416-5 - such a way that each contains three balls We say...

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Denker Spring 2010 416 Stochastic Modeling - Assignment 5 Due Date: Monday, February 22, 2010 NOTE: All problems from assignemnts 1–5 may appear in the Frst exam on ±eb 26!!! Problem 1: (Example 4.6) A gambler beds at each play one third of his present fortune rounded to the next full \$-amount. For example, if his fortune is \$4, he beds \$2. At each play he wins twice the amount of his bed or he loses it. If he is broken he stops playing and if he owns a fortune of \$ N , he stops as well. Model the gambler’s fortune by a Markov chain. Classify the states and ±nd absorbing states. Problem 2: (Example 4.13) Consider the matrix 0 0 1 / 3 0 1 / 3 1 / 3 0 3 / 4 0 0 0 1 / 4 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 3 / 4 1 / 4 0 1 / 6 0 0 0 5 / 6 Which states are transient and which are recurrent? Problem 3: (Problem 6) Let P = p p 1 - p 1 - p p P Find (by mathematical induction) P n . Problem 4: (Problem 1) Three white and three black balls are distributed in two urns in
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Unformatted text preview: such a way that each contains three balls. We say that the system is in state i , if the ±rst urn contains i white balls. At each step, we draw one ball from each urn and place the ball drawn from the ±rst urn into the second, and conversely with the ball drawn from the second urn. Let X n denote the state of the system after the n th step. Explain why X n is a Markov chain and calculate its transition probability. Classify the states, determine transient and recurrent states, and decide whether the chain is irreducible or aperiod. Problem 5: Let X n be a sequence of random variables with values in Z , such that X n = X 2 n-1 . Is X n a Markov chain? If so, determine the transition matrix and ±nd all recurrent states....
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This note was uploaded on 04/15/2010 for the course STAT 416 taught by Professor Denker during the Spring '08 term at Penn State.

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