Definition:Probability Generating Function

Definition
Let $X$ be a discrete random variable whose codomain is a subset of $\N = \left\{{0, 1, 2, \ldots}\right\}$.

The probability generating function (p.g.f.) for (or of) $X$ is denoted $\Pi_X \left({s}\right)$ and defined as:
 * $\Pi_X \left({s}\right) = E \left({s^X}\right)$

where:
 * $s$ is a dummy variable;
 * $E \left({s^X}\right)$ is the expectation of $s^x$ for $x \in X$.

Hence we see that $\Pi_X \left({s}\right)$ is a generating function which can be defined as:
 * $\displaystyle \Pi_X \left({s}\right) = \sum_{x \in \Omega_X} s^x p_X \left({x}\right)$

where $p_X$ is the probability mass function of $X$.

Since $p_X \left({x}\right) = 0$ when $x \notin \Omega_X$, it can also be conveniently written:
 * $\displaystyle \Pi_X \left({s}\right) = \sum_{x \ge 0} p_X \left({x}\right) s^x$

So $\Pi_X \left({s}\right)$ is either a polynomial function or a power series in $s$ (or whatever dummy variable) whose coefficients are the probabilities $p_X \left({x}\right)$.

So, given any probability mass function, we can write down its p.g.f. by determining what the probabilities are of $0, 1, 2, \ldots$ and then writing down:
 * $\Pi_X \left({s}\right) = p_X \left({0}\right) + p_X \left({1}\right) s + p_X \left({2}\right) s^2 + \cdots$

Note that when it is understood what $X$ is, it is common to omit it from the notation:
 * $\displaystyle \Pi \left({s}\right) = \sum_{x \ge 0} p \left({x}\right) s^x = p \left({0}\right) + p \left({1}\right) s + p \left({2}\right) s^2 + \cdots$

Some sources use $G_X$ for $\Pi_X$.