final08soln - Final Stat 515 May 5 2008 1 A prisoner is...

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Final Stat 515 May 5, 2008. 1. A prisoner is trapped in a cell containing three doors. The first door leads to a tunnel that returns him to his cell after two days of travel. The second leads to a tunnel that returns him to his cell after three days of travel. The third door leads immediately (in 0 days) to freedom. Assuming that the prisoner will always select doors 1, 2, and 3 with probabilities 0.5, 0.3, 0.2, what is the expected number of days until he reaches freedom? Soln : Let X denote the number of doors chosen, and let N be the total number of days spent in jail. Conditioning on X , we get E ( N ) = 3 i =1 E ( N | X = i ) P ( X = i ). Process restarts each time prisoner returns to his cell. Hence, E ( N | X = 1) = 2 + E ( N ) E ( N | X = 2) = 3 + E ( N ) E ( N | X = 3) = 0. and, E ( N ) = 0 . 5(2 + E ( N )) + 0 . 3(3 + E ( N )) + 0 . 2(0) . Solving, E ( N ) = 9 . 5. 1
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2. Let X 0 , X 1 , . . . be an iid sequence from the discrete distribution P ( X n = k ) = α k , nonzero, k = 0 , 1 , . . . . For all the questions below, you must justify your answer. (a) Explain why this is a Markov chain. (b) Is it irreducible? (c) Is it periodic ? (d) Classify the states as transient, null recurrent or positive recurrent. (e) What is its stationary distribution? Soln (a) For any n, X n - 1 and X n +1 are independent given X n , and hence also conditionally independent. (b) P ij = α j > 0 so all states communicate in one step. Hence irreducible. (c) P ii = α i > 0 so one step returns have positive probability (strongly aperiodic).
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