Consider the general linear model $Y=X \beta+\epsilon$, where the $n \times p$ matrix $X$ has full rank $p \leqslant n$, and where $\epsilon$ has a multivariate normal distribution with mean zero and covariance matrix $\sigma^{2} I_{n}$. Write down the likelihood function for $\beta, \sigma^{2}$ and derive the maximum likelihood estimators $\hat{\beta}, \hat{\sigma}^{2}$ of $\beta, \sigma^{2}$. Find the distribution of $\hat{\beta}$. Show further that $\hat{\beta}$ and $\hat{\sigma}^{2}$ are independent.