Probability Generating Function of One
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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:
- $\map {\Pi_X} 1 = 1$
Proof
\(\ds \map {\Pi_X} 1\) | \(=\) | \(\ds \map {p_X} 0 + 1^1 \cdot \map {p_X} 1 + 1^2 \cdot \map {p_X} 2 + \cdots\) | Definition of Probability Generating Function | |||||||||||
\(\ds \) | \(=\) | \(\ds \map {p_X} 0 + \map {p_X} 1 + \map {p_X} 2 + \cdots\) | as $\forall n \in \N_{>0}: s^n = 1$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{n \mathop = 0}^\infty \map {p_X} n\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{x \mathop \in \Omega_X} \map {p_X} x\) | by definition of $\Omega_X$ | |||||||||||
\(\ds \) | \(=\) | \(\ds 1\) | Definition of Probability Mass Function |
$\blacksquare$
Sources
- 1986: Geoffrey Grimmett and Dominic Welsh: Probability: An Introduction ... (previous) ... (next): $\S 4.2$: Integer-valued random variables: $(6)$