Parts (a) and (b) are easy, but I'm unsure of (c)
https://i.stack.imgur.com/AETCH.png
I know that $\tilde \theta$ is unbiased and that the expected value of $\hat \theta$, $E(\hat \theta)=e^{\mu e^{-1}-\mu}$. I used the following formula to calculate the respective MSEs:
$$MSE(\hat\theta)=E((\hat\theta-\theta)^2)$$
I obtained that, for $\mu=1$, $MSE(\hat\theta)=0.0656$ and $MSE(\tilde\theta)=0.2325$. For $\mu=2$, I obtianed $MSE(\hat\theta)=0.0451$ and $MSE(\tilde\theta)=0.117$.
The issue is that I'm not really confident in these answers. They took a while for me to obtain. So I'm wondering, is there an easy way to verify these values? For example, could this be simulated with statistical software to compare what I've got to what's empirically observed? If not, how else could I verify them?