The probability generating function P(t) for a discrete random variable Y is

The probability generating function P(t) for a discrete random variable Y is defined to be

P(t) = E(tY) = p+ p1t + p2t+ .....

where pi = P(Y = i).

a. Find P(t) for the Poisson distribution. Write

and note that the quantity being summed is a Poisson probability with mean λt.

b. Use the facts that

to derive the mean and variance of a Poisson random variable.