Probability-generating function

Definition

Let $X$ be a random variable with values in $\{0, 1, 2, \dots\}$ and with mass function $p(k) = P(X = k)$. 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 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.