Past exam of the probability, statistics and options I course of the University of Oxford / 2008 / O / 2 / Solution

Differentiating the least-squares residual norm gives the normal equationsIf $X$ has full column rank, $\hat{\theta }=(X^{T}X{)}^{-1}{X}^{T}Y$, and thereforeHere $\bar{x}=0$ and $\bar{z}=2$, so the design matrix and its Gram matrix areThe centered predictor columns are perpendicular, givingBoth estimates are linear combinations of the jointly normal errors, hence jointly normal. Zero covariance therefore also gives independence. The normality assumption is what permits this last step; orthogonality alone would only prove uncorrelatedness.