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Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 5-1 Chapter 5 Some Important Discrete Probability Distributions Statistics for Managers Using Microsoft ® Excel 4 th Edition
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Chap05 discrete probability distributions

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Page 1: 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

Page 2: Chap05 discrete probability distributions

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

Page 3: Chap05 discrete probability distributions

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

Page 4: Chap05 discrete probability distributions

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)

Page 5: Chap05 discrete probability distributions

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

Page 6: Chap05 discrete probability distributions

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)

Page 7: Chap05 discrete probability distributions

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

Page 8: Chap05 discrete probability distributions

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

Page 9: Chap05 discrete probability distributions

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

Page 10: Chap05 discrete probability distributions

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

Page 11: Chap05 discrete probability distributions

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

Page 12: Chap05 discrete probability distributions

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,

Page 13: Chap05 discrete probability distributions

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

Page 14: Chap05 discrete probability distributions

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

Page 15: Chap05 discrete probability distributions

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

Page 16: Chap05 discrete probability distributions

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

Page 17: Chap05 discrete probability distributions

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

Page 18: Chap05 discrete probability distributions

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

Page 19: Chap05 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

Page 20: Chap05 discrete probability distributions

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

Page 21: Chap05 discrete probability distributions

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

Page 22: Chap05 discrete probability distributions

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)

Page 23: Chap05 discrete probability distributions

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

Page 24: Chap05 discrete probability distributions

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

Page 25: Chap05 discrete probability distributions

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

Page 26: Chap05 discrete probability distributions

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

Page 27: Chap05 discrete probability distributions

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

Page 28: Chap05 discrete probability distributions

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

Page 29: Chap05 discrete probability distributions

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 5-29

Using PHStat

Select PHStat / Probability & Prob. Distributions / Binomial…

Page 30: Chap05 discrete probability distributions

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)

Page 31: Chap05 discrete probability distributions

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

Page 32: Chap05 discrete probability distributions

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

Page 33: Chap05 discrete probability distributions

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

Page 34: Chap05 discrete probability distributions

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

Page 35: Chap05 discrete probability distributions

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

Page 36: Chap05 discrete probability distributions

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%.

Page 37: Chap05 discrete probability distributions

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 5-37

Hypergeometric Distribution in PHStat

Select:PHStat / Probability & Prob. Distributions / Hypergeometric …

Page 38: Chap05 discrete probability distributions

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)

Page 39: Chap05 discrete probability distributions

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

Page 40: Chap05 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)

Page 41: Chap05 discrete probability distributions

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

Page 42: Chap05 discrete probability distributions

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

Page 43: Chap05 discrete probability distributions

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

Page 44: Chap05 discrete probability distributions

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

Page 45: Chap05 discrete probability distributions

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

Page 46: Chap05 discrete probability distributions

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 5-46

Poisson Distribution in PHStat

Select:PHStat / Probability & Prob. Distributions / Poisson…

Page 47: Chap05 discrete probability distributions

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)

Page 48: Chap05 discrete probability distributions

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