= Solution
{c}
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> gives
$$
\widehat E_t=\widehat\alpha+(e^{-t}-c)\widehat\beta.
$$
The two estimates have zero <covariance> by part (ii). Therefore
$$
\boxed{\operatorname{Var}(\widehat E_t)=\sigma^2\left[\frac1n+
\frac{(e^{-t}-c)^2}{\sum_i(e^{-t_i}-c)^2}\right].}
$$
This 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 $E_t$.
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