Sunday, 29 June 2014

probability - Expected value of different types of sample means

Assuming Xi is iid normally distributed with N(μ,σ2)




In summation notation, what is the difference between



1) E(¯X2) and



2)E(X¯2)
(should have the bar over the entire X2)



3)E(¯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(X¯2) = (1n)(ni=1X2i)



(3) E(¯X)2 = (1n2)(ni=1Xi)2



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

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