Let $X_1,X_2,\dots,X_n$ be an i.i.d. random sample from $N(\mu,σ^2)$.
a. I found the estimator MLE of $\sigma^2$ $$\hat{\sigma^2}=\frac{1}{n}\sum_{i=1}^n (X_i-\overline{X})^2$$ But how to calculate: $$Var(\frac{1}{n}\sum_{i=1}^n (X_i-\overline{X})^2)$$ Please help me. I try write that $$\hat{\sigma^2}=\frac{1}{n}\sum_{i=1}^n (X_i-\overline{X})^2=\frac{1}{n}\sum_{i=1}^n X_i^2 - \overline{X}^2$$ and how compute it $$Var(\hat{\sigma^2})$$
Since ${\sum_{i=1}^n(X_i-\bar X)^2\over\sigma^2}\sim \chi^2_{(n-1)},$ we get $V(\hat\sigma^2) = {\sigma^4\over n^2}\,V\left(\chi^2_{(n-1)}\right)=\displaystyle {2\sigma^4(n-1)\over n^2}.$