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Unformatted text preview: 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) [1] the average number of taxis waiting, and (b) [1/2] the proportion of arriving customers that get taxis. 2. Assume { X n } n ≥ is a Markov chain with t.p.m. . 3 0 . 2 0 . 5 . 3 . 5 0 . 2 . 4 0 . 6 . 3 0 . 7 (a) Find the two step transition probability matrix. . 09 0 . 38 0 . 53 . 09 0 . 32 0 . 59 . 34 0 . 66 . 33 0 . 67 (b) [(0, 0.45, 0.325, 0.63)] Suppose that the probability function of X 1 is given by the vector (0 , . 5 , , . 5) . Find the probability function of X 3 . (c) [ { , 1 } , Transient; { 2 , 3 } , Recurrent and Aperiodic] Classify the state space. For each class, determine whether it is recurrent or transient. Determine their periods. (d) [Two Classes implies that MC is reducible] What does it mean by ”irreducible”? Is this MC reducible? (e) [ π = 0 , π 2 = 1 / 3 ] Find the long run proportions of times when the MC is in state 0, in state 2. (f) [8/3] 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) [mean = 1050, variance = 30000] 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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 Fall '08
 Chisholm
 Probability, Variance, Probability theory, PROBABILITY FUNCTION, step transition probability, susceptible individuals

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