Probability Generating Function of Zero

From ProofWiki
Jump to navigation Jump to search

Theorem

Let $X$ be a discrete random variable whose codomain, $\Omega_X$, is a subset of the natural numbers $\N$.

Let $p_X$ be the probability mass function for $X$.

Let $\map {\Pi_X} s$ be the probability generating function for $X$.

Then:

$\map {\Pi_X} 0 = \map {p_X} 0$


Proof

\(\ds \map {\Pi_X} 0\) \(=\) \(\ds \map {p_X} 0 + 0^1 \cdot \map {p_X} 1 + 0^2 \cdot \map {p_X} 2 + \cdots\) Definition of Probability Generating Function
\(\ds \) \(=\) \(\ds \map {p_X} 0 + 0 + 0 + \cdots\) as $\forall n \in \N_{>0}: s^n = 0$
\(\ds \) \(=\) \(\ds \map {p_X} 0\)

$\blacksquare$


Sources