416-7 - X n denote the number of black balls in urn 1 after...

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Denker Spring 2010 416 Stochastic Modeling - Assignment 7 Due Date: Wednesday, March 17, 2010 (exception) Problem 1: (Problem 55, p. 273) Consider a population of individuals each of whom posesses two genes which can be either type A or type a. Call type A dominant and type a recessive. Call an individual dominant if it has a dominant type, otherwise recessive. Let the population have stabilized with probabilities p , q and r . Compute the probability that the oFspring is recessive in case (i) the parents are dominant and (ii) one of the parents is dominant the other recessive. Problem 2: (Problem 66, p. 275) ±or a branching process, calculate the probability π 0 that the process will eventually be extinct when 1. P 0 = 1 4 , P 2 = 3 4 . . 2. P 0 = 1 4 , P 1 = 1 2 , P 2 = 1 4 3. P 0 = 1 6 , P 1 = 1 2 , P 2 = 1 3 . Problem 3: (Problem 70, p. 276) m white balls and m black balls are distributed among two urns, each ball containing exactly m balls. At each stage of the process one ball is selected at random from each urn and the selected balls are interchanched. Let
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Unformatted text preview: X n denote the number of black balls in urn 1 after stage n has been performed. 1. Compute the transition probabilities of the Markov chain X n . 2. Without computation, what are the limiting probabilities of this chain (steady state distribution)? 3. Compute the limiting distribution and show that the chain is reversible. Problem 4: (Problem 78, p. 279) ±or the Markov chain with transition matrix 1 2 1 3 1 6 1 3 2 3 1 2 1 2 suppose that p ( s | j ) is the probability that signal s is emitted when the underlying MC state is j , j = 0 , 1 , 2. 1. What proportion of emissions are signal s ? 2. What proportions of those times in which signal s is emitted is 0 the underlying state? Problem 5: (Problem 64, p. 275) Consider a branching process with μ < 1. Show that if X = 1, then the expected number of individuals that ever exist in this population is given by 1 1-μ ....
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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 Pennsylvania State University, University Park.

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