Variance of Chi-Squared Distribution

Theorem
Let $n$ be a strictly positive integer.

Let $X \sim \chi^2_n$ where $\chi^2_n$ is the chi-squared distribution with $n$ degrees of freedom.

Then the variance of $X$ is given by:


 * $\var X = 2 n$

Proof
By Variance as Expectation of Square minus Square of Expectation, we have:


 * $\var X = \expect {X^2} - \paren {\expect X}^2$

By Expectation of Chi-Squared Distribution, we have:


 * $\expect X = n$

We also have:

So: