Solution
= Solution
Writing $X_1=\mu+Z$ with $Z\sim N(0,1)$ gives
$$
\mathbb E S=\mathbb E X_1^2-1=\mu^2,
$$
so $S$ is unbiased. Its mean square error is therefore its variance. Since
$$
\mathbb E X_1^4
=\mathbb E(\mu+Z)^4
=\mu^4+6\mu^2+3,
$$
we obtain
$$
\operatorname{MSE}(S)
=\operatorname{var}(X_1^2)
=\mu^4+6\mu^2+3-(\mu^2+1)^2
=\boxed{4\mu^2+2}.
$$