STAT 333 a3

STAT 333 a3 - Stat 333 - Assignment 3 - Fall 2011 1. At all...

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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 n-th 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.

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STAT 333 a3 - Stat 333 - Assignment 3 - Fall 2011 1. At all...

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