Consider the normal linear model $Y=X \beta+\varepsilon$ in vector notation, where

$Y=\left(\begin{array}{c}Y_{1} \\ \vdots \\ Y_{n}\end{array}\right), \quad X=\left(\begin{array}{c}x_{1}^{\mathrm{T}} \\ \vdots \\ x_{n}^{\mathrm{T}}\end{array}\right), \quad \beta=\left(\begin{array}{c}\beta_{1} \\ \vdots \\ \beta_{p}\end{array}\right), \quad \varepsilon=\left(\begin{array}{c}\varepsilon_{1} \\ \vdots \\ \varepsilon_{n}\end{array}\right), \quad \varepsilon_{i} \sim$ i.i.d. $N\left(0, \sigma^{2}\right)$,

where $x_{i}^{\mathrm{T}}=\left(x_{i 1}, \ldots, x_{i p}\right)$ is known and $X$ is of full rank $(p<n)$. Give expressions for maximum likelihood estimators $\hat{\beta}$ and $\hat{\sigma}^{2}$ of $\beta$ and $\sigma^{2}$ respectively, and state their joint distribution.

Suppose that there is a new pair $\left(x^{*}, y^{*}\right)$, independent of $\left(x_{1}, y_{1}\right), \ldots,\left(x_{n}, y_{n}\right)$, satisfying the relationship

$$y^{*}=x^{* \mathrm{~T}} \beta+\varepsilon^{*}, \quad \text { where } \quad \varepsilon^{*} \sim N\left(0, \sigma^{2}\right) .$$

We suppose that $x^{*}$ is known, and estimate $y^{*}$ by $\tilde{y}=x^{* \mathrm{~T}} \hat{\beta}$. State the distribution of

$$\frac{\tilde{y}-y^{*}}{\tilde{\sigma} \tau}, \quad \text { where } \quad \tilde{\sigma}^{2}=\frac{n}{n-p} \hat{\sigma}^{2} \quad \text { and } \quad \tau^{2}=x^{* \mathrm{~T}}\left(X^{\mathrm{T}} X\right)^{-1} x^{*}+1$$

Find the form of a $(1-\alpha)$-level prediction interval for $y^{*}$.