Index: The Book of Statistical Proofs ▷ General Theorems ▷ Probability theory ▷ Other probability functions ▷ Probability-generating function

Definition:

1) If $X$ is a discrete random variable taking values in the non-negative integers $\left\lbrace 0, 1, \ldots \right\rbrace$, then the probability-generating function of $X$ is defined as

\[\label{eq:pgf-var} G_X(z) = \sum_{x=0}^{\infty} p(x) \, z^x\]

where $z \in \mathbb{C}$ and $p(x)$ is the probability mass function of $X$.

2) If $X$ is a discrete random vector taking values in the $n$-dimensional integer lattice $x \in \left\lbrace 0, 1, \ldots \right\rbrace^n$, then the probability-generating function of $X$ is defined as

\[\label{eq:cgf-vec} G_X(z) = \sum_{x_1=0}^{\infty} \cdots \sum_{x_n=0}^{\infty} p(x_1,\ldots,x_n) \, {z_1}^{x_1} \cdot \ldots \cdot {z_n}^{x_n}\]

where $z \in \mathbb{C}^n$ and $p(x)$ is the probability mass function of $X$.

 
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Metadata: ID: D69 | shortcut: pgf | author: JoramSoch | date: 2020-05-31, 23:59.