markov_chain_practice_problems

markov_chain_practice_problems - 9 Consider a Markov chain...

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Stat 333 Summer 2007 Markov Chain: Practice Problems The following problems are from Chapter 4 of Ross (plus a few extra thrown in). The problem numbers are the same in both the 8th and 9th editions. Solutions will be provided for most of these problems. 1. #2, #3 2. #8 3. #13 4. #14 Find the classes for each chain and identify which classes are open, which classes are closed. Identify which states are transient, which states are recurrent. 5. #15, #20, #22 6. #25 7. #33, #57 8. Consider a modi±ed symmetric random walk on { 0 , 1 , . . ., N } with re²ecting barriers at both 0 (re²ecting to 1 once reaching 0) and N (re²ecting to N - 1 once reaching N ). For intermediate states 1 j N - 1, we have P j,j +1 = P j,j - 1 = 1 / 2. Find the unique equilibrium distribution of this chain. (Hint: guess at the solution and verify it rather than trying to derive it through a system of equations.) Find the expected number of steps between returns to 0, and the expected number of steps between returns to 1.
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Unformatted text preview: 9. Consider a Markov chain with state space S = { , 1 , . . ., 7 } and transition matrix 1 / 2 1 / 4 1 / 4 1 / 4 3 / 4 1 / 3 2 / 3 1 / 5 1 / 5 1 / 5 1 / 5 1 / 5 1 / 6 1 / 3 1 / 6 1 / 6 1 / 6 1 1 / 4 3 / 4 2 / 3 1 / 3 (a) Determine the classes of this chain, and organize the matrix into simple form. Which states are transient? Determine the period of each class. (b) Find the equilibrium distribution corresponding to each closed class, and write down the general form of all equilibrium distributions for this chain. (c) Find the absorption probability from each transient state into each closed class. (d) If X = 0, describe the long-run behavior of this chain. Do the same for X = 3. 1...
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