 Solution

For a nonnegative integer-valued random variable $X$, its probability generating function isinitially defined at least for $|z|\le 1$. A general real-valued random variable need not have a probability generating function in this sense. For a binomial distribution, the binomial theorem givesDifferentiating the generating function produces factorial moments:Since $X^2=X(X-1)+X$, the mean and variance areThese formulas include the degenerate endpoint probabilities and $n=0$.