Let X be the random variable equal to the proportion of x-chromosome carrying sperm in an individual drawn at random from the male population, so that the probability of an offspring from this individual being female (the event F ) conditional on a given value of X is P[F | X = x] = x. Suppose n female offspring are observed in succession from a given random male individual, denoted as the sequence of events F^1 , F^2 , ... , F^n . Show that the joint probability P[F^1 ∩ F^2 ∩ ... ∩ F^n] is given by the n -th moment of X , i.e. E[X^n].
Derive a general expression for P[X = x | F^1 ∩ F^2 ∩ ... ∩ F^n] using Bayes’ theorem. In the case that X is uniformly distributed on [0,1], discuss the behaviour of this conditional distribution as n → ∞. [Hint: you should realize that, given X = x, the events F^i are independent; however that F^i are dependent.]