Variance of Hat-Check Distribution

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

Then the variance of $X$ is given by:
 * $\var X = 1$

Proof
From the definition of Variance as Expectation of Square minus Square of Expectation:
 * $\var X = \expect {X^2} - \paren {\expect X}^2$

From Expectation of Function of Discrete Random Variable:
 * $\ds \expect {X^2} = \sum_{x \mathop \in \Omega_X} x^2 \, \map \Pr {X = x}$

So:

Then:

Also see

 * Expectation of Hat-Check Distribution