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

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 $X_i^2$.

• Do you mean something like: "Knowing $X \sim Normal(0, \sigma)$, how come $E(\frac{\sum x_i^2}{n}\}= \sigma$ ? " – BBC3 May 22 '17 at 23:52
• Hint: the expected value is a linear operator. Use this to write things in terms of $E(X_i^2)$. – eyeballfrog May 22 '17 at 23:52
• Formatting tips here. – Em. May 23 '17 at 0:30

"What I don't like is $X_i^2$"
$$E[X_i^2] = \operatorname{Var}(X_i)+(E[X_i])^2.$$