ka-fu wong © 2004 econ1003: analysis of economic data lesson5-1 lesson 5: continuous probability...
Post on 22-Dec-2015
217 views
TRANSCRIPT
Lesson5-1 Ka-fu Wong © 2004 ECON1003: Analysis of Economic Data
Lesson 5:
Continuous Probability Distributions
Lesson5-2 Ka-fu Wong © 2004 ECON1003: Analysis of Economic Data
Outline
Continuous probability distributions
Features of univariate probability distribution
Features of bivariate probability distribution
Marginal density and Conditional density
Expectation, Variance, Covariance and Correlation Coefficient
Importance of normal distribution
The normal approximation to the binomial
Lesson5-3 Ka-fu Wong © 2004 ECON1003: Analysis of Economic Data
Types of Probability Distributions
Number of random variables
Joint distribution
1 Univariate probability distribution
2 Bivariate probability distribution
3 Trivariate probability distribution
… …
n Multivariate probability distribution
Probability distribution may be classified according to the number of random variables it describes.
Lesson5-4 Ka-fu Wong © 2004 ECON1003: Analysis of Economic Data
Continuous Probability Distributions
The curve f(x) is the continuous probability distribution (or probability curve or probability density function) of the random variable X if the probability that X will be in a specified interval of numbers is the area under the curve f(x) corresponding to the interval.
Properties of f(x)1. f(x) 0 for all x2. The total area
under the curve of f(x) is equal to 1
Lesson5-5 Ka-fu Wong © 2004 ECON1003: Analysis of Economic Data
Features of a Univariate Continuous Distribution
Let X be a random variable that takes any real values in an interval between a and b. The number of possible outcomes are by definition infinite.
The main features of a probability density function f(x) are: f(x) 0 for all x and may be larger than 1. The probability that X falls into an subinterval
(c,d) is
and lies between 0 and 1. P(X (a,b)) = 1. P(X = x) = 0.
d
c
dxxfdcXP )()),((
Lesson5-6 Ka-fu Wong © 2004 ECON1003: Analysis of Economic Data
The Univariate Uniform Distribution
If c and d are numbers on the real line, the random variable X ~ U(c,d), i.e., has a univariate uniform distribution if
otherwise 0
dxcfor c-d
1=f(x)
The mean and standard deviation of a uniform random variable x are
122
cdand
dcXX
Lesson5-7 Ka-fu Wong © 2004 ECON1003: Analysis of Economic Data
The Uniform Density
Lesson5-8 Ka-fu Wong © 2004 ECON1003: Analysis of Economic Data
The Normal Probability Distribution
The random variable X ~ N(,2), i.e., has a univariate normal distribution if for all x on the real line (-,+ )
e2
1=f(x)
2-x
21
-
and are the mean and standard deviation, = 3.14159 … and e = 2.71828 is the base of natural or Naperian logarithms.
Lesson5-9 Ka-fu Wong © 2004 ECON1003: Analysis of Economic Data
8 10 12 14 16 18 20 22
Learning exercise 4: Part-time Work on Campus
A student has been offered part-time work in a laboratory. The professor says that the work will vary from week to week. The number of hours will be between 10 and 20 with a uniform probability density function, represented as follows:
How tall is the rectangle? What is the probability of
getting less than 15 hours in a week?
Given that the student gets at least 15 hours in a week, what is the probability that more than 17.5 hours will be available?
Lesson5-10 Ka-fu Wong © 2004 ECON1003: Analysis of Economic Data
8 10 12 14 16 18 20 22
Learning exercise 4: Part-time Work on Campus
How tall is the rectangle? (20-10)*h = 1 h=0.1
What is the probability of getting less than 15 hours in a week? 0.1*(15-10) = 0.5
Given that the student gets at least 15 hours in a week, what is the probability that more than 17.5 hours will be available? 0.1*(20-17.5) = 0.25 0.25/0.5 = 0.5P(hour>17.5)/P(hour>15)
Lesson5-11 Ka-fu Wong © 2004 ECON1003: Analysis of Economic Data
Features of a Bivariate Continuous Distribution
Let X1 and X2 be a random variables that takes any real values in a region (rectangle) of (a,b,c,d). The number of possible outcomes are by definition infinite.
The main features of a probability density function f(x1,x2) are: f(x1,x2) 0 for all (x1,x2) and may be larger than 1. The probability that (X1,X2) falls into a region
(rectangle) or (p,q,r,s) is
and lies between 0 and 1. P((X1,X2) (a,b,c,d)) = 1. P((X1,X2) = (x1,x2) ) = 0.
q
p
s
r
dxdxxxfsrqpXXP 212121 ),()),,,(),((
Lesson5-12 Ka-fu Wong © 2004 ECON1003: Analysis of Economic Data
The Bivariate Uniform Distribution
If a, b, c and d are numbers on the real line, , the random variable (X1,X2) ~ U(a,b,c,d), i.e., has a bivariate uniform distribution if
otherwise 0
dxc and bxa for c)-a)(d-(b
1
=)x,f(x 2121
Lesson5-13 Ka-fu Wong © 2004 ECON1003: Analysis of Economic Data
The Marginal Density
The marginal density functions are:
y)dxf(x, f(y)
y)dyf(x, f(x)
Lesson5-14 Ka-fu Wong © 2004 ECON1003: Analysis of Economic Data
The Conditional Density
The conditional density functions are:
y)/f(y)f(x, y)|f(x
y)/f(x)f(x, x)|f(y
Lesson5-15 Ka-fu Wong © 2004 ECON1003: Analysis of Economic Data
The Expectation (Mean) of Continuous Probability Distribution
For univariate probability distribution, the expectation or mean E(X) is computed by the formula:
For bivariate probability distribution, the the expectation or mean E(X) is computed by the formula:
xf(x)dxE(X)
dyy)dx xf(x, E(X)
Lesson5-16 Ka-fu Wong © 2004 ECON1003: Analysis of Economic Data
Conditional Mean of Bivariate Discrete Probability Distribution
For bivariate probability distribution, the conditional expectation or conditional mean E(X|Y) is computed by the formula:
Unconditional expectation or mean of X, E(X)
dxy)Y|E(X
)|( yxxf
][
[
)|(
XμE
Y)]|E(XE
f(y)dydx E(X)
yxxf
Lesson5-17 Ka-fu Wong © 2004 ECON1003: Analysis of Economic Data
Expectation of a linear transformed random variable
If a and b are constants and X is a random variable, then E(a) = aE(bX) = bE(X)E(a+bX) = a+bE(X)
bE(x)a
dx f(x)x bdx f(x)a
dx f(x)bx dx f(x) a
dx f(x) bx)(a
dx bx)f(a bx)(abx)E(a
Lesson5-18 Ka-fu Wong © 2004 ECON1003: Analysis of Economic Data
The Variance of a Continuous Probability Distribution
For univariate continuous probability distribution
]μ)E[(XXV 2)(
If a and b are constants and X is a random variable, then V(a) = 0V(bX) = b2V(X)V(a+bX) = b2V(X)
Lesson5-19 Ka-fu Wong © 2004 ECON1003: Analysis of Economic Data
The Covariance of a Bivariate Discrete Probability Distribution
)]μ)(YμE[(XC YX ),( YX
Covariance measures how two random variables co-vary.
If a and b are constants and X is a random variable, then C(a,b) = 0C(a,bX) = 0C(a+bX,Y) = bC(X,Y)
Lesson5-20 Ka-fu Wong © 2004 ECON1003: Analysis of Economic Data
Variance of a sum of random variables
If a and b are constants and X and Y are random variables, then
V(X+Y) = V(X) + V(Y) + 2C(X,Y)V(aX+bY) =a2V(X) + b2V(Y) + 2abC(X,Y)
Y)C(X,
)]μ)(YμE[(X)μ(YE ]) μ(X E[
)]μ)(Yμ(X)μ(Y ) μ(X E[
)]μ(Y)μ(X E[
] )μ μY XE[YXV
YX2
Y2
X
YX2
Y2
X
2YX
2YX
2)()(
2[
2
()(
YVXV
Y)C(X,a
)]μ)(YμE[(X)μ(YE ]) μ(X E[a
)]μ)(bYμ(aX)μ(Y ) μ(Xa E[
)]μ(bY)μ(aX E[
] )μ μYaX E[YXV
22
YX2
Y22
X2
YX2
Y22
X2
2YX
2YX
abYVbXV
abb
bab
ba
babba
2)()(
2[
2
()(
Lesson5-21 Ka-fu Wong © 2004 ECON1003: Analysis of Economic Data
Correlation coefficient
The strength of the dependence between X and Y is measured by the correlation coefficient:
V(X)V(Y)Y)C(X,
Y)rr(X,C o
Lesson5-22 Ka-fu Wong © 2004 ECON1003: Analysis of Economic Data
Importance of Normal Distribution
1. Describes many random processes or continuous phenomena
2. Basis for Statistical Inference
Lesson5-23 Ka-fu Wong © 2004 ECON1003: Analysis of Economic Data
Characteristics of a Normal Probability Distribution
1. bell-shaped and single-peaked (unimodal) at the exact center of the distribution.
Lesson5-24 Ka-fu Wong © 2004 ECON1003: Analysis of Economic Data
Characteristics of a Normal Probability Distribution
2. Symmetrical about its mean. The arithmetic mean, median, and mode of the distribution are equal and located at the peak. Thus half the area under the curve is above the mean and half is below it.
Lesson5-25 Ka-fu Wong © 2004 ECON1003: Analysis of Economic Data
Characteristics of a Normal Probability Distribution
The normal probability distribution is asymptotic. That is the curve gets closer and closer to the X-axis but never actually touches it.
Lesson5-26 Ka-fu Wong © 2004 ECON1003: Analysis of Economic Data
N(0,2)
Symmetric
Mean=median = mode
Unimodal
Bell-shaped
Asymptotic
Lesson5-27 Ka-fu Wong © 2004 ECON1003: Analysis of Economic Data
N(,2)
x
x
x
(a)
(b)
(c)
Lesson5-28 Ka-fu Wong © 2004 ECON1003: Analysis of Economic Data
Normal Distribution Probability
Probability is the area under the curve!
c dX
f(X) A table may be constructed to help us find the probability
Lesson5-29 Ka-fu Wong © 2004 ECON1003: Analysis of Economic Data
Infinite Number of Normal Distribution Tables
Normal distributions differ by mean & standard deviation.
Each distribution would require its own table.
X
f(X)
Ka-fu Wong © 2004 ECON1003: Analysis of Economic Data
The Standard Normal Probability Distribution -- N(0,1)
The standard normal distribution is a normal distribution with a mean of 0 and a standard deviation of 1.
It is also called the z distribution.
A z-value is the distance between a selected value, designated X, and the population mean , divided by the population standard deviation, . The formula is:
X
z
Ka-fu Wong © 2004 ECON1003: Analysis of Economic Data
Transform to Standard Normal Distribution -- A numerical example
Any normal random variable can be transformed to a standard normal random variable
x x- (x-)/σ x/σ
0 -2 -1.4142 0
1 -1 -0.7071 0.7071
2 0 0 1.4142
3 1 0.7071 2.1213
4 2 1.4142 2.8284
Mean 2 0 0 1.4142
std 1.4142 1.4142 1 1
Ka-fu Wong © 2004 ECON1003: Analysis of Economic Data
The Standard Normal Probability Distribution
Any normal random variable can be transformed to a standard normal random variable
Suppose X ~ N(µ, 2) Z=(X-µ)/ ~ N(0,1)
P(X<k) = P [(X-µ)/ < (k-µ)/ ]
Lesson5-33 Ka-fu Wong © 2004 ECON1003: Analysis of Economic Data
Standardize the Normal Distribution
Z
= 0
z = 1
Z
Because we can transform any normal random variable into standard normal random variable, we need only one table!
Normal Distribution
Standardized Normal Distribution
X
XZ
Lesson5-34 Ka-fu Wong © 2004 ECON1003: Analysis of Economic Data
Standardizing Example
ZZ
= 0
Z = 1
.12
Normal Distribution
Standardized Normal Distribution
X = 5
= 10
6.2
12.010
52.6
XZ
Lesson5-35 Ka-fu Wong © 2004 ECON1003: Analysis of Economic Data
Obtaining the Probability
ZZ
= 0
Z = 1
0.12
Z .00 .01
0.0 .0000 .0040 .0080
.0398 .0438
0.2 .0793 .0832 .0871
0.3 .1179 .1217 .1255
0.0478
.02
0.1 .0478
Standardized Normal Probability Table (Portion)
ProbabilitiesShaded Area Exaggerated
Lesson5-36 Ka-fu Wong © 2004 ECON1003: Analysis of Economic Data
Example P(3.8 X 5)
Z Z = 0
Z = 1
-0.12
Normal Distribution
0.0478
Standardized Normal Distribution
Shaded Area Exaggerated
X = 5
= 10
3.8
12.010
58.3
XZ
Lesson5-37 Ka-fu Wong © 2004 ECON1003: Analysis of Economic Data
Example (2.9 X 7.1)
0
Z
= 1
-.21 Z.21
Normal Distribution
.1664
.0832.0832
Standardized Normal Distribution
5
= 10
2.9 7.1 X
ZX
ZX
2 9 510
21
7 1 5
1021
..
..
Shaded Area Exaggerated
Lesson5-38 Ka-fu Wong © 2004 ECON1003: Analysis of Economic Data
Example (2.9 X 7.1)
0
Z
= 1
-.21 Z.21
Normal Distribution
.1664
.0832.0832
Standardized Normal Distribution
5
= 10
2.9 7.1 X
ZX
ZX
2 9 510
21
7 1 5
1021
..
..
Shaded Area Exaggerated
Lesson5-39 Ka-fu Wong © 2004 ECON1003: Analysis of Economic Data
Example P(X 8)
ZZ
= 0
Z
= 1
.30
Normal Distribution
Standardized Normal Distribution
.1179
.5000 .3821
ZX
8 5
1030.
X = 5
= 10
8
Shaded Area Exaggerated
Lesson5-40 Ka-fu Wong © 2004 ECON1003: Analysis of Economic Data
Example P(7.1 X 8)
z = 0
Z = 1
.30 Z.21
Normal Distribution
.0832
.1179 .0347
Standardized Normal Distribution
ZX
ZX
71 510
21
8 5
1030
..
.
= 5
= 10
87.1 XShaded Area Exaggerated
Lesson5-41 Ka-fu Wong © 2004 ECON1003: Analysis of Economic Data
Normal Distribution Thinking Challenge
You work in Quality Control for GE. Light bulb life has a normal distribution with µ= 2000 hours & = 200 hours. What’s the probability that a bulb will last between 2000 & 2400 hours? less than 1470 hours?
Lesson5-42 Ka-fu Wong © 2004 ECON1003: Analysis of Economic Data
Solution P(2000 X 2400)
ZZ
= 0
Z
= 1
2.0
Normal Distribution
.4772
Standardized Normal DistributionZ
X
2400 2000
2002 0.
X = 2000
= 200
2400
P(2000<X<2400) = P [(2000-µ)/ <(X-µ)/ < (2400-µ)/ ]= P[(X-µ)/ < (2400-µ)/ ] – P [(X-µ)/ < (2000-µ)/ ]= P[(X-µ)/ < (2400-µ)/ ] – 0.5
Shaded Area Exaggerated
Lesson5-43 Ka-fu Wong © 2004 ECON1003: Analysis of Economic Data
Solution P(X 1470)
Z Z= 0
Z = 1
-2.65
Normal Distribution
.4960 .0040
.5000
Standardized Normal Distribution
ZX
1470 2000
2002 65.
X = 2000
= 200
1470
P(X<1470) = P [(X-µ)/ < (1470-µ)/ ]
Shaded Area Exaggerated
Lesson5-44 Ka-fu Wong © 2004 ECON1003: Analysis of Economic Data
Finding Z Values for Known Probabilities
Z .00 .02
0.0 .0000 .0040 .0080
0.1 .0398 .0438 .0478
0.2 .0793 .0832 .0871
.1179 .1255
Z Z = 0
Z = 1
.31
.1217 .01
0.3 .1217
Standardized Normal Probability Table (Portion)
What Is Z Given P(Z) = 0.1217?
Shaded Area Exaggerated
Lesson5-45 Ka-fu Wong © 2004 ECON1003: Analysis of Economic Data
Finding X Values for Known Probabilities
Z Z = 0
Z = 1
.31X = 5
= 10
?
Normal Distribution Standardized Normal Distribution
.1217 .1217
1.810)31.0(5 ZXShaded Area Exaggerated
Lesson5-46 Ka-fu Wong © 2004 ECON1003: Analysis of Economic Data
EXAMPLE 1
The bi-monthly starting salaries of recent MBA graduates follows the normal distribution with a mean of $2,000 and a standard deviation of $200. What is the z-value for a salary of $2,400?
00.2200$
000,2$400,2$
Xz
Lesson5-47 Ka-fu Wong © 2004 ECON1003: Analysis of Economic Data
EXAMPLE 1 continued
A z-value of 2 indicates that the value of $2,400 is one standard deviation above the mean of $2,000.
A z-value of –1.50 indicates that $1,900 is 1.5 standard deviation below the mean of $2000.
50.1200$
200,2$900,1$
Xz
What is the z-value of $1,900 ?
Lesson5-48 Ka-fu Wong © 2004 ECON1003: Analysis of Economic Data
Areas Under the Normal Curve
About 68 percent of the area under the normal curve is within one standard deviation of the mean.
± P( - < X < + ) = 0.6826
About 95 percent is within two standard deviations of the mean. ± 2 P( - 2 < X < + 2 ) = 0.9544
Practically all is within three standard deviations of the mean. ± 3 P( - 3 < X < + 3 ) = 0.9974
Lesson5-49 Ka-fu Wong © 2004 ECON1003: Analysis of Economic Data
EXAMPLE 2
The daily water usage per person in New Providence, New Jersey is normally distributed with a mean of 20 gallons and a standard deviation of 5 gallons.
About 68 percent of those living in New Providence will use how many gallons of water?
About 68% of the daily water usage will lie between 15 and 25 gallons.
Lesson5-50 Ka-fu Wong © 2004 ECON1003: Analysis of Economic Data
EXAMPLE 2 continued
What is the probability that a person from New Providence selected at random will use between 20 and 24 gallons per day?
00.05
2020
X
z
80.05
2024
X
z
P(20<X<24)=P[(20-20)/5 < (X-20)/5 < (24-20)/5 ] =P[ 0<Z<0.8 ]
The area under a normal curve between a z-value of 0 and a z-value of 0.80 is 0.2881. We conclude that 28.81 percent of the residents use between 20 and 24 gallons of water per day.
Lesson5-51 Ka-fu Wong © 2004 ECON1003: Analysis of Economic Data
How do we find P(0<z<0.8)
P(0<z<0.8) = P(z<0.8) – P(z<0)=0.7881 – 0.5=0.2881
P(z<c)
c
P(0<z<c)
c0
P(0<z<0.8) = 0.2881
Lesson5-52 Ka-fu Wong © 2004 ECON1003: Analysis of Economic Data
EXAMPLE 2 continued
What percent of the population use between 18 and 26 gallons of water per day?
40.05
2018
X
z
20.15
2026
X
z
Suppose X ~ N(µ, 2) Z=(X-µ)/ ~ N(0,1)
P(X<k) = P [(X-µ)/ < (k-µ)/ ]
Lesson5-53 Ka-fu Wong © 2004 ECON1003: Analysis of Economic Data
How do we find P(-0.4<z<1.2)
P(z<c)
c
P(0<z<c)
c0
P(-0.4<z<1.2) = P(-0.4<z<0) + P(0<z<1.2)=P(0<z<0.4) + P(0<z<1.2)=0.1554+0.3849=0.5403
P(-0.4<z<1.2) = P(z<1.2) - P(z<-0.4)= P(z<1.2) - P(z>0.4) = P(z<1.2) – [1- P(z<0.4)]=0.8849 – [1- 0.6554]=0.5403
P(-0.4<z<0) =P(0<z<0.4) because of symmetry of the z distribution.
Lesson5-54 Ka-fu Wong © 2004 ECON1003: Analysis of Economic Data
EXAMPLE 3
Professor Mann has determined that the scores in his statistics course are approximately normally distributed with a mean of 72 and a standard deviation of 5. He announces to the class that the top 15 percent of the scores will earn an A.
What is the lowest score a student can earn and still receive an A?
Lesson5-55 Ka-fu Wong © 2004 ECON1003: Analysis of Economic Data
Example 3 continued
To begin let k be the score that separates an A from a B.
15 percent of the students score more than k, then 35 percent must score between the mean of 72 and k.
Write down the relation between k and the probability: P(X>k) = 0.15 and P(X<k) =1-P(X>k) = 0.85
Transform X into z: P[(X-72)/5) < (k-72)/5 ] = P[z < (k-72)/5] P[0<z < s] =0.85 -0.5 = 0.35
Find s from table: P[0<z<1.04]=0.35
Compute k: (k-72)/5=1.04 implies K=77.2Those with a score of 77.2 or more earn an A.
Lesson5-56 Ka-fu Wong © 2004 ECON1003: Analysis of Economic Data
The Normal Approximation to the Binomial
The normal distribution (a continuous distribution) yields a good approximation of the binomial distribution (a discrete distribution) for large values of n.
The normal probability distribution is generally a good approximation to the binomial probability distribution when n and n(1- ) are both greater than 5.
Why can we approximate binomial by normal?Because of the Central Limit Theorem.
Lesson5-57 Ka-fu Wong © 2004 ECON1003: Analysis of Economic Data
The Normal Approximation continued
Recall for the binomial experiment: There are only two mutually exclusive outcomes
(success or failure) on each trial. A binomial distribution results from counting the
number of successes. Each trial is independent. The probability is fixed from trial to trial, and the
number of trials n is also fixed.
Lesson5-58 Ka-fu Wong © 2004 ECON1003: Analysis of Economic Data
The Normal Approximation
normal
binomial
Lesson5-59 Ka-fu Wong © 2004 ECON1003: Analysis of Economic Data
Continuity Correction Factor
Because the normal distribution can take all real numbers (is continuous) but the binomial distribution can only take integer values (is discrete), a normal approximation to the binomial should identify the binomial event "8" with the normal interval "(7.5, 8.5)" (and similarly for other integer values). The figure below shows that for P(X > 7) we want the magenta region which starts at 7.5.
Lesson5-60 Ka-fu Wong © 2004 ECON1003: Analysis of Economic Data
Continuity Correction Factor
Example: If n=20 and p=.25, what is the probability that X is greater than or equal to 8?
The normal approximation without the continuity correction factor yields z=(8-20 × .25)/(20 × .25 × .75)0.5 = 1.55, P(X ≥ 8) is approximately .0606 (from the table).
The continuity correction factor requires us to use 7.5 in order to include 8 since the inequality is weak and we want the region to the right. z = (7.5 - 20 × .25)/(20 × .25 × .75)0.5 = 1.29, P(X ≥ 7.5) is .0985.
The exact solution from binomial distribution function is .1019.
The continuity correct factor is important for the accuracy of the normal approximation of binomial.
The approximation is quite good.
Lesson5-61 Ka-fu Wong © 2004 ECON1003: Analysis of Economic Data
EXAMPLE 4
A recent study by a marketing research firm showed that 15% of American households owned a video camera. For a sample of 200 homes, how many of the homes would you expect to have video cameras?
30)200)(15(. n
What is the variance?
5.25)15.1)(30()1(2 n
0498.55.25
What is the standard deviation?
What is the mean?
Lesson5-62 Ka-fu Wong © 2004 ECON1003: Analysis of Economic Data
What is the probability that less than 40 homes in the sample have video cameras?
“Less than 40” means “less or equal to 39”. We use the correction factor, so X is 39.5.
The value of z is 1.88.
88.10498.5
0.305.39
X
z
EXAMPLE 5 continued
Lesson5-63 Ka-fu Wong © 2004 ECON1003: Analysis of Economic Data
Example 4 continued
From Standard Normal Table the area between 0 and 1.88 on the z scale is .4699.
So the area to the left of 1.88 is .5000 + .4699 = .9699.
The likelihood that less than 40 of the 200 homes have a video camera is about 97%.
Lesson5-64 Ka-fu Wong © 2004 ECON1003: Analysis of Economic Data
- END -
Lesson 5: Lesson 5: Continuous Probability Distributions