2.4.2 Let N have a Poisson distribution with parameter  > 0. Suppose that, conditioned on N D n, the random variable X is binomially distributed with parameters N D n and p. Set Y D N ????X. Show that X and Y have Poisson distributions with respective parameters p and .1????p/ and that X and Y are independent.

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