21 48 chapter 1 markov chains to check this note that

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Unformatted text preview: e is the binomial(N, x/N ) distribution, i.e., the number of successes in N trials when success has probability x/N , so the mean number of successes is x. From this it follows that if we define h(x) = x/N , then X h(x) = p(x, y )h(y ) y Taking a = N and b = 0, we have h(a) = 1 and h(b) = 0. Since Px (Va ^ Vb < 1) > 0 for all 0 < x < N , it follows from Lemma 1.27 that Px (VN < V0 ) = x/N (1.18) i.e., the probability of fixation to all A’s is equal to the fraction of the genes that are A. Our next topic is non-fair games. Example 1.43. Gambler’s ruin. Consider a gambling game in which on any turn you win $1 with probability p 6= 1/2 or lose $1 with probability 1 p. Suppose further that you will quit playing if your fortune reaches $N . Of course, if your fortune reaches $0, then the casino makes you stop. Let h(x) = Px (VN < V0 ) be the happy event that our gambler reaches the goal of $N before going bankrupt when starting with $x. Thanks to our definition of Vx as the minimum of n 0 with Xn = x we have h(0) = 0, and h(N...
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This document was uploaded on 03/06/2014 for the course MATH 4740 at Cornell University (Engineering School).

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