We have data $x_0, x_1, \ldots, x_{N-1}$ where the $x_n$'s are independent and identically distributed as ${\rm Normal}(0,\sigma^2)$. The estimate of $\sigma^2$ is
$$\hat \sigma^2 = \frac{1}{N} \sum_{n=0}^{N-1} x_n^2 $$
To find its expected value, I don't understand how mean of $x_n^2$ comes. moreover, how to find variance of above equation, what is central moment?