160a_review

160a_review - Winter 2008 Pstat160a Practice exam Problem 1...

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Unformatted text preview: Winter 2008 Pstat160a Practice exam Problem 1 1. For each of the following transition matrices, determine the communication classes, which ones are transient, recurrent, and find the period. a ) . 3 . 2 . 5 1 . 2 . 8 1 . 2 . 6 . 2 b ) . 3 . 5 . 2 1 . 2 . 8 1 . 6 . 4 c ) . 3 . 5 . 2 1 . 2 . 8 1 . 2 . 6 . 2 d ) . 3 . 5 . 2 1 . 2 . 8 1 . 2 . 6 . 2 2. Let P be the transition matrix of c) above, and X n be the associated Markov chain on the state space 1,2,3,4,5. In all the following questions, we assume the chain starts at 1, ( X = 1 ) . It would be helpful to carefully write the labels of the states on the matrices. a) What is the probability that X 4 = 5? X 4 = 3? b)What is the probability that the chain reaches 5 in 4 or less steps? Compare with a) and explain. c) What is the probability that X 4 = 3 and never hit 5 before? Compare with a) and explain. c)What is the probability that the chain never hit 5 in the first 4 steps? d) What is the probability that X 4 = 3 given that the chain never hit 5 before? Compare with c) e) What is the expected number of steps to reach 5? f) What is the probability that the chain hits 4 before 5?...
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160a_review - Winter 2008 Pstat160a Practice exam Problem 1...

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