> [!definition] > > Let $X$ be a discrete [[Random Variable|random variable]] taking non-negative integer values. Then the **probability generating function** of $X$ is > $ > G_X(z) = \ev(z^X) = \sum_{n =1}^{\infty}z^nP(X = n) > $ > [!theorem] Factorisation Property > > Let $\bracs{X_i}_1^n$ be a sequence of [[Probabilistic Independence|independent]] random variables with PGFs. Then > $ > G_{\sum_{i}X_i}(z) = \ev\paren{z^{\sum_i X_i}} = > \prod_{i}\ev(z^{X_i}) = \prod_{i}G_{X_i}(z) > $ > [!theorem] > > Let $X$ be a [[Random Variable|random variable]] with a PGF. Then its PGF has a radius of convergence of at least 1. > [!theorem] > > Let $X$ be a random variable with a PGF. Then > $ > P(X = k) = \frac{\frac{d^{k}G_X(z)}{dz^{k}}(0)}{k!} > $ > *Proof*. Since > $ > G_X(z) = \sum_{n = 0}^{\infty}z^{n}P(X = n) > $ > We have > $ > \frac{d^{k}}{dz^{k}}G_X(z) = \sum_{n = k}^{\infty}n(n-1) \cdots(n - k + 1) \cdot z^{n - k}P(X = n) > $ > and > $ > \paren{\frac{d^{k}}{dz^{k}}G_X(z)}(0)= k! P(x = k) > $ > [!theorem] > > Let $X$ be a random variable with a PGF, then > $ > \frac{d^{k}G_X}{dz^{k}}(1) = \ev(X(X - 1)(X - 2) \cdots (X - k + 1)) > $ > where the right-hand side is the $k$-th **factorial moment** of $X$. > > *Proof*. > $ > \begin{align*} > \frac{d^kG_X}{dz^k} &= \sum_{n = k}^{\infty}n(n - 1) \cdots (n - k + 1)z^{n - k}P(X = n) \\ > \frac{d^kG_X}{dz^k}(1) &= \sum_{n = k}^{\infty}n(n - 1) \cdots (n - k + 1)P(X = n) \\ > &= \ev(X(X - 1)(X - 2) \cdots (X - k + 1)) > \end{align*} > $