Probability Generating Function of Scalar Multiple of Random Variable

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 $3$