A sequence ofN independent Bernoulli trials is performed, where N in a non-
negative integer-valued random variable, and the probability of success on any one trial is p. Let S be the total number of success and F be the total number of failures (S + F = N). Show that the joint probability generating function of S and F is given by GS,F (s,t) = GN(ps + (1 -p)t), where GN is the probability generating function of N.
Hint: use conditional expectation, E( .) = E(E( . |N)).
Also, show that if N has a Poisson distribution, then S and F are independent.
What is E( . )? I dont recall ever seeing that. I have no chance in hell of doing this question without help