If X is a nonnegative integer-valued random variable then the function P(z), defined for |z| 1 by 00 P(z) = E{z^]=ZzP{X=j}, 1-0 is called the probability generating function of X

(a) Show that dk dzk P(z) = = k!P{X = k}.

(b) With 0 being considered even, show that P{X is even} P(-1)+P(1) 2

(c) If X is binomial with parameters n and p, show that P{X is even} 1+(1-2p) 2

(d) If X is Poisson with mean A, show that 1+e-2A P{X is even} 2

(e) If X is geometric with parameter p, show that 1-P P{X is even} 2-p

(f) If X is a negative binomial random variable with parameters r and p, show that P{X is even} = [1 + (-1) (