(Wright-Fisher model) The Wright-Fisher model describes the evolution of a fixed population of k genes. Genes...

 

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(Wright-Fisher model) The Wright-Fisher model describes the evolution of a fixed population of k genes. Genes can be one of two types, called alleles: A or a. Let X, denote the number of A alleles in the population at time n, where time is measured by generations. Under the model, the number of A alleles at time n +1 is obtained by drawing with replacement from the gene population at time n. Thus, conditional on there being i alleles of type A at time n, the number of A alleles at time n + 1 has a binomial distribution with parameters k and p = i/k. This gives a Markov chain with transition matrix defined by P- () ((1-D for 0 s ijsk. = Observe that Poo= P = 1. As the chain progresses, the population is eventually made up of all a alleles (state 0) or all A alleles (state k). A question of interest is what is the probability that the population evolves to the all-A state? (Wright-Fisher model) The Wright-Fisher model describes the evolution of a fixed population of k genes. Genes can be one of two types, called alleles: A or a. Let X, denote the number of A alleles in the population at time n, where time is measured by generations. Under the model, the number of A alleles at time n +1 is obtained by drawing with replacement from the gene population at time n. Thus, conditional on there being i alleles of type A at time n, the number of A alleles at time n + 1 has a binomial distribution with parameters k and p = i/k. This gives a Markov chain with transition matrix defined by P- () ((1-D for 0 s ijsk. = Observe that Poo= P = 1. As the chain progresses, the population is eventually made up of all a alleles (state 0) or all A alleles (state k). A question of interest is what is the probability that the population evolves to the all-A state?