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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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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: Compute the mean and standard deviation for a discrete probability distribution Use the binomial, hypergeometric and Poisson discrete probability distributions to find probabilities Describe when to apply the binomial, hypergeometric and Poisson distributions

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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 event Random Variables Discrete Random Variable Continuous Random Variable Ch. 5Ch. 6

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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 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 = 0, 1, 2, 3, 4, or 5)

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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 HH Probability Distribution 0 1 2 X X Value Probability 0 1/4 =.25 1 2/4 =.50 2 1/4 =.25.50.25 Probability

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Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 5-6 Discrete Random Variable Summary Measures Expected Value 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

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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 X i = the i th outcome of X P(X i ) = Probability of the i th occurrence of X Discrete Random Variable Summary Measures (continued)

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Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 5-8 Computing the Mean for Investment Returns Return per $1,000 for two types of investments P(X i Y i ) 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) = μ X = (-25)(.2) +(50)(.5) + (100)(.3) = 50 E(Y) = μ Y = (-200)(.2) +(60)(.5) + (350)(.3) = 95

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Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 5-9 Computing the Standard Deviation for Investment Returns P(X i Y i ) Economic condition Passive Fund X Aggressive Fund Y.2 Recession- $ 25 - $200.5 Stable Economy+ 50 + 60.3 Expanding Economy + 100 + 350 Investment

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Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 5-10 Interpreting the Results for Investment Returns The aggressive fund has a higher expected return, but much more risk μ Y = 95 > μ X = 50 but σ Y = 193.21 > σ X = 43.30

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Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 5-11 Probability Distributions Continuous Probability Distributions Binomial Hypergeometric Poisson Probability Distributions Discrete Probability Distributions Normal Uniform Exponential Ch. 5Ch. 6

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Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 5-12 The Binomial Distribution Binomial Hypergeometric Poisson Probability Distributions Discrete Probability Distributions

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Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 5-13 Binomial Probability Distribution A fixed number of observations, n e.g.: 15 tosses of a coin; ten light bulbs taken from a shipment 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

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Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 5-14 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

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Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 5-15 Examples 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 buy” New job applicants either accept the offer or reject it

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Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 5-16 Rule of Combinations The number of combinations of selecting X objects out of n objects is where: n! =n(n - 1)(n - 2)... (2)(1) X! = X(X - 1)(X - 2)... (2)(1) 0! = 1 (by definition)

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Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 5-17 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 ! nX p(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

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Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 5-18 Example: Calculating a Binomial Probability What is the probability of one success in four flips if the probability of success is.5? X = 1, n = 4, and p =.5

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Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 5-19 n = 5 p = 0.1 n = 5 p = 0.5 Mean 0.2.4.6 012345 X P(X).2.4.6 012345 X 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 (all distributions for p=.5 are symmetrical)

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Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 5-20 Binomial Distribution Characteristics Mean Variance and Standard Deviation Wheren = sample size p = probability of success (1 – p) = probability of failure

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Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 5-21 n = 5 p = 0.1 n = 5 p = 0.5 Mean 0.2.4.6 012345 X P(X).2.4.6 012345 X P(X) 0 Binomial Characteristics Examples

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Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 5-22 Using PHStat Select PHStat / Probability & Prob. Distributions / Binomial…

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Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 5-23 Using PHStat 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 for additional output (continued)

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Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 5-24 P(X = 3 | n = 10, p =.35) =.2522 PHStat Output P(X > 5 | n = 10, p =.35) =.0949

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Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 5-25 The Hypergeometric Distribution Binomial Poisson Probability Distributions Discrete Probability Distributions Hypergeometric

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Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 5-26 The Hypergeometric Distribution “n” trials in a sample taken from a finite population of size N Sample taken without replacement Trials are dependent Concerned with finding the probability of “X” successes in the sample where there are “A” successes in the population

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Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 5-27 Hypergeometric Distribution Formula Where N = Population size A = number of successes in the population N – A = number of failures in the population n = sample size X = number of successes in the sample n – X = number of failures in the sample (Two possible outcomes per trial)

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Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 5-28 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

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Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 5-29 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 = 10n = 3 A = 4 X = 2 The probability that 2 of the 3 selected computers will have illegal software loaded is.30

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Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 5-30 Hypergeometric Distribution in PHStat Select: PHStat / Probability & Prob. Distributions / Hypergeometric …

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Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 5-31 Hypergeometric Distribution in PHStat Complete dialog box entries and get output … N = 10 n = 3 A = 4 X = 2 P(X = 2) = 0.3 (continued)

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Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 5-32 The Poisson Distribution Binomial Hypergeometric Poisson Probability Distributions Discrete Probability Distributions

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Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 5-33 The Poisson Distribution Apply the Poisson Distribution when: You wish to count the number of times an event occurs in a given interval The probability that an event occurs in the interval is the same for all intervals of equal size The number of events that occur in one interval is independent of the number of events that occur in the other intervals The average number of events per interval or unit is (lambda)

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Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 5-34 Poisson Distribution Formula where: X = number of successes per unit = expected number of successes per interval e = base of the natural logarithm system (2.71828...)

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Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 5-35 Poisson Distribution Characteristics Mean Variance and Standard Deviation where = expected number of successes per unit

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Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 5-36 Graph of Poisson Probabilities X = 0.50 0123456701234567 0.6065 0.3033 0.0758 0.0126 0.0016 0.0002 0.0000 P(X = 2) =.0758 Graphically: =.50

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Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 5-37 Poisson Distribution Shape The shape of the Poisson Distribution depends on the parameter : = 0.50 = 3.00

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Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 5-38 Poisson Distribution in PHStat Select: PHStat / Probability & Prob. Distributions / Poisson…

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Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 5-39 Poisson Distribution in PHStat Complete dialog box entries and get output … P(X = 2) = 0.0758 (continued)

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Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 5-40 Chapter Summary Addressed the probability of a discrete random variable Discussed the Binomial distribution Discussed the Poisson distribution Discussed the Hypergeometric distribution

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