4.5.1—Probability generating functions

For a non-negative integer-valued X, the probability generating function is G_X(s)=E(s^X)=ΣP(X=r)s^r. The coefficient of s^r is P(X=r).

Check G(1)=1. Derivatives at s=1 give factorial moments: G′(1)=E(X), while higher derivatives combine to produce variance.

For X~Bin(n,p), G_X(s)=(1−p+ps)^n, because each independent trial contributes 1−p+ps and products combine the trials.

A PGF is not an ordinary moment-generating function; its variable is a formal probability marker and the support must be non-negative integers.