Answer Happy

6. (*) Parasitologists studying wild populations have observed that some hosts are in- fected by many parasites while most hosts are only infected by a few parasites. They call this phenomenon 'aggregation'. One measure of aggregation is the ratio Var(X) Ix = 9 EX where X is the random variable modelling the number of parasites in a single host. Greater values of Ix indicate a higher level of aggregation. (a) Consider the pair of random variables (X, A). The random variable X models the number of parasites in a host and A models the host's susceptibility of infection. Suppose that the conditional distribution of X given {A = X} is Poi (A). Show that Ix ≥ 1 with equality if and only if the marginal distribution of X is a Poisson distribution. [3 marks] (b) It can be difficult to get an accurate count of the number of parasites in a host. Consider the pair of random variables (X,Y). As before, the random variable X models the number of parasites in a host. When counting the parasites, some can be missed. To model this suppose that the conditional distribution of Y given X = x is Bin(x, p). Give an expression for Iy in terms of Ix and p. [2 marks]