416-6sol - Denker Spring 2010 416 Stochastic Modeling -...

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Unformatted text preview: Denker Spring 2010 416 Stochastic Modeling - Assignment 6 Solution Problem 1: (Problem 44, p. 271) Suppose that a population consists of a fixed number of genes, say m genes. If any generation has exactly i of its m genes of type 1, then the next generation will have j type 1 genes (and hence m j type 2 genes) with probability parenleftbigg m j parenrightbigg i j ( m i ) m- j m- m . Let X n denote the number of type 1 genes in the n-th generation, and assume that X = i . 1. Find E [ X n ]. 2. What is the probability that eventually all the genes will be type 1? Solution: We have that P ( X n +1 = j | X n = i ) = parenleftbigg m j parenrightbiggparenleftbigg i m parenrightbigg j parenleftbigg m i m parenrightbigg m- j . 1. Hence E [ x n +1 | X n = i ] = m i m = i = X n , and E [ X n +1 ] = summationdisplay i E [ X n +1 | X n = i ] P ( X n = i ) = E [ X n ] . Therefore E [ X n ] = E [ X ] = i , since the chain starts in X = 1 with probability one....
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416-6sol - Denker Spring 2010 416 Stochastic Modeling -...

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