Consider the normal linear model $Y \mid X \sim \mathrm{N}\left(X \beta, \sigma^{2} I\right)$, where $X$ is a $n \times p$ design matrix, $Y$ is a vector of responses, $I$ is the $n \times n$ identity matrix, and $\beta, \sigma^{2}$ are unknown parameters.

Derive the maximum likelihood estimator of the pair $\beta$ and $\sigma^{2}$. What is the distribution of the estimator of $\sigma^{2}$ ? Use it to construct a $(1-\alpha)$-level confidence interval of $\sigma^{2}$. [You may use without proof the fact that the "hat matrix" $H=X\left(X^{T} X\right)^{-1} X^{T}$ is a projection matrix.]