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.

The  problem is, that $E[Y^2] \neq \mu^2$ - take $Y$, such that $$P(Y=-1)=P(Y=1)=0.5,$$ clearly $EY=0$, but $E[Y^2]=1$. 
From this example you can can see that $\frac{1}{N}\sum_{i=1}^{n}(Y_i)^2$ does not estimate $\mu^2$.
Edit:
You've (nearly) proved in your edit that $\frac{1}{N}\sum_{i=1}^{n}(Y_i)^2$ is unbiased estimate of $E[Y^2]$. But then once again $E[Y^2] \neq (EY)^2.$
