6 chapter 1 markov chains in this model the states x 0

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Unformatted text preview: he states x = 0 and N that correspond to fixation of the population in the all a or all A states are absorbing states, that is, p(x, x) = 1. So it is natural to ask: Q1. Starting from i of the A alleles and N i of the a alleles, what is the probability that the population fixates in the all A state? To make this simple model more realistic we can introduce the possibility of mutations: an A that is drawn ends up being an a in the next generation with probability u, while an a that is drawn ends up being an A in the next generation with probability v . In this case the probability an A is produced by a given draw is i Ni ⇢i = (1 u) + v N N but the transition probability still has the binomial form ✓◆ N p(i, j ) = (⇢i )j (1 ⇢i )N j j If u and v are both positive, then 0 and N are no longer absorbing states, so we ask: Q2. Does the genetic composition settle down to an equilibrium distribution as time t ! 1? As the next example shows it is easy to extend the notion of a Markov chain to cover situations in which the future evolution is independent of the past when we know the last two states. Example 1.10. Two-stage M...
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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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