Help with mean squared error

Question

I am doing this question on mean squared error, but I don't know how to do any of the parts.

This is the question:

Any help? Thanks!

Answer 1

A small MSE means that on average $\hat{\theta}$ is close to $\theta$. Thus from this point of view it is a "good" estimator.

For the second question I'll just give you a hint: $\hat{\theta}-\theta = (\hat{\theta} - E(\hat{\theta})) + (E(\hat{\theta}) - \theta)$

The last question is a simple computation. Expand the square and use linearity of expectation. The only trick here is to see that $E[\bar{X}^2] = V(\bar{X}) + E[\bar{X}]$, which you can express in terms of $\mu$ and $V(X_1)$.