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HW35Answers_MarkovChain2

# HW35Answers_MarkovChain2 - Math331 Spring 2008 Instructor...

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Math331, Spring 2008 Instructor: David Anderson 2nd Markov Chain Homework 1. Consider a Markov chain transition matrix P = 1 / 2 1 / 3 1 / 6 3 / 4 0 1 / 4 0 1 0 . (a) Show the this is a regular Markov chain. (b) If the process is started in state 1, find the probability that it is in state 3 after two steps. (c) Find the limiting probability vector w . (d) Find lim n →∞ p n ( x,y ) for all x,y S . Why do you know these limits exist? Solution: (a) We have that P 3 = 1 / 2 1 / 3 1 / 6 9 16 1 / 4 3 / 16 3 / 8 1 / 2 1 / 8 and so the Markov chain is regular. (b) We have P 2 = 1 / 2 1 / 3 1 / 6 3 / 8 1 / 2 1 / 8 3 / 4 0 1 / 4 . Thus P ( X 2 = 3 | X 0 = 1) = P 2 (1 , 3) = 1 / 6. (c) We have to solve the system of equations xP = x = [ x 1 ,x 2 ,x 3 ] 1 / 2 1 / 3 1 / 6 3 / 4 0 1 / 4 0 1 0 = [ x 1 ,x 2 ,x 3 ] . Subject to x 1 + x 2 + x 3 = 1. Therefore, we must solve the linear system of equations (1 / 2) x 1 + (3 / 4) x 2 = x 1 (1 / 3) x 1 + x 3 = x 2 (1 / 6) x 1 + (1 / 4) x 2 = x 3 x 1 + x 2 + x 3 = 1 .

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HW35Answers_MarkovChain2 - Math331 Spring 2008 Instructor...

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