Probability Generating Function of Degenerate Distribution

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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:

$\map {\Pi_X} x = s^k$


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.