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

Theorem: Let $X$ be a random variable with probability-generating function $G_X(z)$ and probability mass function $f_X(x)$. Then, the value of the probability-generating function at zero is equal to the value of the probability mass function at zero:

\[\label{eq:pgf-zero} G_X(0) = f_X(0) \; .\]

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 $z = 0$, we obtain:

\[\label{eq:pgf-zero-qed} \begin{split} G_X(0) &= \sum_{x=0}^{\infty} f_X(x) \cdot 0^x \\ &= f_X(0) + 0^1 \cdot f_X(1) + 0^2 \cdot f_X(2) + \ldots \\ &= f_X(0) + 0 + 0 + \ldots \\ &= f_X(0) \; . \end{split}\]
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Metadata: ID: P361 | shortcut: pgf-zero | author: JoramSoch | date: 2022-10-11, 08:06.