Sunday 29 June 2014

probability - 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)?

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