4.5.2—PGF mean and variance

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.