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4.2 The Poisson Distribution 4.2.16. A tool and die press that stamps out cams used in small gasoline engines tends to break down once every ï¬ve hours. The machine can be repaired and put back on line quickly, but each such incident costs $50. What is the probability that maintenance expenses for the press will be no more than $100 on a typical eight-hour workday? 4.2.17. In a new ï¬ber-optic communication system, transmission errors occur at the rate of 1.5 per ten seconds. What is the probability that more than two errors will occur during the next half-minute? 4.2.18. Assume that the number of hits, X , that a baseball team makes in a nine-inning game has a Poisson distribution. If the probability that a team makes zero hits is 13 , what are their chances of getting two or more hits? 4.2.19. Flaws in metal sheeting produced by a hightemperature roller occur at the rate of one per ten square feet. What is the probability that three or more ï¬aws will appear in a ï¬ve-by-eight-foot panel? 4.2.20. Suppose a radioactive source is metered for two hours, during which time the total number of alpha particles counted is 482. What is the probability that exactly three particles will be counted in the next two minutes? Answer the question two waysâï¬rst, by deï¬ning X to be the number of particles counted in two minutes, and 235 second, by deï¬ning X to be the number of particles counted in one minute. 4.2.21. Suppose that on-the-job injuries in a textile mill occur at the rate of 0.1 per day. (a) What is the probability that two accidents will occur during the next (ï¬ve-day) workweek? (b) Is the probability that four accidents will occur over the next two workweeks the square of your answer to part (a)? Explain. 4.2.22. Find P(X = 4) if the random variable X has a Poisson distribution such that P(X = 1) = P(X = 2). 4.2.23. Let X be a Poisson random variable with parameter Î». Show that the probability that X is even is 12 (1 + eâ2Î» ). 4.2.24. Let X and Y be independent Poisson random variables with parameters Î» and Î¼, respectively. Example 3.12.10 established that X + Y is also Poisson with parameter Î» + Î¼. Prove that same result using Theorem 3.8.3. 4.2.25. If X 1 is a Poisson random variable for which E(X 1 ) = Î» and if the conditional pdf of X 2 given that X 1 = x1 is binomial with parameters x1 and p, show that the marginal pdf of X 2 is Poisson with E(X 2 ) = Î»p. Intervals Between Events: The Poisson/Exponential Relationship Situations sometimes arise where the time interval between consecutively occurring events is an important random variable. Imagine being responsible for the maintenance on a network of computers. Clearly, the number of technicians you would need to employ in order to be capable of responding to service calls in a timely fashion would be a function of the âwaiting timeâ from one breakdown to another. Figure 4.2.3 shows the relationship between the random variables X and Y , where X denotes the number of occurrences in a unit of time and Y denotes the interval between consecutive occurrences. Pictured are six intervals: X = 0 on one occasion, X = 1 on three occasions, X = 2 once, and X = 3 once. Resulting from those eight occurrences are seven measurements on the random variable Y . Obviously, the pdf for Y will depend on the pdf for X . One particularly important special case of that dependence is the Poisson/exponential relationship outlined in Theorem 4.2.3. Figure 4.2.3 Y values: y1 X=1 y2 X=1 y3 X=2 y4 X=1 y5 X=0 y6 y7 X=3 Unit time Theorem 4.2.3 Suppose a series of events satisfying the Poisson model are occurring at the rate of Î» per unit time. Let the random variable Y denote the interval between consecutive events. Then Y has the exponential distribution f Y (y) = Î»eâÎ»y , y >0