Consider a linear model for $Y=\left(Y_{1}, \ldots, Y_{n}\right)^{T}$ given by

$$Y=X \beta+\epsilon,$$

where $X$ is a known $n \times p$ matrix of full rank $p<n$, where $\beta$ is an unknown vector and $\epsilon \sim N_{n}\left(0, \sigma^{2} I\right)$. Derive an expression for the maximum likelihood estimator $\hat{\beta}$ of $\beta$, and write down its distribution.

Find also the maximum likelihood estimator $\hat{\sigma}^{2}$ of $\sigma^{2}$, and derive its distribution.

[You may use Cochran's theorem, provided that it is stated carefully. You may also assume that the matrix $P=X\left(X^{T} X\right)^{-1} X^{T}$ has rank $p$, and that $I-P$ has rank $n-p$.]