Probability Generating Function of Shifted 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 $Y = X + m$.

Then
 * $\map {\Pi_Y} s = s^m \map {\Pi_X} s$

where $\map {\Pi_Y} s$ is the probability generating function of $Y$.

Proof
From the definition of p.g.f:


 * $\ds \map {\Pi_X} s = \sum_{k \mathop \ge 0} \map \Pr {X = k} s^k$

By hypothesis:
 * $\map \Pr {Y = k + m} = \map \Pr {X = k}$

Thus:

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