Past exam of the probability, statistics and options I course of the University of Oxford / 2010 / O / 2 / b / ii / Solution

Centering gives ${\sum }_i{z}_i=0$, and henceThe least-squares covariance formula from part (a) therefore givesExplicitly $\hat{\alpha }=\overline{Y}$ and $\hat{\beta }={\sum }_i{z}_i{Y}_i/{\sum }_i{z}_i^2$. Their zero covariance follows directly from the centered predictor, and their joint normality additionally makes them independent.