Hat-Check Distribution Gives Rise to Probability Mass Function

Theorem
Let $X$ be a discrete random variable on a probability space $\struct {\Omega, \Sigma, \Pr}$.

Let $X$ have the hat-check distribution with parameter $n$ (where $n > 0$).

Then $X$ gives rise to a probability mass function.

Proof
By definition:


 * $\Img X = \set {0, 1, \ldots, n}$


 * $\ds \map \Pr {X = k} = \dfrac 1 {\paren {n - k }!} \sum_{s \mathop = 0}^k \dfrac {\paren {-1}^s} {s!}$

Then:

So $X$ satisfies $\map \Pr \Omega = 1$, and hence the result.