Q2 (5 points)
Let $X_1, \dots, X_n$ be independent and identically distributed random variables. The common distribution is the Normal distribution with mean zero and variance $\sigma^2 > 0$, i.e., $\mathcal{N}(0, \sigma^2)$, where $\sigma^2$ is the parameter of interest.
(a) Show that $X_1^2$ is an unbiased estimator for $\sigma^2$.
(b) Show that $S_n = \sum_{i=1}^n X_i^2$ is a sufficient statistic for $\sigma^2$.
(b) Compute $\hat{\theta}_n := E[X_1^2 | S_n]$. Hint: use the result in Question 1.
