TITLE: Mean and variance of Squared Gaussian: $Y=X^2$ where: $X\sim\mathcal{N}(0,\sigma^2)$?
QUESTION [31 upvotes]: What is the mean and variance of Squared Gaussian: $Y=X^2$ where: $X\sim\mathcal{N}(0,\sigma^2)$?
It is interesting to note that Gaussian R.V here is zero-mean and non-central Chi-square Distribution doesn't work.
Thanks.

REPLY [37 votes]: We can avoid using the fact that $X^2\sim\sigma^2\chi_1^2$, where $\chi_1^2$ is the chi-squared distribution with $1$ degree of freedom, and calculate the expected value and the variance just using the definition. We have that
$$
\operatorname E X^2=\operatorname{Var}X=\sigma^2
$$
since $\operatorname EX=0$ (see here).
Also,
$$
\operatorname{Var}X^2=\operatorname EX^4-(\operatorname EX^2)^2.
$$
The fourth moment $\operatorname EX^4$ is equal to $3\sigma^4$ (see here). Hence,
$$
\operatorname{Var}X^2=3\sigma^4-\sigma^4=2\sigma^4.
$$