Knowing $X \sim N(0, \sigma)$, how come $E\left(\frac{\sum X_i^2}{n}\right)= \sigma?$

Question

Knowing $X \sim \operatorname{Normal}(0, \sigma)$, how come $$E\left(\frac{\sum X_i^2}{n}\right)= \sigma?$$

I had already thought of using linearity. What I don't like is $X_i^2$.

Answer 1

"What I don't like is $X_i^2$"

Hint:

$$E[X_i^2] = \operatorname{Var}(X_i)+(E[X_i])^2.$$