I am trying to find the mean squared error of the sample mean $\frac{Y_1 + \dots + Y_n}{n}$, but I am a little stuck. Here is what I've done so far.
$$MSE[\hat{\theta}] = E\bigg[ \bigg(\frac{Y_1 + \dots + Y_n}{n} - \theta^2 \bigg) \bigg] = E\bigg[ \bigg(\frac{Y_1 + \dots + Y_n}{n} \bigg)^2 \bigg] - 2\mu E\bigg[ \frac{Y_1 + \dots + Y_n}{n} \bigg] + \mu^2 $$
I tried factoring (as shown above), but I don't know how to simplify everything to get the mean squared error of the sample mean. Any suggestions?
Suppose your estimator for $\mu$ is $\bar Y$. Write $$\begin{split}MSE(\bar Y)&=E(\bar Y^2)-2\mu E(\bar Y)+\mu^2\\&=Var(\bar Y)+[E(\bar Y)]^2-2\mu E(\bar Y)+\mu^2\end{split}$$
If each of the Y's has expected value $\mu$ and variance $\sigma^2$, we have $\frac{\sigma^2}{n}+\mu^2-2\mu^2+\mu^2=\frac{\sigma^2}{n}$
This is in the form of $Var(\bar Y)+Bias(\bar Y, \mu)=\frac{\sigma^2}{n}+0$ as $\bar Y$ is an unbiased estimator of $\mu$.