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# hmwk7new - X 4 = 3,X 1 = 5 | X 2 = 6 3 A six-sided die is...

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ISyE 3232 Stochastic Manufacturing and Service Systems Spring 2011 H. Ayhan & J. Dai Homework 7 March 2, 2011 (Due: at the start of class on Tuesday, March 8) 1. Let X 0 , X 1 , . . . be a Markov chain with state space { 0 , 1 , 2 } , initial distribution α = (1 / 5 , ? , 2 / 5), and transition matrix P = 1 / 5 4 / 5 ? 2 / 5 1 / 2 ? 0 1 / 10 ? Fill in the entries for P and α , and compute the following: (a) Compute Pr { X 1 = 0 | X 0 = 1 } . (b) The row vector α describes the distribution of X 0 . What is the row vector describing the distri- bution of X 1 ? (c) What is the row vector describing the distribution of X 15 ? 2. For the discrete time Markov chain X in Problem 4a) of Assignment 6, compute the following: (a) Pr { X 2 = 6 | X 0 = 5 } . (b) Pr { X 2 = 5 , X 3 = 4 , X 5 = 6 | X 0 = 3 } . (c) E( X 2 | X 0 = 6). (d) Assume the initial distribution is (0 , 0 , . 5 , . 5). Find P ( X 2 = 6). (e) With the same initial distribution in part (d), find Pr
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Unformatted text preview: { X 4 = 3 ,X 1 = 5 | X 2 = 6 } 3. A six-sided die is rolled repeatedly. After each roll n = 1 , 2 ,... , let X n be the largest number rolled in the ﬁrst n rolls. Is { X n ,n ≥ 1 } a discrete-time Markov chain? If it’s not, show that it is not. If it is, answer the following questions: (a) What is the state space and the transition probabilities of the Markov chain? (b) What is the distribution of X 1 ? 4. Redo the previous problem except replace X n with Y n where Y n is the number of sixes among the ﬁrst n rolls. (So the ﬁrst question will be, is { Y n ,n ≥ 1 } a discrete-time Markov chain?)...
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