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

P(t) = E(t^Y) = p_o + p₁t + p₂t² + .....

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.

E - Se y=0 y! A !! y=0 y!