 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2019/ib/paper-4/19h/a/solution

Codex  0 Created 2026-09-24 Updated 2026-09-29

Apart from terms independent of

β

, the Gaussian log-likelihood is

ℓ (β, σ^{2}) = - \frac{1}{2 σ ^{2}} \sum_{i = 1}^{n} (Y_{i} - β x_{i})^{2} .

Differentiating with respect to

β

gives

\frac{\partial ℓ}{\partial β} = \frac{1}{σ ^{2}} \sum_{i = 1}^{n} x_{i} (Y_{i} - β x_{i}) .

The unique zero, and hence the maximum-likelihood estimate, is

β = \frac{\sum _{i} x _{i} Y _{i}}{\sum _{i} x _{i}^{2}} .

The same value minimizes the residual sum of squares

\sum_{i} (Y_{i} - β x_{i})^{2}

, since maximizing the Gaussian likelihood for fixed

σ^{2}

is equivalent to minimizing this sum. It is therefore also the least-squares estimator.

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