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Unformatted text preview: Stat 333  Assignment 3  Fall 2011 1. At all times, a container holds a mixture of N balls, some white and the rest black. At each stage, a coin having probability p , < p < 1 , of landing heads is tossed. If heads is the result of the coin toss, then a ball is chosen at random from the container and is replaced by a white ball; if tails is the result, then a ball is chosen at random from the container and is replaced by a black ball. Let X n denote the number of white balls in the container after the nth coin toss and associated substitution of the selected ball has occurred. (a) What are the equivalence classes for this Markov chain? What are their periods? Are they transient or recurrent ? (b) Compute the transition probabilities P ij . (c) Let N = 2 . Find the limiting proportion of time that { X n } spends in each state. (d) Based on your answer in part (c) and your intuition, guess the answer for the limit ing probability when N is any positive integer . (e) Prove your guess in part (d) by showing that the equilibrium equations are satisfied. 2. Suppose that { N ( t ) , t ≥ } is a Poisson process....
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This note was uploaded on 01/21/2012 for the course STAT 333 taught by Professor Chisholm during the Fall '08 term at Waterloo.
 Fall '08
 Chisholm
 Probability

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