Download - Chap05 discrete probability distributions
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 5-1
Chapter 5
Some Important Discrete Probability Distributions
Statistics for ManagersUsing Microsoft® Excel
4th Edition
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 5-2
Chapter Goals
After completing this chapter, you should be able to:
Interpret the mean and standard deviation for a discrete probability distribution
Explain covariance and its application in finance Use the binomial probability distribution to find
probabilities Describe when to apply the binomial distribution Use the hypergeometric and Poisson discrete
probability distributions to find probabilities
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 5-3
Introduction to Probability Distributions
Random Variable Represents a possible numerical value from
an uncertain eventRandom Variables
Discrete Random Variable
ContinuousRandom Variable
Ch. 5 Ch. 6
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 5-4
Discrete Random Variables Can only assume a countable number of values
Examples:
Roll a die twiceLet X be the number of times 4 comes up (then X could be 0, 1, or 2 times)
Toss a coin 5 times. Let X be the number of heads
(then X = 0, 1, 2, 3, 4, or 5)
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 5-5
Experiment: Toss 2 Coins. Let X = # heads.
T
T
Discrete Probability Distribution
4 possible outcomes
T
T
H
H
H H
Probability Distribution
0 1 2 X
X Value Probability
0 1/4 = .25
1 2/4 = .50
2 1/4 = .25
.50
.25
Prob
abili
ty
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 5-6
Discrete Random Variable Summary Measures
Expected Value (or mean) of a discrete distribution (Weighted Average)
Example: Toss 2 coins, X = # of heads, compute expected value of X:
E(X) = (0 x .25) + (1 x .50) + (2 x .25) = 1.0
X P(X)
0 .25
1 .50
2 .25
N
1iii )X(PX E(X)
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 5-7
Variance of a discrete random variable
Standard Deviation of a discrete random variable
where:E(X) = Expected value of the discrete random variable X
Xi = the ith outcome of XP(Xi) = Probability of the ith occurrence of X
Discrete Random Variable Summary Measures
N
1ii
2i
2 )P(XE(X)][Xσ
(continued)
N
1ii
2i
2 )P(XE(X)][Xσσ
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 5-8
Example: Toss 2 coins, X = # heads, compute standard deviation (recall E(X) = 1)
Discrete Random Variable Summary Measures
)P(XE(X)][Xσ i2
i
.707.50(.25)1)(2(.50)1)(1(.25)1)(0σ 222
(continued)
Possible number of heads = 0, 1, or 2
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 5-9
The Covariance
The covariance measures the strength of the linear relationship between two variables
The covariance:
)YX(P)]Y(EY)][(X(EX[σN
1iiiiiXY
where: X = discrete variable XXi = the ith outcome of XY = discrete variable YYi = the ith outcome of YP(XiYi) = probability of occurrence of the condition affecting
the ith outcome of X and the ith outcome of Y
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 5-10
Computing the Mean for Investment Returns
Return per $1,000 for two types of investments
P(XiYi) Economic condition Passive Fund X Aggressive Fund Y
.2 Recession - $ 25 - $200
.5 Stable Economy + 50 + 60
.3 Expanding Economy + 100 + 350
Investment
E(X) = μX = (-25)(.2) +(50)(.5) + (100)(.3) = 50
E(Y) = μY = (-200)(.2) +(60)(.5) + (350)(.3) = 95
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 5-11
Computing the Standard Deviation for Investment Returns
P(XiYi) Economic condition Passive Fund X Aggressive Fund Y
.2 Recession - $ 25 - $200
.5 Stable Economy + 50 + 60
.3 Expanding Economy + 100 + 350
Investment
43.30
(.3)50)(100(.5)50)(50(.2)50)(-25σ 222X
71.193
)3(.)95350()5(.)9560()2(.)95200-(σ 222Y
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 5-12
Computing the Covariance for Investment Returns
P(XiYi) Economic condition Passive Fund X Aggressive Fund Y
.2 Recession - $ 25 - $200
.5 Stable Economy + 50 + 60
.3 Expanding Economy + 100 + 350
Investment
8250
95)(.3)50)(350(100
95)(.5)50)(60(5095)(.2)200-50)((-25σ YX,
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 5-13
Interpreting the Results for Investment Returns
The aggressive fund has a higher expected return, but much more risk
μY = 95 > μX = 50 butσY = 193.21 > σX = 43.30
The Covariance of 8250 indicates that the two investments are positively related and will vary in the same direction
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 5-14
The Sum of Two Random Variables
Expected Value of the sum of two random variables:
Variance of the sum of two random variables:
Standard deviation of the sum of two random variables:
XY2Y
2X
2YX σ2σσσY)Var(X
)Y(E)X(EY)E(X
2YXYX σσ
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 5-15
Portfolio Expected Return and Portfolio Risk
Portfolio expected return (weighted average return):
Portfolio risk (weighted variability)
Where w = portion of portfolio value in asset X (1 - w) = portion of portfolio value in asset Y
)Y(E)w1()X(EwE(P)
XY2Y
22X
2P w)σ-2w(1σ)w1(σwσ
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 5-16
Portfolio Example
Investment X: μX = 50 σX = 43.30 Investment Y: μY = 95 σY = 193.21 σXY = 8250
Suppose 40% of the portfolio is in Investment X and 60% is in Investment Y:
The portfolio return and portfolio variability are between the values for investments X and Y considered individually
77)95()6(.)50(4.E(P)
04.133
8250)2(.4)(.6)((193.21))6(.(43.30)(.4)σ 2222P
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 5-17
Probability Distributions
Continuous Probability
Distributions
Binomial
Hypergeometric
Poisson
Probability Distributions
Discrete Probability
Distributions
Normal
Uniform
Exponential
Ch. 5 Ch. 6
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 5-18
The Binomial Distribution
Binomial
Hypergeometric
Poisson
Probability Distributions
Discrete Probability
Distributions
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 5-19
Binomial Probability Distribution
A fixed number of observations, n e.g., 15 tosses of a coin; ten light bulbs taken from a warehouse
Two mutually exclusive and collectively exhaustive categories
e.g., head or tail in each toss of a coin; defective or not defective light bulb
Generally called “success” and “failure” Probability of success is p, probability of failure is 1 – p
Constant probability for each observation e.g., Probability of getting a tail is the same each time we toss
the coin
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 5-20
Binomial Probability Distribution(continued)
Observations are independent The outcome of one observation does not affect the outcome
of the other Two sampling methods
Infinite population without replacement Finite population with replacement
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 5-21
Possible Binomial Distribution Settings
A manufacturing plant labels items as either defective or acceptable
A firm bidding for contracts will either get a contract or not
A marketing research firm receives survey responses of “yes I will buy” or “no I will not”
New job applicants either accept the offer or reject it
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 5-22
Rule of Combinations
The number of combinations of selecting X objects out of n objects is
)!Xn(!X!n
Xn
where:n! =n(n - 1)(n - 2) . . . (2)(1)
X! = X(X - 1)(X - 2) . . . (2)(1) 0! = 1 (by definition)
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 5-23
P(X) = probability of X successes in n trials, with probability of success p on each trial
X = number of ‘successes’ in sample, (X = 0, 1, 2, ..., n)
n = sample size (number of trials or observations)
p = probability of “success”
P(X)n
X ! n Xp (1-p)X n X!
( )!
Example: Flip a coin four times, let x = # heads:
n = 4
p = 0.5
1 - p = (1 - .5) = .5
X = 0, 1, 2, 3, 4
Binomial Distribution Formula
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 5-24
Example: Calculating a Binomial Probability
What is the probability of one success in five observations if the probability of success is .1?
X = 1, n = 5, and p = .1
32805.
)9)(.1)(.5(
)1.1()1(.)!15(!1
!5
)p1(p)!Xn(!X
!n)1X(P
4
151
XnX
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 5-25
n = 5 p = 0.1
n = 5 p = 0.5
Mean
0.2.4.6
0 1 2 3 4 5X
P(X)
.2
.4
.6
0 1 2 3 4 5X
P(X)
0
Binomial Distribution
The shape of the binomial distribution depends on the values of p and n
Here, n = 5 and p = .1
Here, n = 5 and p = .5
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 5-26
Binomial Distribution Characteristics
Mean
Variance and Standard Deviation
npE(x)μ
p)-np(1σ2
p)-np(1σ
Where n = sample sizep = probability of success(1 – p) = probability of failure
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 5-27
n = 5 p = 0.1
n = 5 p = 0.5
Mean
0.2.4.6
0 1 2 3 4 5X
P(X)
.2
.4
.6
0 1 2 3 4 5X
P(X)
0
0.5(5)(.1)npμ
0.6708
.1)(5)(.1)(1p)-np(1σ
2.5(5)(.5)npμ
1.118
.5)(5)(.5)(1p)-np(1σ
Binomial CharacteristicsExamples
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 5-28
Using Binomial Tablesn = 10
x … p=.20 p=.25 p=.30 p=.35 p=.40 p=.45 p=.50
012345678910
……………………………
0.10740.26840.30200.20130.08810.02640.00550.00080.00010.00000.0000
0.05630.18770.28160.25030.14600.05840.01620.00310.00040.00000.0000
0.02820.12110.23350.26680.20010.10290.03680.00900.00140.00010.0000
0.01350.07250.17570.25220.23770.15360.06890.02120.00430.00050.0000
0.00600.04030.12090.21500.25080.20070.11150.04250.01060.00160.0001
0.00250.02070.07630.16650.23840.23400.15960.07460.02290.00420.0003
0.00100.00980.04390.11720.20510.24610.20510.11720.04390.00980.0010
109876543210
… p=.80 p=.75 p=.70 p=.65 p=.60 p=.55 p=.50 x
Examples: n = 10, p = .35, x = 3: P(x = 3|n =10, p = .35) = .2522
n = 10, p = .75, x = 2: P(x = 2|n =10, p = .75) = .0004
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 5-29
Using PHStat
Select PHStat / Probability & Prob. Distributions / Binomial…
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 5-30
Using PHStat
Enter desired values in dialog box
Here: n = 10p = .35
Output for X = 0 to X = 10 will be generated by PHStat
Optional check boxesfor additional output
(continued)
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 5-31
P(X = 3 | n = 10, p = .35) = .2522
PHStat Output
P(X > 5 | n = 10, p = .35) = .0949
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 5-32
The Hypergeometric Distribution
Binomial
Poisson
Probability Distributions
Discrete Probability
Distributions
Hypergeometric
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 5-33
The Hypergeometric Distribution
“n” trials in a sample taken from a finite population of size N
Sample taken without replacement Outcomes of trials are dependent Concerned with finding the probability of “X”
successes in the sample where there are “A” successes in the population
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 5-34
Hypergeometric Distribution Formula
n
NXn
AN
X
A
)X(P
WhereN = population sizeA = number of successes in the population
N – A = number of failures in the populationn = sample sizeX = number of successes in the sample
n – X = number of failures in the sample
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 5-35
Properties of the Hypergeometric Distribution
The mean of the hypergeometric distribution is
The standard deviation is
Where is called the “Finite Population Correction Factor” from sampling without replacement from a finite population
NnAE(x)μ
1- Nn-N
NA)-nA(Nσ 2
1- Nn-N
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 5-36
Using the Hypergeometric Distribution
■ Example: 3 different computers are checked from 10 in the department. 4 of the 10 computers have illegal software loaded. What is the probability that 2 of the 3 selected computers have illegal software loaded?
N = 10 n = 3 A = 4 X = 2
0.3120
(6)(6)
3
101
6
2
4
n
NXn
AN
X
A
2)P(X
The probability that 2 of the 3 selected computers have illegal software loaded is .30, or 30%.
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 5-37
Hypergeometric Distribution in PHStat
Select:PHStat / Probability & Prob. Distributions / Hypergeometric …
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 5-38
Hypergeometric Distribution in PHStat
Complete dialog box entries and get output …
N = 10 n = 3A = 4 X = 2
P(X = 2) = 0.3
(continued)
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 5-39
The Poisson Distribution
Binomial
Hypergeometric
Poisson
Probability Distributions
Discrete Probability
Distributions
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 5-40
The Poisson Distribution
Apply the Poisson Distribution when: You wish to count the number of times an event
occurs in a given area of opportunity The probability that an event occurs in one area of
opportunity is the same for all areas of opportunity The number of events that occur in one area of
opportunity is independent of the number of events that occur in the other areas of opportunity
The probability that two or more events occur in an area of opportunity approaches zero as the area of opportunity becomes smaller
The average number of events per unit is (lambda)
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 5-41
Poisson Distribution Formula
where:X = number of successes per unit = expected number of successes per unite = base of the natural logarithm system (2.71828...)
!Xe)X(P
x
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 5-42
Poisson Distribution Characteristics
Mean
Variance and Standard Deviation
λμ
λσ2
λσ
where = expected number of successes per unit
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 5-43
Using Poisson Tables
X
0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90
01234567
0.90480.09050.00450.00020.00000.00000.00000.0000
0.81870.16370.01640.00110.00010.00000.00000.0000
0.74080.22220.03330.00330.00030.00000.00000.0000
0.67030.26810.05360.00720.00070.00010.00000.0000
0.60650.30330.07580.01260.00160.00020.00000.0000
0.54880.32930.09880.01980.00300.00040.00000.0000
0.49660.34760.12170.02840.00500.00070.00010.0000
0.44930.35950.14380.03830.00770.00120.00020.0000
0.40660.36590.16470.04940.01110.00200.00030.0000
Example: Find P(X = 2) if = .50
.07582!(0.50)e
!Xe)2X(P
20.50X
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 5-44
Graph of Poisson Probabilities
0.00
0.10
0.20
0.30
0.40
0.50
0.60
0.70
0 1 2 3 4 5 6 7
x
P(x)X
=0.50
01234567
0.60650.30330.07580.01260.00160.00020.00000.0000
P(X = 2) = .0758
Graphically: = .50
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 5-45
Poisson Distribution Shape
The shape of the Poisson Distribution depends on the parameter :
0.00
0.05
0.10
0.15
0.20
0.25
1 2 3 4 5 6 7 8 9 10 11 12
x
P(x
)
0.00
0.10
0.20
0.30
0.40
0.50
0.60
0.70
0 1 2 3 4 5 6 7
x
P(x)
= 0.50 = 3.00
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 5-46
Poisson Distribution in PHStat
Select:PHStat / Probability & Prob. Distributions / Poisson…
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 5-47
Poisson Distribution in PHStat
Complete dialog box entries and get output …
P(X = 2) = 0.0758
(continued)
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 5-48
Chapter Summary
Addressed the probability of a discrete random variable
Defined covariance and discussed its application in finance
Discussed the Binomial distribution Discussed the Hypergeometric distribution Reviewed the Poisson distribution