Probability Generating Functions

The probability generating function (PGF) is a tool for working with discrete random variables that take non-negative integer values. It encodes the entire probability distribution and provides elegant methods for finding means, variances, and distributions of sums.

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Definition

For a discrete random variable $X$X taking values $0, 1, 2, \ldots$0,1,2,…, the PGF is:

$G_X(t) = E(t^X) = \sum_{r=0}^{\infty} P(X = r) \cdot t^r$GX​(t)=E(tX)=∑r=0∞​P(X=r)⋅tr

This is a power series in $t$t, where the coefficient of $t^r$tr is $P(X = r)$P(X=r).

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Recovering Probabilities

The probabilities can be recovered from the PGF:

$P(X = r) = \frac{G_X^{(r)}(0)}{r!}$P(X=r)=r!GX(r)​(0)​

In particular:

• $P(X = 0) = G_X(0)$P(X=0)=GX​(0)
• $P(X = 1) = G_X'(0)$P(X=1)=GX′​(0)
• $P(X = 2) = G_X''(0)/2$P(X=2)=GX′′​(0)/2

Also, $G_X(1) = \sum P(X = r) = 1$GX​(1)=∑P(X=r)=1 (always).

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Standard PGFs

Distribution	PGF $G_X(t)$GX​(t)
$\text{Bernoulli}(p)$Bernoulli(p)	$1 - p + pt = q + pt$1−p+pt=q+pt
$B(n, p)$B(n,p)	$(q + pt)^n$(q+pt)n
$\text{Geometric}(p)$Geometric(p) (starting from 1)	$\frac{pt}{1 - qt}$1−qtpt​, $
$\text{Po}(\lambda)$Po(λ)	$e^{\lambda(t - 1)}$eλ(t−1)

Derivation for Poisson: