4.5 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.

For a PGF G, E(X)=G′(1). Also E[X(X−1)]=G″(1), so Var(X)=G″(1)+G′(1)−[G′(1)]².

Differentiate before substituting s=1, simplify carefully, and use the non-negative variance check. The second derivative is not itself E(X²); add the first factorial moment.

For a Bernoulli PGF G=1−p+ps, G′(1)=p and G″(1)=0, giving Var(X)=p−p²=p(1−p).

Confusing G″(1) with E(X²) drops the E(X) term and gives the wrong variance.

If X and Y are independent non-negative integer-valued variables, G_{X+Y}(s)=G_X(s)G_Y(s). Independence makes the generating function factorise because E(s^{X+Y})=E(s^X)E(s^Y).

Multiply the functions, then read coefficients or differentiate the product to obtain the sum’s distribution or moments. Different distributions can be combined when their supports are compatible.

The sum of independent Bin(n₁,p) and Bin(n₂,p) variables has PGF (1−p+ps)^{n₁+n₂}, hence Bin(n₁+n₂,p).

The product rule fails without independence; adding means is always valid under finite expectations, but multiplying PGFs is not.