Probability Generating Function of Shifted Random Variable

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 $Y = X + m$.

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

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

Thus:


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

From the definition of a probability generating function:
 * $\Pi_Y \left({s}\right) = s^m \Pi_X \left({s}\right)$