If X is a random variable where X takes only non-negative integer values, then G_X(s) =  E(s^X) is the probability generating function (pgf). Then

The mean and variance of X can also be computed using the pgf from

The pgf of the sum X + Y of two independent random variables is given by

and for random sampling, where Z = X₁  + … + X_N, with N itself a random variable and the X_i are iid, then $G_Z\left(s\right)=G_N\left(G_X\left(s\right)\right).$