14.10 [Wright–Fisher population model] In a population we have two types of individuals, type A and type B. Type A is purely dominant and type B carries a particular type of recessive allele. Let Xn be the proportion of population of type A in generation n. Suppose the population is always constant of size N. Each individual in generation n + 1 chooses its ancestor at random from one of the N individuals existing at time n. The parent genetic type is always preserved. At time t = 0, the initial proportion X0 is 0.5.

a) Find the distribution of NXn+1 | Xn, or the number of individuals of type A in generation n + 1, given the proportion of individuals of type A in generation n.

b) Show that Xn is a martingale.

c) Show that the process Zn = Xn(1 − Xn)

(1 − N)n is a martingale.