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

Theorem: Let $X$ be a random variable with probability-generating function $G_X(z)$ and set of possible values $\mathcal{X}$. Then, the value of the probability-generating function at one is equal to one:

\[\label{eq:pgf-one} G_X(1) = 1 \; .\]

Proof: The probability-generating function of $X$ is defined as

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

where $f_X(x)$ is the probability mass function of $X$. Setting $z = 1$, we obtain:

\[\label{eq:pgf-zero-s1} \begin{split} G_X(1) &= \sum_{x=0}^{\infty} f_X(x) \cdot 1^x \\ &= \sum_{x=0}^{\infty} f_X(x) \cdot 1 \\ &= \sum_{x=0}^{\infty} f_X(x) \; . \end{split}\]

Because the probability mass function sums up to one, this becomes:

\[\label{eq:pgf-zero-s2} \begin{split} G_X(1) &= \sum_{x \in \mathcal{X}} f_X(x) \\ &= 1 \; . \end{split}\]
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Metadata: ID: P362 | shortcut: pgf-one | author: JoramSoch | date: 2022-10-11, 08:17.