Solution
= Solution
Since $\widehat\beta$ is an affine transformation of $Y$,
$$
\boxed{
\widehat\beta\sim
N_p\!\left(\beta,\sigma^2(X^TX)^{-1}\right)
}.
$$
The residual projection $I-H$ has rank $n-p$. <Cochran's theorem> therefore gives
$$
\boxed{
\frac{n\widehat\sigma^2}{\sigma^2}
=\frac{\|(I-H)Y\|^2}{\sigma^2}
\sim\chi^2_{n-p}
}.
$$