# Expected value of different types of sample means

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 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 $$\mu$$.