Assuming $X_i$ is iid normally distributed with $N(\mu , \sigma ^2) $

In summation notation, what is the difference between

1) $ E(\overline X ^2 )$ and

2)$ E(\mathrm{X}^\overline2) $ (should have the bar over the entire $X^2$)

3)$E(\overline X)^2$

so basically the difference between the expected value of the sample mean squared (1), the expected value of the RV squared's sample mean (2)(not sure how to put #2 into words sorry), and the square of the expected value of the sample mean (3).

I know

(2) $ E(\mathrm{X}^\overline2) $ = $(\frac{1}{n}$)$(\sum_{i=1}^{n} X_i^2)$

(3) $E(\overline X)^2$ = $(\frac{1}{n^2}$)$(\sum_{i=1}^{n} X_i)^2$

But I'm confused on what (1) would be? How is it different from (3)?


\begin{align} E(\bar{X}^2)&=E\left( \left(\frac{\sum_{i=1}^n X_i}{n}\right)^2\right)\\ &=\frac1{n^2} E\left( \sum_{i=1}^n X_i^2+2\sum_{i<j}X_iX_j\right)\\ &=\frac1{n^2} \left( \sum_{i=1}^n E(X_i^2)+2\sum_{i<j}E[X_i]E[X_j]\right)\\ \end{align} In the third line, I have used the property that the $X_i$ are independent.

Hopefully you can take it from here to simplify the expression in terms of $\sigma$ and $\mu$.

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