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This has been a brief introduction to four of the most commonly occurring discrete random variables There are many others that occur in practical problems, but those discussed here are the most important
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5
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Random Variables and Discrete Probability Distributions
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We will soon return to random variables, but only to random variables whose probability distributions are continuous For now, we pause and consider an application of our work so far by considering seven-game series in sports Then we will return to discrete probability distribution functions
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1 For a geometric random variable with parameters p and q, let r denote the probability that the rst success occurs in no more than n trials (a) Show that r = 1 qn (b) Now let r vary and show a graph of r and n 2 A sample of size 2 has been selected from the uniform distribution 1, 2, , n, but it is not known whether n = 5 or n = 6 It is agreed that if the sum of the sample is 6 or greater, then it will be decided that n = 6 This decision rule is subject to two kinds of errors: we could decide that n = 6 while in reality n = 5, or we could decide that n = 5 while in reality n = 6 Find the probabilities of each of these errors 3 A lot of 100 manufactured items contains an unknown number, k, of defective items Items are selected from the lot and inspected, and the inspected items are not replaced before the next item is drawn The second defective item is the fth item drawn What is k (Try various values of k and select the value that makes the event most likely) 4 A hypergeometric distribution has N = 100, with D = 10 special items Samples of size 4 are selected Find the probability distribution of the number of special items in the sample and then compare these probabilities with those from a binomial distribution with p = 010 5 Flip a fair coin, or use a computer to select random numbers 0 or 1, and verify that the expected waiting time for a head to appear is 2
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Seven-Game Series in Sports
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to consider seven-game play-off series in sports to discover when the winner of the series is in fact the better team to nd the in uence of winning the rst game on winning the series to discuss the effect of extending the series beyond seven games
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We pause now in our formal development of probability and statistics to concentrate on a particular application of the theory and ideas we have developed so far Sevengame play-off series in sports such as basketball play-offs and the World Series present a waiting time problem In this case, we wait until one team has won a given number of games We analyze this problem in some depth
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SEVEN-GAME SERIES
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In a seven-game series, the series ends when one team has won four games This is another waiting time problem and gives us a nite sample space as opposed to the in nite sample spaces we have considered Let the teams be A and B with probabilities of wining an individual game as p and q, where p + q = 1 We also assume that p and q remain constant throughout the series and that the games are independent; that is, winning or losing a game has no effect on winning or losing the next game We look rst at the sample space The series can last four, ve, six, or seven games We show here the ways in which team A can win the series
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