Consider the model $Y=X \beta+\epsilon$, where $Y$ is an $n$-dimensional observation vector, $X$ is an $n \times p$ matrix of rank $p, \epsilon$ is an $n$-dimensional vector with components $\epsilon_{1}, \ldots, \epsilon_{n}$, and $\epsilon_{1}, \ldots, \epsilon_{n}$ are independently and normally distributed, each with mean 0 and variance $\sigma^{2}$

(a) Let $\hat{\beta}$ be the least-squares estimator of $\beta$. Show that

$$\left(X^{T} X\right) \hat{\beta}=X^{T} Y$$

and find the distribution of $\hat{\beta}$.

(b) Define $\hat{Y}=X \hat{\beta}$. Show that $\hat{Y}$ has distribution $N\left(X \beta, \sigma^{2} H\right)$, where $H$ is a matrix that you should define.

[You may quote without proof any results you require about the multivariate normal distribution.]