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

Theorem: Let $X$ be a discrete random variable whose set of possible values $\mathcal{X}$ is a subset of the natural numbers $\mathbb{N}$. Then, the probability-generating function of $X$ can be expressed in terms of an expected value of a function of $X$

\[\label{eq:pgf-mean} G_X(z) = \mathrm{E}\left[ z^X \right]\]

where $z \in \mathbb{C}$.

Proof: The law of the unconscious statistician states that

\[\label{eq:mean-lotus} \mathrm{E}[g(X)] = \sum_{x \in \mathcal{X}} g(x) f_X(x)\]

where $f_X(x)$ is the probability mass function of $X$. Here, we have $g(X) = z^X$, such that

\[\label{eq:E-zX-s1} \mathrm{E}\left[ z^X \right] = \sum_{x \in \mathcal{X}} z^x f_X(x) \; .\]

By the definition of $X$, this is equal to

\[\label{eq:E-zX-s2} \mathrm{E}\left[ z^X \right] = \sum_{x=0}^{\infty} f_X(x) \, z^x \; .\]

The right-hand side is equal to the probability-generating function of $X$:

\[\label{eq:pgf-mean-qed} \mathrm{E}\left[ z^X \right] = G_X(z) \; .\]
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Metadata: ID: P360 | shortcut: pgf-mean | author: JoramSoch | date: 2022-10-11, 02:01.