S The Wright–Fisher model in population genetics. Consider a particular gene with the two alleles A and a in a population of constant size N. Suppose for simplicity that the individuals have a haploid set of chromosomes. So the gene occurs N times in each generation.

Assume each generation is created from the previous generation by random mating: Each gene of the offspring generation ‘selects’, independently of all others, a gene of the parental generation and adopts its allele.

(a) Let „n be the random set of A-individuals in the nth generation. Describe the evolution of „n by a sequence of i.i.d. random mappings and conclude that .„n/ is a Markov chain. Determine the transition matrix.

(b) Show that also Xn WD j„nj is a Markov chain, and find its transition matrix.

(c) Show that limN!1 Px.Xn D N/ D x=N for each x 2 ¹0; : : : ; N º.