 Solution

ID: past-exam-of-the-options-i-course-of-the-university-of-oxford/2010/o/o2/b/iii/solution

The expected activity is

E_{t} = α + β (e^{- t} - c)

. Under normal errors, least squares is maximum likelihood, and invariance of maximum-likelihood estimation gives

E_{t} = α + β (e^{- t} - c) .

The covariance term vanishes by part (ii), so

Var (E_{t}) = σ^{2} [\frac{1}{n} + \frac{( e ^{- t} - c ) ^{2}}{\sum _{i} ( e ^{- t_{i}} - c ) ^{2}}] .

This is uncertainty in the estimated mean activity; the variance of a future individual observation would include its additional measurement-error variance.

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