Probability Generating Function of Degenerate Distribution

Theorem
Let $X$ be the degenerate distribution:
 * $\forall x \in \N: \map {p_X} x = \begin{cases}

1 & : x = k \\ 0 & : x \ne k \end{cases}$ where $k \in \N$.

Then the p.g.f. of $X$ is:
 * $\map {\Pi_X} x = s^k$

Proof
Follows directly from the definition:


 * $\ds \map {\Pi_X} s = \sum_{x \mathop \ge 0} \map {p_X} x s^x$

As $\map {p_X} x \ne 0$ for only one value of $x$, all the terms vanish except that one.

Hence the result.