4.2. Let N have a Poisson distribution with parameter A > 0. Suppose that, conditioned on N = n, the random variable X is binomially distributed with parameters N = n and p. Set Y = N - X. Show that X and Y have Poisson distributions with respective parameters Ap and A(1 - p)

and that X and Y are independent.

Note: Conditionally dependent random variables may become independent through randomization.