AMS342final_sample - AMS 342 Stochastic Models Sample Final...

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AMS 342: Stochastic Models Sample Final 1. Let X and Y be two continuous r.v.’s with joint density function f ( x, y ) = ( 3 2 ( x 2 + y 2 ) if 0 x 1, 0 y 1, 0 otherwise (a) Find the marginal distribution f X ( x ). (b) Find P ( X 1 2 ). (c) Find the probability P ( X 1 2 , Y 1 2 ). (d) Are X and Y independent? Justify your answer. 2. Stores A, B, and C have 50, 75, and 100 employees, and, respectively, 50, 60, and 70 percent of these are women. Resignations are equally likely among all employees, regardless of sex. One employee resigns and this is a woman. What is the probability that she works in store C? 3. Suppose that customers arrive at a bank according to a PP ( λ ) with λ = 12 per hour. Compute the following: (a) The mean and variance of the customers who enter the bank during 5 hours. (b) Probability that more than 5 customers enter the bank during an hour. (c) Probability that nobody enters the bank during the last hour of the working day.
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