Probability Generating Function¶

Definition¶

For a nonnegative integer-valued random variable $N$ , its probability generating function (PGF) is $\displaystyle G_N(z)=\mathbb{E}[z^N]=\sum_{k=0}^\infty\mathbb{P}(N=k)z^k$ $k) z^{k}$ , at least for $|z|\le1$ $1$ . In this power-series definition $z^0=1$ $1$ , including at $z=0$ $0$ .

Interpretation¶

The coefficient of each power stores a probability. PGFs are especially useful for counting claims and for composing claim frequency with claim severity.

PGF of a Poisson(2) count, e^(2(z−1)), on 0 ≤ z ≤ 1, starting at e^−2 = P(N = 0), ending at 1, with the dashed tangent at z = 1 of slope 2.

Here $N\sim\operatorname{Pois}(\lambda)$ with $\lambda=2$ $2$ , so $G_N(z)=\mathrm{e}^{2(z-1)}$ $e^{2 (z - 1)}$ . The height at $z=0$ is the constant coefficient, $G_N(0)=\mathbb{P}(N=0)=\mathrm{e}^{-2}$ $e^{- 2}$ , and the curve ends at $G_N(1)=1$ because the probabilities sum to one. The dashed tangent at $z=1$ has slope $G_N'(1)=\mathbb{E}[N]=2$ $2$ , the mean read off by differentiating at one.

Pick a count law and move $z$ : $G_N(0)=\mathbb P(N=0)$ $0)$ , the slope at $z=1$ is $\mathbb E[N]$ and the power-series coefficients are the probabilities shown on the right. The default $N\sim\operatorname{Poisson}(2)$ gives $G_N'(1)=2$ and $G_N''(1)=4$ $4$ .

For a discrete random variable taking nonnegative integer values, expectation becomes a sum of $z^k$ weighted by the count probabilities. Substituting $z=\mathrm{e}^t$ connects the PGF with the moment generating function wherever the transform is finite.

Key Results¶

$\mathbb{P}(N=k)=G_N^{(k)}(0)/k!$ $G_{N}^{(k)} (0) / k!$ .
$G_N'(1-)=\mathbb{E}[N]$ and $G_N''(1-)=\mathbb{E}[N(N-1)]$ $1)]$ , allowing infinite values.
Independent nonnegative integer-valued sums have product PGFs.
For $z>0$ where finite, $G_N(z)=M_N(\log z)$ $M_{N} (lo g z)$ . A general real-valued severity need not have a PGF.

Examples¶

For $N\sim\operatorname{Pois}(\lambda)$ $Pois (λ)$ , $G_N(z)=\mathrm{e}^{\lambda(z-1)}$ $e^{λ (z - 1)}$ . Differentiating at one gives mean $\lambda$ and second factorial moment $\lambda^2$ .

Prerequisites¶

Sources¶

Risk Insurance · Lecture 1

Aliases: PGF, probability generating transform, probability-generating function Keywords: G_N(z) = E[z^N], P(N = k) = G_N^(k)(0)/k!, factorial moments, Poisson PGF e^(λ(z - 1)) Symbols: infinity, lambda, sum

Last updated: 14 October 2026 08:21