0
$\begingroup$

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)?

$\endgroup$
0
$\begingroup$

\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 $\mu$.

| cite | improve this answer | |
$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.