Consider the linear model $Y=X \beta+\varepsilon$, where $Y$ is a $n \times 1$ random vector, $\varepsilon \sim N_{n}\left(0, \sigma^{2} I\right)$, and where the $n \times p$ nonrandom matrix $X$ is known and has full column rank $p$. Derive the maximum likelihood estimator $\hat{\sigma}^{2}$ of $\sigma^{2}$. Without using Cochran's theorem, show carefully that $\hat{\sigma}^{2}$ is biased. Suggest another estimator $\tilde{\sigma}^{2}$ for $\sigma^{2}$ that is unbiased.