# 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:

 $\ds \map \Pr \Omega$ $=$ $\ds \sum_{k \mathop = 0}^n \dfrac 1 {\paren {n - k }!} \sum_{s \mathop = 0}^k \dfrac {\paren {-1}^s} {s!}$ $\ds$ $=$ $\ds \sum_{k \mathop = 0}^n \dfrac 1 {\paren {n - k }! } \dfrac {n! k!} {n! k!} \sum_{s \mathop = 0}^k \dfrac {\paren {-1}^s} {s!}$ multiplying by $1$ $\ds$ $=$ $\ds \sum_{k \mathop = 0}^n \dbinom n k \dfrac {k!} {n!} \sum_{s \mathop = 0}^k \dfrac {\paren {-1}^s} {s!}$ Definition of Binomial Coefficient $\ds$ $=$ $\ds \dfrac 1 {n!} \sum_{k \mathop = 0}^n \dbinom n k k! \sum_{s \mathop = 0}^k \dfrac {\paren {-1}^s} {s!}$ $\ds$ $=$ $\ds \dfrac 1 {n!} n!$ Sum over k of r Choose k by -1^r-k by Polynomial $\ds$ $=$ $\ds 1$

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

$\blacksquare$