Let X and X2 be independent random variables with Poisson distribu- tions {p(k; } and {p(k; 2)}.

(a) Prove that X + X2 has the Poisson distribution {p(k; + 2)}.

(b) Show that the conditional distribution of X given X + X2 is binomial, namely (9.1) P{Xk X + X = n} = b (k; n, 21