227 a math professor waits at the bus stop at the

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Unformatted text preview: ime they will stay is exponentially distributed with means of 1, 1/2, and 1/3 hour. (a) What is the expected time until only one student remains? (b) For each student find the probability they are the last student left. (c) What is the expected time until all three students are gone? 2.17. Let Ti , i = 1, 2, 3 be independent exponentials with rate that for any numbers t1 , t2 , t3 max{t1 , t2 , t3 } = t1 + t2 + t3 min{t1 , t2 } min{t2 , t3 } + min{t1 , t2 , t3 } i. (a) Show min{t1 , t3 } (b) Use (a) to find E max{T1 , T2 , T3 }. (c) Use the formula to give a simple solution of part (c) of Exercise 2.16. Poisson approximation to binomial 2.18. Compare the Poisson approximation with the exact binomial probabilities of 1 success when n = 20, p = 0.1. 2.19. Compare the Poisson approximation with the exact binomial probabilities of no success when (a) n = 10, p = 0.1, (b) n = 50, p = 0.02. 2.20. The probability of a three of a kind in poker is approximately 1/50. Use the Poisson approximation to estimate the probability you will get at least one three of a kind if yo...
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