17. Let Y be a gamma random variable with parameters (s,). That is, its density is fY (y) = Cey ys1, y> 0 where C

17. Let Y be a gamma random variable with parameters (s,α). That is, its density is fY (y) = Ce−αy ys−1, y> 0 where C is a constant that does not depend on y. Suppose also that the conditional distribution of X given that Y = y is Poisson with mean y. That is, P{X = i|Y = y} = e−y yi

/i!, i 0 Show that the conditional distribution of Y given that X = i is the gamma distribution with parameters (s + i,α + 1).