Definition:Hat-Check Distribution

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

Then $X$ has the hat-check distribution with parameter $n$ :


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


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

Given a totally ordered set with $n$ elements, $X$ represents the number of elements that are not in the correct order.

Also see

 * Hat-Check Distribution Gives Rise to Probability Mass Function satisfying $\map \Pr \Omega = 1$.