discrete random variable with probability function p(x) (probability density function f(x) if X is continuous) then the expected value of g(X), E[g(X)] is defined to be:
i ix i
E g X g x p x g x p x
E g X g x f x dx
and if X is continuous with probability density function f(x)
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Example:
The Uniform distribution
Suppose X has a uniform distribution from 0 to b.
Then:
1 0
0 0,b x b
f xx x b
Find the expected value of A = X2 . If X is the length of a side of a square (chosen at random form 0 to b) then A is the area of the square
2 2 2 1b
b aa
E X x f x dx x dx
3 3 3 2
1
0
0
3 3 3
b
b
x b b
b
= 1/3 the maximum area of the square
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Example:
The Geometric distribution
Suppose X (discrete) has a geometric distribution with parameter p.
Then: 1
1 for 1, 2,3,x
p x p p x
Find the expected value of X A and the expected value of X2.
2 32 2 2 2 2 2
1
1 2 1 3 1 4 1x
E X x p x p p p p p p p
2 3
1
1 2 1 3 1 4 1x
E X xp x p p p p p p p
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Recall: The sum of a geometric Series
Differentiating both sides with respect to r we get:
2 3
1
aa ar ar ar
r
2 3 1or with 1, 1
1a r r r
r
22 32
11 2 3 4 1 1 1
1r r r r
r
Differentiating both sides with respect to r we get:
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Thus
This formula could also be developed by noting:
2 3
2
11 2 3 4
1r r r
r
2 31 2 3 4r r r 2 31 r r r 2 3r r r 2 3r r
3r 2 31
1 1 1 1
r r r
r r r r
2 32
1 1 1 11
1 1 1 1r r r
r r r r
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This formula can be used to calculate:
2 3
1
1 2 1 3 1 4 1x
E X xp x p p p p p p p
2 31 2 3 4 where 1p r r r r p
2 31 2 3 4 where 1p r r r r p 22
1 1 1= =
1p p
r p p
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To compute the expected value of X2.
2 32 2 2 2 2 2
1
1 2 1 3 1 4 1x
E X x p x p p p p p p p
2 2 2 2 2 31 2 3 4p r r r
we need to find a formula for
2 2 2 2 2 3 2 1
1
1 2 3 4 x
x
r r r x r
Note 2 31
0
11
1x
x
r r r r S rr
Differentiating with respect to r we get
1 2 3 1 11 2
0 1
11 2 3 4
1x x
x x
S r r r r xr xrr
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Differentiating again with respect to r we get
2 31 2 3 2 4 3 5 4S r r r r
32 23
1 2
21 1 2 1 1
1x x
x x
x x r x x r rr
23
2
21
1x
x
x x rr
Thus
2 2 2
32 2
2
1x x
x x
x r xrr
2 2 2
32 2
2
1x x
x x
x r xrr
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implies
2 2 23
2 2
2
1x x
x x
x r xrr
2 2 2 2 2 3 2
3
22 3 4 5 2 3 4
1r r r r r
r
Thus
2 2 2 2 3 2
3
21 2 3 4 1 2 3 4
1r r r r r r
r
2 3
3
21 2 3 4
1
rr r r
r
3 2 3 3
2 1 2 1 1
1 1 1 1
r r r r
r r r r
3
2 if 1
pr p
p
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Thus
2 2 2 2 2 31 2 3 4E X p r r r
2
2 p
p
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Moments of Random Variables
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Definition
Let X be a random variable (discrete or continuous), then the kth moment of X is defined to be:
kk E X
-
if is discrete
if is continuous
k
x
k
x p x X
x f x dx X
The first moment of X , = 1 = E(X) is the center of gravity of the distribution of X. The higher moments give different information regarding the distribution of X.
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Definition
Let X be a random variable (discrete or continuous), then the kth central moment of X is defined to be:
0 k
k E X
-
if is discrete
if is continuous
k
x
k
x p x X
x f x dx X
where = 1 = E(X) = the first moment of X .
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The central moments describe how the probability distribution is distributed about the centre of gravity, .
01 E X
and is denoted by the symbol var(X).
= 2nd central moment. 202 E X
depends on the spread of the probability distribution of X about .
202 E X is called the variance of X.
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202 E X is called the standard
deviation of X and is denoted by the symbol .
20 22var X E X
The third central moment
contains information about the skewness of a distribution.
303 E X
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The third central moment
contains information about the skewness of a distribution.
303 E X
Measure of skewness
0 03 3
1 3 23 02
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0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0 5 10 15 20 25 30 35
03 0, 0
Positively skewed distribution
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0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0 5 10 15 20 25 30 35
03 10, 0
Negatively skewed distribution
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0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0 5 10 15 20 25 30 35
03 10, 0
Symmetric distribution
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The fourth central moment
Also contains information about the shape of a distribution. The property of shape that is measured by the fourth central moment is called kurtosis
404 E X
The measure of kurtosis
0 04 4
2 24 02
3 3
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0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0 5 10 15 20 25 30 35
040, moderate in size
Mesokurtic distribution
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0
0 20 40 60 80
040, small in size
Platykurtic distribution
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0
0 20 40 60 80
040, large in size
leptokurtic distribution
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Example: The uniform distribution from 0 to 1
1 0 1
0 0, 1
xf x
x x
11 1
0 0
11
1 1
kk k
k
xx f x dx x dx
k k
Finding the moments
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Finding the central moments:
1
0 12
0
1kk
k x f x dx x dx
12making the substitution w x
1122
1 12 2
1 11 1 12 20
1 1
k kkk
k
ww dw
k k
1
1
1if even1 1
2 12 1
0 if odd
kk
k
kk
kk
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Thus
0 0 02 3 42 4
1 1 1 1Hence , 0,
2 3 12 2 5 80
02
1var
12X
02
1var
12X
03
1 30
The standard deviation
The measure of skewness
04
1 24
1 803 3 1.2
1 12
The measure of kurtosis
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Rules for expectation
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Rules:
if is discrete
if is continuous
x
g x p x X
E g Xg x f x dx X
1. where is a constantE c c c
if then g X c E g X E c cf x dx
c f x dx c
Proof
The proof for discrete random variables is similar.
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2. where , are constantsE aX b aE X b a b
if then g X aX b E aX b ax b f x dx
Proof
The proof for discrete random variables is similar.
a xf x dx b f x dx
aE X b
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2023. var X E X
2 22x x f x dx
Proof
The proof for discrete random variables is similar.
22 22 1E X E X
2 2var X E X x f x dx
2 22x f x dx xf x dx f x dx
2 2
2 2 12E X E X
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24. var varaX b a X
Proof
2var aX baX b E aX b
2E aX b a b
22E a X
22 2 vara E X a X
aX b E aX b aE X b a b
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Moment generating functions
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Definition
if is discrete
if is continuous
x
g x p x X
E g Xg x f x dx X
Let X denote a random variable, Then the moment generating function of X , mX(t) is defined by:
Recall
if is discrete
if is continuous
tx
xtX
Xtx
e p x X
m t E ee f x dx X
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Examples
1 0,1,2, ,n xxn
p x p p x nx
The moment generating function of X , mX(t) is:
1. The Binomial distribution (parameters p, n)
tX txX
x
m t E e e p x
0
1n
n xtx x
x
ne p p
x
0 0
1n nx n xt x n x
x x
n ne p p a b
x x
1nn ta b e p p
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0,1,2,!
x
p x e xx
The moment generating function of X , mX(t) is:
2. The Poisson distribution (parameter )
tX txX
x
m t E e e p x 0 !
xntx
x
e ex
0 0
using ! !
t
xt xe u
x x
e ue e e e
x x
1te
e
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0
0 0
xe xf x
x
The moment generating function of X , mX(t) is:
3. The Exponential distribution (parameter )
0
tX tx tx xXm t E e e f x dx e e dx
0 0
t xt x e
e dxt
undefined
tt
t
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2
21
2
x
f x e
The moment generating function of X , mX(t) is:
4. The Standard Normal distribution ( = 0, = 1)
tX txXm t E e e f x dx
2 22
1
2
x tx
e dx
2
21
2
xtxe e dx
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2 2 2 22 2
2 2 21 1
2 2
x tx t x tx t
Xm t e dx e e dx
We will now use the fact that 2
221
1 for all 0,2
x b
ae dx a ba
We have completed the square
22 2
2 2 21
2
x tt t
e e dx e
This is 1
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1 0
0 0
xx e xf x
x
The moment generating function of X , mX(t) is:
4. The Gamma distribution (parameters , )
tX txXm t E e e f x dx
1
0
tx xe x e dx
1
0
t xx e dx
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We use the fact
1
0
1 for all 0, 0a
a bxbx e dx a b
a
1
0
t xXm t x e dx
1
0
t xtx e dx
tt
Equal to 1
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Properties of Moment Generating Functions
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1. mX(0) = 1
0, hence 0 1 1tX XX Xm t E e m E e E
v) Gamma Dist'n Xm tt
2
2iv) Std Normal Dist'n t
Xm t e
iii) Exponential Dist'n Xm tt
1ii) Poisson Dist'n
te
Xm t e
i) Binomial Dist'n 1nt
Xm t e p p
Note: the moment generating functions of the following distributions satisfy the property mX(0) = 1
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2 33212. 1
2! 3! !kk
Xm t t t t tk
We use the expansion of the exponential function:2 3
12! 3! !
ku u u u
e uk
tXXm t E e
2 32 31
2! 3! !
kkt t t
E tX X X Xk
2 3
2 312! 3! !
kkt t t
tE X E X E X E Xk
2 3
1 2 312! 3! !
k
k
t t tt
k
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0
3. 0k
kX X kk
t
dm m t
dt
Now 2 332
112! 3! !
kkXm t t t t t
k
2 1321 2 3
2! 3! !kk
Xm t t t ktk
2 13
1 2 2! 1 !kkt t t
k
1and 0Xm
24
2 3 2! 2 !kk
Xm t t t tk
2and 0Xm continuing we find 0kX km
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i) Binomial Dist'n 1nt
Xm t e p p
Property 3 is very useful in determining the moments of a random variable X.
Examples
11
nt tXm t n e p p pe
10 010 1
n
Xm n e p p pe np
2 11 1 1
n nt t t t tXm t np n e p p e p e e p p e
2
2
1 1 1
1 1
nt t t t
nt t t
npe e p p n e p e p p
npe e p p ne p p
2 221np np p np np q n p npq
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1ii) Poisson Dist'n
te
Xm t e
1 1t te e ttXm t e e e
1 1 2 121t t te t e t e tt
Xm t e e e e
1 2 12 2 1t te t e tt t
Xm t e e e e
1 2 12 3t te t e tte e e
1 3 1 2 13 23t t te t e t e t
e e e
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0 1 0
1 0e
Xm e
0 01 0 1 02 22 0
e e
Xm e e
3 0 2 0 0 3 23 0 3 3t
Xm e e e
To find the moments we set t = 0.
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iii) Exponential Dist'n Xm tt
1
X
d tdm t
dt t dt
2 21 1t t
3 32 1 2Xm t t t
4 42 3 1 2 3Xm t t t
5 54 2 3 4 1 4!Xm t t t
1
!kk
Xm t k t
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Thus
2
1
10Xm
3
2 2
20 2Xm
1 !
0 !kk
k X k
km k
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2 33211
2! 3! !kk
Xm t t t t tk
The moments for the exponential distribution can be calculated in an alternative way. This is note by expanding mX(t) in powers of t and equating the coefficients of tk to the coefficients in:
2 31 11
11Xm t u u u
tt u
2 3
2 31
t t t
Equating the coefficients of tk we get:
1 ! or
!k
kk k
k
k
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2
2t
Xm t e
The moments for the standard normal distribution
We use the expansion of eu.2 3
0
1! 2! 3! !
k ku
k
u u u ue u
k k
2 2 2
22
2
2 3
2 2 2
212! 3! !
t
kt t t
tXm t e
k
2 4 6 212 2 3
1 1 11
2 2! 2 3! 2 !k
kt t t t
k
We now equate the coefficients tk in:
2 222
112! ! 2 !
k kk kXm t t t t t
k k
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If k is odd: k = 0.
2 1
2 ! 2 !k
kk k
For even 2k:
2
2 !or
2 !k k
k
k
1 2 3 4 2
2! 4!Thus 0, 1, 0, 3
2 2 2!