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Unformatted text preview: Massachusetts Institute of Technology Department of Electrical Engineering & Computer Science 6.041/6.431: Probabilistic Systems Analysis (Fall 2010) Problem Set 8 Due November 15, 2010 1. Oscar goes for a run each morning. When he leaves his house for his run, he is equally likely to go out either the front or back door; and similarly, when he returns, he is equally likely to go to either the front or back door. Oscar owns only five pairs of running shoes which he takes off immediately after the run at whichever door he happens to be. If there are no shoes at the door from which he leaves to go running, he runs barefooted. We are interested in determining the long-term proportion of time that he runs barefooted. (a) Set the scenario up as a Markov chain, specifying the states and transition probabilities. (b) Determine the long-run proportion of time Oscar runs barefooted. 2. Consider a Markov chain X 1 ,X 2 ,... modeling a symmetric simple random walk with barriers , as shown below: 1 / 2 1 / 2 1 / 2 1 / 2 1 / 2 1 / 2 1 / 2 1 / 2 1 / 2 1 / 2 1 / 2 1 / 2 1 / 2 1 / 2 1 m − 1 − m (a) Explain why | X 1 | , | X 2 | , | X 3 | ,... also satisfies the Markov property and draw the associated chain....
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- Fall '10
- Electrical Engineering, Probability theory, Markov chain, Andrey Markov, Random walk, Law of total variance, Probabilistic Systems Analysis