Probability Generating Function of Scalar Multiple of Random Variable
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Theorem
Let $X$ be a discrete random variable whose probability generating function is $\map {\Pi_X} s$.
Let $k \in \Z_{\ge 0}$ be a positive integer.
Let $Y$ be a discrete random variable such that $Y = m X$.
Then
- $\map {\Pi_Y} s = \map {\Pi_X} {s^m}$
where $\map {\Pi_Y} s$ is the probability generating function of $Y$.
Proof
From the definition of p.g.f:
- $\ds \map {\Pi_X} s = \sum_{k \mathop \ge 0} \map \Pr {X = k} s^k$
We have by hypothesis:
- $\map \Pr {Y = m k} = \map \Pr {X = k}$
Thus:
- $\ds \map {\Pi_Y} s = \sum_{m k \mathop \ge 0} \map \Pr {X = k} s^{m k}$
From the definition of a probability generating function:
- $\map {\Pi_Y} s = \map {\Pi_X} {s^m}$
$\blacksquare$
Sources
- 1986: Geoffrey Grimmett and Dominic Welsh: Probability: An Introduction ... (previous) ... (next): $\S 4.2$: Integer-valued random variables: Exercise $3$