---
tag: 0Z8PSSAVBA
kind: definition
name: Probability-generating function
msc: ["60E05"]
defines: 0EBCWQ8BVE
uses: [0WF4EJP91W, 0DCT0QZVQ1, 0HH6JX5WZZ]
sources:
  - url: https://en.wikipedia.org/wiki/Probability_generating_function
    locator: Univariate case
---

## Definition

Let $X$ be a random variable with values in $\{0, 1, 2, \dots\}$ and with [[mass function|0WF4EJP91W]] $p(k) = P(X = k)$.
The probability-generating function of $X$ is
$$G_X(z) = E\left[z^X\right] = \sum_{k=0}^{\infty} p(k) \, z^k,$$
for each complex $z$ with $|z| \le 1$.

## Remarks

$G_X$ is the [[generating function|0HH6JX5WZZ]] of the sequence $p(0), p(1), \dots$. The series converges absolutely for $|z| \le 1$, and $G_X(1) = 1$.
$G_X$ determines the law of $X$: $P(X = k) = G_X^{(k)}(0)/k!$.
If $E[X] < \infty$, then $E[X] = G_X'(1)$, with the left derivative at $1$. Also $E[X(X - 1)] = G_X''(1)$ when it is finite.
For $X \sim \operatorname{Poisson}(\lambda)$, $G_X(z) = e^{\lambda(z - 1)}$. For $X \sim \operatorname{Bin}(n, p)$, $G_X(z) = (1 - p + pz)^n$.

## Explanations

### L1 human

The probability-generating function packs all the probabilities of a count into one function.
Derivatives of this function at $0$ give back the probabilities, and derivatives at $1$ give the moments.