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416-6sol

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

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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 0 = 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 parenrightbigg parenleftbigg 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 0 ] = i , since the chain starts in X 0 = 1 with probability one. 2. 0 and m are absorbing states, all other states are transient. Hence lim X n = 0 or = m .

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

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