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

P(t) = E(t^{Y}) = p_{o }+ p_{1}t + p_{2}t^{2 }+ .....

where p_{i = }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.