Consider the Wright-Fisher model with no selection but with mutation from allele A1 to allele A2 at rate 1 and from A2 to A1 at

Consider the Wright-Fisher model with no selection but with mutation from allele A1 to allele A2 at rate η1 and from A2 to A1 at rate

η2. With constant population size N, prove that the frequency of the A1 allele follows the beta distribution f(x) = Γ[4N(η1 + η2)]

Γ(4Nη2)Γ(4Nη1)

x4Nη2−1(1 − x)

4Nη1−1 at equilibrium. (Hint: Substitute p(x) = x in formula (15.7) defining the infinitesimal variance σ2(t, x).)