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

With independent Gaussian errors, maximizing the likelihood over $\alpha ,\beta$ minimizes the residual sum of squares, so the maximum-likelihood estimators are the same as least squares. The invariance property of maximum likelihood estimation givesThe two estimates have zero covariance by part (ii). ThereforeThis is the uncertainty in the estimated expected response. Estimating an additional noisy observation would add its own error variance, but the quantity requested is the mean response $E_t$.