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Chap05 discrete probability distributions

This chapter discusses important discrete probability distributions used in statistics. It begins with an introduction to discrete random variables and probability distributions. It then covers the key concepts of mean, variance, standard deviation, and covariance for discrete distributions. The chapter focuses on explaining the binomial, hypergeometric, and Poisson distributions and how to calculate probabilities using them. It concludes with examples of how to apply these distributions to areas like finance.

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Statistics for Managers Using Microsoft® Excel 4th Edition Chapter 5 Some Important Discrete Probability Distributions Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 5-1
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 5-2
Introduction to Probability Distributions  Random Variable  Represents a possible numerical value from an uncertain event Random Variables Ch. 5 Discrete Random Variable Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Continuous Random Variable Ch. 6 Chap 5-3
Discrete Random Variables  Can only assume a countable number of values Examples:  Roll a die twice Let 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 = Using Statistics for Managers0, 1, 2, 3, 4, or 5)  Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 5-4
Discrete Probability Distribution Experiment: Toss 2 Coins. 4 possible outcomes T Let X = # heads. Probability Distribution T T H H Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. 1 2/4 = .50 1/4 = .25 Probability H 1/4 = .25 2 H Probability 0 T X Value .50 .25 0 1 2 X Chap 5-5
Discrete Random Variable Summary Measures  Expected Value (or mean) of a discrete distribution (Weighted Average) N µ = E(X) = ∑ Xi P( Xi ) i=1  Example: Toss 2 coins, X = # of heads, compute expected value of X: X P(X) 0 .25 1 .50 2 .25 E(X) = (0 x .25) + (1 x .50) + (2 x .25) Statistics for Managers Using = 1.0 Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 5-6
Discrete Random Variable Summary Measures (continued)  Variance of a discrete random variable N σ 2 = ∑ [Xi − E(X)]2 P(Xi ) i=1  Standard Deviation of a discrete random variable σ = σ2 = N [Xi − E(X)]2 P(Xi ) ∑ i=1 where: E(X) = Expected value of the Statistics for Managers Using discrete random variable X Xi = the ith outcome of X Microsoft Excel, 4e © 2004ith occurrence of X P(Xi) = Probability of the Prentice-Hall, Inc. Chap 5-7
Discrete Random Variable Summary Measures (continued)  Example: Toss 2 coins, X = # heads, compute standard deviation (recall E(X) = 1) σ= ∑ [X − E(X)] P(X ) 2 i i σ = (0 − 1)2 (.25) + (1 − 1)2 (.50) + (2 − 1)2 (.25) = .50 = .707 Possible number of heads = 0, 1, or 2 Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 5-8
The Covariance  The covariance measures the strength of the linear relationship between two variables  The covariance: N σ XY = ∑ [ Xi − E( X)][( Yi − E( Y )] P( Xi Yi ) i=1 where: X = discrete variable X Xi = the ith outcome of X Y = discrete variable Y Yi = the ith outcome of Y Statistics for Managers Using P(XiYi) = probability of occurrence of the condition affecting Microsoft Excel, 4e ©ith2004 of X and the ith outcome of Y the outcome Prentice-Hall, Inc. Chap 5-9
Computing the Mean for Investment Returns Return per $1,000 for two types of investments P(XiYi) Economic condition Investment Passive Fund X Aggressive Fund Y .2 Recession - $ 25 - $200 .5 Stable Economy + 50 + 60 .3 Expanding Economy + 100 + 350 E(X) = μX = (-25)(.2) +(50)(.5) + (100)(.3) = 50 Statistics for Managers Using E(Y) = μY = (-200)(.2) +(60)(.5) + (350)(.3) = 95 Microsoft Excel, 4e © 2004 Chap 5-10 Prentice-Hall, Inc.
Computing the Standard Deviation for Investment Returns P(XiYi) Economic condition Investment Passive Fund X Aggressive Fund Y .2 Recession - $ 25 - $200 .5 Stable Economy + 50 + 60 .3 Expanding Economy + 100 + 350 σ X = (-25 − 50)2 (.2) + (50 − 50)2 (.5) + (100 − 50)2 (.3) = 43.30 σ Y = (-200 − 95)2 (.2) + (60 − 95)2 (.5) + (350 − 95)2 (.3) Statistics for Managers Using = 193 71 Microsoft Excel, .4e © 2004 Chap 5-11 Prentice-Hall, Inc.
Computing the Covariance for Investment Returns P(XiYi) Economic condition Investment Passive Fund X Aggressive Fund Y .2 Recession - $ 25 - $200 .5 Stable Economy + 50 + 60 .3 Expanding Economy + 100 + 350 σ X,Y = (-25 − 50)(-200 − 95)(.2) + (50 − 50)(60 − 95)(.5) + (100 − 50)(350 − 95)(.3) = 8250 Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 5-12
Interpreting the Results for Investment Returns  The aggressive fund has a higher expected return, but much more risk μY = 95 > μX = 50 but σY = 193.21 > σX = 43.30 The Covariance of 8250 indicates that the two investments are positively related and will vary Statistics for Managers Using in the same 2004 Microsoft Excel, 4e © direction  Prentice-Hall, Inc. Chap 5-13
The Sum of Two Random Variables  Expected Value of the sum of two random variables: E(X + Y) = E( X) + E( Y )  Variance of the sum of two random variables: Var(X + Y) = σ 2 + Y = σ 2 + σ 2 + 2σ XY X X Y  Standard deviation of the sum of two random variables: Statistics for Managers Using σ Microsoft Excel, 4e © 2004 X + Y = Prentice-Hall, Inc. σ2 +Y X Chap 5-14
Portfolio Expected Return and Portfolio Risk  Portfolio expected return (weighted average return): E(P) = w E( X) + (1 − w ) E( Y )  Portfolio risk (weighted variability) σ P = w 2σ 2 + (1 − w )2 σ 2 + 2w(1 - w)σ XY X Y Where w = portion of portfolio Statistics for Managers Using value in asset X (1 - w) = portion of portfolio value in asset Y Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 5-15
Portfolio Example Investment X: Investment Y: μX = 50 σX = 43.30 μY = 95 σY = 193.21 σXY = 8250 Suppose 40% of the portfolio is in Investment X and 60% is in Investment Y: E(P) = .4 (50) + (.6) (95) = 77 σ P = (.4)2 (43.30)2 + (.6)2 (193.21) 2 + 2(.4)(.6)(8250) = 133.04 Statistics for Managers Using The Excel, return and portfolio variability are between the values Microsoft portfolio 4e © 2004 for investments X and Y considered individually Chap 5-16 Prentice-Hall, Inc.
Probability Distributions Probability Distributions Ch. 5 Discrete Probability Distributions Continuous Probability Distributions Binomial Normal Hypergeometric Ch. 6 Uniform Statistics for Poisson Using Managers Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Exponential Chap 5-17
The Binomial Distribution Probability Distributions Discrete Probability Distributions Binomial Hypergeometric Statistics for Poisson Using Managers Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 5-18
Binomial Probability Distribution  A fixed number of observations, n   Two mutually exclusive and collectively exhaustive categories     e.g., 15 tosses of a coin; ten light bulbs taken from a warehouse 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 5-19
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 5-20
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 Statistics for Managers Using or reject it  Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 5-21
Rule of Combinations  The number of combinations of selecting X objects out of n objects is n n!  =  X  X! (n − X)!   where: n! =n(n - 1)(n - 2) . . . (2)(1) X! = X(X - 1)(X - 2) . . . (2)(1) Statistics for Managers 0! = 1 (by definition) Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 5-22
Binomial Distribution Formula n! X n−X P(X) = p (1-p) X ! (n − X)! 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” Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. 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 Chap 5-23
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 P( X = 1) = n! p X (1 − p)n− X X! (n − X)! 5! = (.1)1(1 − .1)5−1 1 (5 − 1)! ! = (5)(.1)(.9)4 Statistics for Managers Using Microsoft Excel, 4e © 2004 = .32805 Prentice-Hall, Inc. Chap 5-24
Binomial Distribution  The shape of the binomial distribution depends on the values of p and n Mean  Here, n = 5 and p = .1 .6 .4 .2 0 P(X) X 0  Here, n = 5 and p = .5 Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. .6 .4 .2 0 n = 5 p = 0.1 P(X) 1 2 3 4 5 n = 5 p = 0.5 X 0 1 2 3 4 5 Chap 5-25
Binomial Distribution Characteristics   Mean μ = E(x) = np Variance and Standard Deviation σ = np(1 - p) 2 σ = np(1 - p) Where n = sample size p = probability Statistics for Managers of success Using (1 – p) = probability of failure Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 5-26
Binomial Characteristics Examples μ = np Mean = (5)(.1) = 0.5 σ = np(1 - p) = (5)(.1)(1 − .1) = 0.6708 μ = np = (5)(.5) = 2.5 σ = np(1 - p) = (5)(.5)(1 − .5) Statistics for Managers Using = 1.118 Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. .6 .4 .2 0 P(X) X 0 .6 .4 .2 0 n = 5 p = 0.1 P(X) 1 2 3 4 5 n = 5 p = 0.5 X 0 1 2 3 4 5 Chap 5-27
Using Binomial Tables n = 10 x … p=.20 p=.25 p=.30 p=.35 p=.40 p=.45 p=.50 0 1 2 3 4 5 6 7 8 9 10 … … … … … … … … … … … 0.1074 0.2684 0.3020 0.2013 0.0881 0.0264 0.0055 0.0008 0.0001 0.0000 0.0000 0.0563 0.1877 0.2816 0.2503 0.1460 0.0584 0.0162 0.0031 0.0004 0.0000 0.0000 0.0282 0.1211 0.2335 0.2668 0.2001 0.1029 0.0368 0.0090 0.0014 0.0001 0.0000 0.0135 0.0725 0.1757 0.2522 0.2377 0.1536 0.0689 0.0212 0.0043 0.0005 0.0000 0.0060 0.0403 0.1209 0.2150 0.2508 0.2007 0.1115 0.0425 0.0106 0.0016 0.0001 0.0025 0.0207 0.0763 0.1665 0.2384 0.2340 0.1596 0.0746 0.0229 0.0042 0.0003 0.0010 0.0098 0.0439 0.1172 0.2051 0.2461 0.2051 0.1172 0.0439 0.0098 0.0010 10 9 8 7 6 5 4 3 2 1 0 … 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 Statistics for Managers Using n= Microsoft 10, p = .75, x =2004 P(x = 2|n =10, p = .75) = .0004 Excel, 4e © 2: Chap 5-28 Prentice-Hall, Inc.
Using PHStat  Select PHStat / Probability & Prob. Distributions / Binomial… Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 5-29
Using PHStat (continued)  Enter desired values in dialog box Here: n = 10 p = .35 Output for X = 0 to X = 10 will be generated by PHStat Optional check boxes Statistics for Managers Using for additional output Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 5-30
PHStat Output P(X = 3 | n = 10, p = .35) = .2522 Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. P(X > 5 | n = 10, p = .35) = .0949 Chap 5-31
The Hypergeometric Distribution Probability Distributions Discrete Probability Distributions Binomial Hypergeometric Statistics for Poisson Using Managers Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 5-32
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 5-33
Hypergeometric Distribution Formula  A  N − A      X  n − X   P( X) =   N   n    Where N = population size A = number of successes in the population N – A = number of failures in the population n = sample size Statistics for Managers Using X = number of successes in the sample Microsoft Excel, = number of failures in the sample n – X 4e © 2004 Prentice-Hall, Inc. Chap 5-34
Properties of the Hypergeometric Distribution  The mean of the hypergeometric distribution is nA μ = E(x) = N  The standard deviation is nA(N - A) N - n σ= ⋅ 2 N N -1 Where N-n is called the “Finite Population Correction Factor” N -1 from Using Statistics for Managers sampling without replacement from a Microsoft Excel, 4efinite population © 2004 Chap 5-35 Prentice-Hall, Inc.
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 A=4 n=3 X=2  A  N − A   4  6         X  n − X   2 1  (6)(6)  =    = P(X = 2) =   = 0.3 120 N 10      n  3      Statistics for Managers2Using 3 selected computers have illegal The probability that of the Microsoft Excel, 4e © .30, or 30%. software loaded is 2004 Chap 5-36 Prentice-Hall, Inc.
Hypergeometric Distribution in PHStat  Select: PHStat / Probability & Prob. Distributions / Hypergeometric … Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 5-37
Hypergeometric Distribution in PHStat (continued)  Complete dialog box entries and get output … N = 10 A=4 n=3 X=2 Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. P(X = 2) = 0.3 Chap 5-38
The Poisson Distribution Probability Distributions Discrete Probability Distributions Binomial Hypergeometric Statistics for Poisson Using Managers Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 5-39
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 Statisticsopportunity becomes smaller for Managers Using  MicrosoftThe average 2004 Excel, 4e © number of events per unit is λ (lambda) Chap 5-40 Prentice-Hall, Inc. 
Poisson Distribution Formula −λ x e λ P( X) = X! where: X = number of successes per unit λ = expected number of successes per unit e = base of the natural logarithm system (2.71828...) Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 5-41
Poisson Distribution Characteristics   Mean μ=λ Variance and Standard Deviation σ2 = λ σ= λ where λ = expected number of successes per unit Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 5-42
Using Poisson Tables λ X 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 0 1 2 3 4 5 6 7 0.9048 0.0905 0.0045 0.0002 0.0000 0.0000 0.0000 0.0000 0.8187 0.1637 0.0164 0.0011 0.0001 0.0000 0.0000 0.0000 0.7408 0.2222 0.0333 0.0033 0.0003 0.0000 0.0000 0.0000 0.6703 0.2681 0.0536 0.0072 0.0007 0.0001 0.0000 0.0000 0.6065 0.3033 0.0758 0.0126 0.0016 0.0002 0.0000 0.0000 0.5488 0.3293 0.0988 0.0198 0.0030 0.0004 0.0000 0.0000 0.4966 0.3476 0.1217 0.0284 0.0050 0.0007 0.0001 0.0000 0.4493 0.3595 0.1438 0.0383 0.0077 0.0012 0.0002 0.0000 0.4066 0.3659 0.1647 0.0494 0.0111 0.0020 0.0003 0.0000 Example: Find P(X = 2) if λ = .50 e − λ λX e −0.50 (0.50)2 P( X = 2) = = = .0758 Statistics for Managers Using X! 2! Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 5-43
Graph of Poisson Probabilities 0.70 Graphically: 0.60 λ = .50 0 1 2 3 4 5 6 Statistics 7 λ= 0.50 P(x) X 0.50 0.40 0.30 0.6065 0.20 0.3033 0.10 0.0758 0.0126 0.00 0 1 0.0016 0.0002 0.0000 for Managers UsingP(X 0.0000 Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. 2 3 4 5 6 7 x = 2) = .0758 Chap 5-44
Poisson Distribution Shape The shape of the Poisson Distribution depends on the parameter λ :  λ = 0.50 λ = 3.00 0.70 0.25 0.60 0.20 0.15 0.40 P(x) P(x) 0.50 0.30 0.10 0.20 0.05 0.10 0.00 0.00 0 1 2 3 4 5 6 7 x Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. 1 2 3 4 5 6 7 8 9 10 x Chap 5-45 11 12
Poisson Distribution in PHStat  Select: PHStat / Probability & Prob. Distributions / Poisson… Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 5-46
Poisson Distribution in PHStat  (continued) Complete dialog box entries and get output … Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. P(X = 2) = 0.0758 Chap 5-47
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 Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 5-48