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A function G(t) of the form

p0+p1t+p2t2+...

The coefficients p_i are the probabilities of some random variable X taking the value i.
This means that G(1)=1, G^′ (1) = E(X) , G ^′′ (1) = E(X²) , which can be used to find the variance.

Let X be a random variable taking values in N₀ = {0,1,2, …} , and p_r = P(X=r) for r ∈ N₀ . Then the probability generating function of X is

p(z) = E(zX) = ∞ ∑ r=0  zr P(X=r) = ∞ ∑ r=0  przr.

Then

E X = lim z → 1  p′(z)

(Abel's lemma) and

E (X(X−1) ) = lim z → 1  p′′(z).