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STAT 333 finalreview

STAT 333 finalreview - Final Exam Review 1 Consider a taxi...

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Final Exam Review 1. Consider a taxi station where taxis and customers arrive in Poisson processes with respective rates of one and two per minute. A taxi will wait no matter how many other taxis are present. However, if an arriving customer does not find a taxi awaiting, she leaves. Find: (a) the average number of taxis waiting, and (b) the proportion of arriving customers that get taxis. 2. Assume { X n } n 0 is a Markov chain with t.p.m. 0 0 . 3 0 . 2 0 . 5 0 . 3 0 0 . 5 0 . 2 0 0 0 . 4 0 . 6 0 0 0 . 3 0 . 7 (a) Find the two step transition probability matrix. (b) Suppose that the probability function of X 1 is given by the vector (0 , 0 . 5 , 0 , 0 . 5) . Find the probability function of X 3 . (c) Classify the state space. For each class, determine whether it is recurrent or transient. Determine their periods. (d) What does it mean by ”irreducible”? Is this MC reducible? (e) Find the long run proportions of times when the MC is in state 0, in state 2. (f) Calculate lim n →∞ E ( X n ) . 3. The number of claims received at an insurance company during a week is a random variable with mean 20 and variance 120. The amount paid in each claim is a random variable with mean 350 and variance 10000. Assume that the amounts of different claims are independent. (a) Suppose this company received exactly 3 claims in a particular week. The amount of each claim is still random as already specified. What are the mean and variance of the total amount paid to these 3 claims in this week?
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