HW9Soln - runner runs barefooted is 1 / ( k + 1). 1 26. Let...

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IEOR 161 - Introduction to Stochastic Processes Spring 2010 HW9 Solutions ** Note that the numbering is from Ross 9th Edition 24. Let the state be the color of the last ball selected. Call it 0 if it was red, 1 if white, and 2 if blue. The transition probability matrix is P = 1 / 5 0 4 / 5 2 / 7 3 / 7 2 / 7 3 / 9 4 / 9 2 / 9 The limiting probabilities are obtained from π 0 = 1 / 5 π 0 + 2 / 7 π 1 + 1 / 3 π 2 π 1 = 3 / 7 π 1 + 4 / 9 π 2 π 0 + π 1 + π 2 = 1 which can be solved to get π 0 = 25 / 89, π 1 = 28 / 89, π 2 = 36 / 89. 25. Let the state X n be the number of pairs of shoes at the door that the runner de- parts from at the beginning of day n , then { X n } is a Markov Chain with transition probabilities, P i,i = 1 / 4 , 0 < i < k P i,i - 1 = 1 / 4 , 0 < i < k P i,k - i = 1 / 4 , 0 < i < k P i,k - i +1 = 1 / 4 , 0 < i < k The first equation refers to the situation where the runner returns to the same door she left from and then chooses the same door the next day, so on and so forth. Also, P 0 , 0 = 1 / 2 P 0 ,k = 1 / 2 P k,k = 1 / 4 P k, 0 = 1 / 4 P k, 1 = 1 / 4 P k,k - 1 = 1 / 4 It’s now easy to check that this Markov Chain is doubly stochastic, and so that the long run proportions in each state is the same. Hence, the proportion of time the
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Unformatted text preview: runner runs barefooted is 1 / ( k + 1). 1 26. Let the state be the ordering of the deck of n cards, so there are n ! states. The transition probabilities are P ( i 1 ,i 2 , ,i n ) , ( i j ,i 1 ,i 2 , ,i j-1 ,i j +1 , ,i n ) = 1 n This Markov Chain is doubly stochastic, so in the limit all n ! states are equally likely. 35. The transition probabilities are P = 1 1 1 / 2 1 / 2 1 / 3 1 / 3 1 / 3 1 / 4 1 / 4 1 / 4 1 / 4 0 The limiting probabilities are obtained from = 1 + 1 / 2 2 + 1 / 3 3 + 1 / 4 4 1 = 1 / 2 2 + 1 / 3 3 + 1 / 4 4 2 = 1 / 3 3 + 1 / 4 4 3 = 1 / 4 4 4 = + 1 + 2 + 3 + 4 = 1 The solution is, = 4 = 12 / 37 , 1 = 6 / 37 , 2 = 4 / 37 , 3 = 3 / 37 . 2...
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HW9Soln - runner runs barefooted is 1 / ( k + 1). 1 26. Let...

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