Expectation of Hat-Check Distribution

Theorem
Let $X$ be a discrete random variable with the Hat-Check distribution with parameter $n$.

Then the expectation of $X$ is given by:
 * $\expect X = n - 1$

Proof
From the definition of expectation:


 * $\ds \expect X = \sum_{x \mathop \in \Img X} x \map \Pr {X = x}$

By definition of hat-check distribution:


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

Then:

Also see

 * Variance of Hat-Check Distribution