HW34Answers_MarkovChain1

# HW34Answers_MarkovChain1 - Math331 Spring 2008 Instructor...

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Math331, Spring 2008 Instructor: David Anderson 1st Markov Chain Homework 1. Suppose there are three white and three black balls in two urns distributed so that each urn contains three balls. We say the system is in state i , i = 0 , 1 , 2 , 3, if there are i white balls in urn one. At each stage one ball is drawn at random from each urn and interchanged. Let X n denote the state of the system after the n th draw, and compute the transition matrix for the Markov chain { X n : n 0 } . Solution : Consider a system in state 0. Then there are zero white balls in urn 1, and therefore, three white balls in urn 2. Thus, after interchanging one ball from each urn, the next state is necessarily 1. Therefore, p (0 , 0) = 0 , p (0 , 1) = 1 , p (0 , 2) = 0 , and p (0 , 3) = 0 . Now suppose that the system is in state 1. Then urn one has one white ball and urn two has 2 white balls. Therefore, letting X be one if a white ball is drawn from urn one and zero otherwise, and Y be one if a white ball is drawn from urn two and zero

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HW34Answers_MarkovChain1 - Math331 Spring 2008 Instructor...

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