Suppose N Poisson() with > 0. Conditioned on N = n, X binomial(n, p), i.e., X|N binomial(N, p). Let Y = N X.

(a) Show that X and Y each have Poisson distributions with respective parameters p and (1 p).

(b) Show that X and Y are, in fact, independent random variables! This shows that conditionally dependent random variables can be made independent through randomization.