Probability Generating Function of Degenerate Distribution

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

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

Then the p.g.f. of $X$ is:
 * $\Pi_X \left({s}\right) = s^k$

Proof
Follows directly from the definition:


 * $\displaystyle \Pi_X \left({s}\right) = \sum_{x \ge 0} p_X \left({x}\right) s^x$

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

Hence the result.