Probability Generating Functions

Combining two independent random counts — the number of trades filled in the morning session plus the afternoon session — normally means computing a full convolution, sliding one distribution across the other value by value. There is a shortcut that turns this fiddly sliding-and-summing into ordinary multiplication of two simple algebraic expressions. It works by packaging an entire distribution into a single function, in a way that makes "combine two independent counts" as easy as multiplying two polynomials.

The analogy: a probability distribution written as a shopping receipt

Imagine writing a receipt where the "price" of buying $k$ units of something is $P(X=k)$, and the "quantity" is a placeholder variable $s^k$. Add up every line item: $P(X{=}0)s^0 + P(X{=}1)s^1 + P(X{=}2)s^2 + \cdots$. That polynomial (or infinite series) is the distribution, just rewritten with each probability attached to a marker $s^k$ that remembers which outcome it belongs to. Now here is the payoff: if you have two independent receipts like this and you multiply them together as ordinary polynomials, the coefficient of $s^n$ in the product automatically sums up every way the two counts could add to $n$ — which is exactly the convolution formula, done for free by polynomial multiplication.

Writing it down

For a non-negative integer-valued random variable $X$, the probability generating function (PGF) is

$$G_X(s) = E\big[s^X\big] = \sum_{k=0}^{\infty} P(X=k)\, s^k .$$

In words: $G_X(s)$ is the sum of $s$ raised to each possible outcome, weighted by that outcome's probability — precisely the "receipt" described above. Two properties do the heavy lifting.

Recovering probabilities. $P(X=k)$ is the coefficient of $s^k$ in the series, retrievable by differentiating $k$ times and evaluating at $s=0$: $P(X=k) = \dfrac{G_X^{(k)}(0)}{k!}$. In words: the whole distribution is stored inside $G_X$, and you can read any individual probability back out.

Sums of independent variables. If $X$ and $Y$ are independent, then

$$G_{X+Y}(s) = G_X(s) \, G_Y(s) .$$

In words: the PGF of a sum of independent counts is just the product of their individual PGFs — no convolution sum required, only ordinary multiplication.

Worked example 1: combining two independent counts by multiplying PGFs

$X$ takes values $0,1,2$ with probabilities $0.5, 0.3, 0.2$, so $G_X(s) = 0.5 + 0.3s + 0.2s^2$. $Y$ takes values $0,1$ with probabilities $0.6, 0.4$, so $G_Y(s) = 0.6+0.4s$. Multiply them out term by term: $G_X(s)G_Y(s) = (0.5)(0.6) + \big[(0.5)(0.4)+(0.3)(0.6)\big]s + \big[(0.3)(0.4)+(0.2)(0.6)\big]s^2 + (0.2)(0.4)s^3$. Compute each bracket: constant term $0.30$; $s^1$ term $0.20+0.18=0.38$; $s^2$ term $0.12+0.12=0.24$; $s^3$ term $0.08$. So $G_{X+Y}(s) = 0.30+0.38s+0.24s^2+0.08s^3$, meaning $P(X{+}Y{=}0)=0.30$, $P(X{+}Y{=}1)=0.38$, $P(X{+}Y{=}2)=0.24$, $P(X{+}Y{=}3)=0.08$. Check one entry directly: $P(X{+}Y{=}1)$ needs $(X,Y)=(0,1)$ or $(1,0)$, probability $0.5{\times}0.4 + 0.3{\times}0.6 = 0.20+0.18=0.38$. Matches, with none of the manual convolution bookkeeping.

Worked example 2: extracting the mean from the PGF

The PGF also hands you moments for free. Differentiate $G_X(s)$ once and evaluate at $s=1$: $G_X'(s) = 0.3 + 0.4s$, so $G_X'(1) = 0.3+0.4 = 0.7 = E[X]$. Check directly: $E[X] = 0(0.5)+1(0.3)+2(0.2) = 0.3+0.4=0.7$. Matches. This works because differentiating $s^k$ gives $k\,s^{k-1}$, and setting $s=1$ collapses every $s^{k-1}$ to 1, leaving $\sum_k k\,P(X{=}k) = E[X]$ — the derivative "brings down" the exponent, which is exactly the count you're averaging.

What this means in practice

Aggregate claim counts in insurance, order-arrival counts in market microstructure, and any model built from sums of independent discrete counts (a Poisson process's increments, a binomial tree's up-move count) use PGFs to avoid manual convolution. Multiplying two Poisson PGFs, for instance, gives you the closed-form fact that the sum of two independent Poissons is itself Poisson with summed rates — a fact that takes one line to prove with PGFs and considerably more with direct convolution.