Normal linear model maximum-likelihood sampling distributions ($\hat{\beta }\sim N_p(\beta ,{\sigma }^2(X^{T}X{)}^{-1}),n{\hat{\sigma }}^2/{\sigma }^2\sim {\chi }_{n-p}^2$)

For a full-column-rank design $X$ and $Y=X\beta +\epsilon$ with $\epsilon \sim N_n(0,{\sigma }^{2}I)$, the maximum-likelihood estimators are $\hat{\beta }=(X^{T}X{)}^{-1}{X}^{T}Y$ and ${\hat{\sigma }}^2=\Vert Y-X\hat{\beta }{\Vert }^2/n$. Orthogonal Gaussian projections give their displayed distributions and independence. The variance maximum-likelihood estimator has expectation $(n-p){\sigma }^2/n$; the unbiased residual variance divides by $n-p$ instead.