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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 $X_i^2$.

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  • $\begingroup$ Do you mean something like: "Knowing $X \sim Normal(0, \sigma)$, how come $E(\frac{\sum x_i^2}{n}\}= \sigma$ ? " $\endgroup$ – BBC3 May 22 '17 at 23:52
  • $\begingroup$ Hint: the expected value is a linear operator. Use this to write things in terms of $E(X_i^2)$. $\endgroup$ – eyeballfrog May 22 '17 at 23:52
  • $\begingroup$ Formatting tips here. $\endgroup$ – Em. May 23 '17 at 0:30
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"What I don't like is $X_i^2$"

Hint:

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

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