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:


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

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

Hence the result.