PS8 - 15 April 2009 IE 304 OPERATIONS RESEARCH III:...

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Unformatted text preview: 15 April 2009 IE 304 OPERATIONS RESEARCH III: STOCHASTIC MODELS PS # 8 Question 1) Let X1 and X2 be independent exponential random variables, each having rate . (a) (b) (c) (d) Question 2) Customers arrive at a bank at a Poisson rate . Suppose two customers arrived during the first hour. (a) What is the probability that both arrived during the first 20 minutes? (b) What is the probability that at least one arrived during the first 20 minutes? Question 3) There are three jobs and a single worker who works first on job 1, then on job 2 and finally on job 3. The amounts of time that he spends on each job are independent exponential random variables be the sum of with mean 1. Let Ci be the time at which job i is completed, i = 1,2,3, and let X = these completion times. Find, (a) E[X], (b) Var[X]. Question 4) Two individuals, A and B, both require kidney transplants. If she does not receive a new kidney, then A will die after an exponential time with rate A, and B after an exponential time with rate B. New kidneys arrive in accordance with a Poisson process having rate . It has been decided that the first kidney will go to A (or to B if B is alive and A is not at that time) and next one to B (if still living). (a) What is the probability that A obtains a new kidney? (b) What is the probability that B obtains a new kidney? Question 5) An insurance company pays out claims on its life insurance policies in accordance with a Poisson process having rate = 5 per week. If the amount of money paid on each policy is exponentially distributed with mean $2000, what is the mean and the variance of the amount of money paid by the insurance company in a four week span? Find E[min{ X1, X2}]. Find Var[min{ X1, X2}]. Find E[max{ X1, X2}]. Find Var[max{ X1, X2}]. ...
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