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 $\Pi_X \left({s}\right)$.

Let $k \in \Z_{\ge 0}$ be a positive integer.

Let $Y$ be a discrete random variable such that $Y = m X$.


Then

$\Pi_Y \left({s}\right) = \Pi_X \left({s^m}\right)$.

where $\Pi_Y \left({s}\right)$ is the probability generating function of $Y$.


Proof

From the definition of p.g.f:

$\displaystyle \Pi_X \left({s}\right) = \sum_{k \mathop \ge 0} \Pr \left({X = k}\right) s^k$

By hypothesis:

$\Pr \left({Y = m k}\right) = \Pr \left({X = k}\right)$

Thus:

$\displaystyle \Pi_Y \left({s}\right) = \sum_{m k \mathop \ge 0} \Pr \left({X = k}\right) s^{m k}$


From the definition of a probability generating function:

$\Pi_Y \left({s}\right) = \Pi_X \left({s^m}\right)$

$\blacksquare$


Sources