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Unformatted text preview: Homework 2: Solutions Chapter 1 9.9 (a) Indicating each child by their initial and rearranging to make the structure more clear M H D S J T M 1 H 1 / 4 1 / 4 1 / 4 1 / 4 D 1 / 4 1 / 4 1 / 4 1 / 4 S 1 / 4 1 / 4 1 / 4 1 / 4 J 1 T 1 States D, H, S communicate with T which does not communicate with them so they are all transient. { J,T } and { M } are closed irreducible sets so those states are recurrent. (b) Letting p D , p H , p S be the probabilities that Mark ends up with the ball when Dick, Helen, or Sam have it first we have p H = 1 4 p D + 1 4 p S p D = 1 4 + 1 4 p H + 1 4 p S P S = 1 4 + 1 4 p H + 1 4 p D Symmetry implies p D = p S = c . Using this in the first equation gives p H = c/ 2. Plugging these relations into the second (or third) equation gives c = 1 / 4 + c/ 4 + c/ 8 so 5 c/ 8 = 2 / 8 and c = 2 / 5. 9.11 (a) The equations for a stationary distribution say: . 4 π (1) + . 2 π (2) = π (1) . 6 π (1) + . 4 π (2) + . 3 π (3) = π (2) . 4 π (2) + . 7 π (3) = π (3) 1 Suppose...
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 Spring '10
 DURRETT
 DICK, Markov chain, Helen, 4 j

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