Skewness of Hat-Check Distribution

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

Then the skewness $\gamma_1$ of $X$ is given by:


 * $\gamma_1 = n^2 - 4n + 3$

Proof
From Skewness in terms of Non-Central Moments:


 * $\gamma_1 = \dfrac {\expect {X^3} - 3 \mu \sigma^2 - \mu^3} {\sigma^3}$

where $\mu$ is the mean of $X$, and $\sigma$ the standard deviation.

We have, by Expectation of Hat-Check Distribution:


 * $\expect X = n - 1$

By Variance of Hat-Check Distribution:


 * $\var X = \sigma^2 = 1$

so:


 * $\sigma = \sqrt 1 = 1$

To now calculate $\gamma_1$, we must calculate $\expect {X^3}$.

So: