\begin{definition}[Definition:Hat-Check Distribution]
Let $X$ be a discrete random variable on a probability space $\struct {\Omega, \Sigma, \Pr}$.
Let $\Img X = \set {0, 2, 3, \ldots, n}$
Let $X$ represent the number of elements in a a totally ordered set with $n$ elements that are not in the correct order.
Then $X$ has the '''Hat-Check distribution with parameter $n$''' (where $n > 0$) {{iff}}:
\end{definition}