Probability Generating Function

Polynomial

For \(X\), a random variable with possible values \(0, 1, \ldots, N\) of probability \(p_k := P(X=k)\), the PGF of \(X\) is,

\[ G_X(s) = \sum_{k=0}^N p_ks^k \]

Which is a degree \(N\) polynomial with coefficients as the probability of \(X\). Notice that the PGF is also equivalent to \(E(s^X)\).

Independent Joint Distribution : The joint PGF for two independent non-negative random variables \(X\) and \(Y\) are,

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

Because $G_{X+Y}(s) = E(s^Xs^Y)$ is separable by independence.

Sums of an IID Sample : For \(S_n = X_1 + X_2 + \ldots X_n\) where the set of \(X_k\) are iid random variables, the PGF is the \(n\)th power any one PGF.

$$
G_{S_n} = G_{X_1}^n
$$