Prove that the Expected value of Y bar ^2 = µ^2 I'm trying to show whether or not $\bar(Y^2)$ = $\\µ^2$  Or the mean of the sample squared) is a biased or unbiased estimator of the population mean squared.
I can prove that Ybar is an unbiased estimator of the population mean, but it's not clear how to prove the same for Ybar squared. 
So I have something like:
$E[\bar(Y)^2]$ = $\frac{1}{N}E[\sum_{i=1}^{n}(Y_i)^2]$
I'm wondering where to go from here.  I can swap the sum and the expected value components, but it doesn't seem to simplify anything.

Suppose that $\bar{Y} = \frac 1n \sum_{i=1}^n Y_i$ where the $Y_i$ are
independent random variables with the same mean $\mu$ and (positive) variance $\sigma^2$.  The $Y_i$ need not be identically distributed 
but you can
make this assumption if it makes you feel more comfortable.
Then, $\bar{Y}$ is a random variable with mean $\mu$ and variance
$\frac{\sigma^2}{n}$ and so
$\displaystyle E\left[\left(\bar{Y}\right)^2\right] = \mu^2 + \frac{\sigma^2}{n} > \mu^2.$
