pgf-zero.md

layout	mathjax	author	affiliation	e_mail	date	title	chapter	section	topic	theorem	sources	proof_id	shortcut	username
proof	true	Joram Soch	BCCN Berlin	joram.soch@bccn-berlin.de	2022-10-11 01:06:00 -0700	Value of the probability-generating function for argument zero	General Theorems	Probability theory	Other probability functions	Probability-generating function of zero	authors year title in pages url ProofWiki 2022 Probability Generating Function of Zero ProofWiki retrieved on 2022-10-11 https://proofwiki.org/wiki/Probability_Generating_Function_of_Zero	P361	pgf-zero	JoramSoch

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:

$$ \label{eq:pgf-zero-qed} \begin{split} G_X(0) &= \sum_{x=0}^{\infty} f_X(x) \cdot 0^x \\ &= f_X(0) + 0^1 \cdot f_X(1) + 0^2 \cdot f_X(2) + \ldots \\ &= f_X(0) + 0 + 0 + \ldots \\ &= f_X(0) ; . \end{split} $$