Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 219 / 1 / d

Table of contents
- Solution d

Solution

All three sets of statistics are independent. With

s_{1} = S_{1}

,

s_{2 g} = S_{2}^{g}

, and

s_{3 i} = S_{3}^{i}

, their likelihood function, up to a parameter-independent factor, is

L (M_{0}^{*}, Δ M, M) \propto exp [- \frac{( s _{1} - M _{0}^{*} ) ^{2}}{2 V _{1}} - \sum_{g = 1}^{G} \frac{( s _{2 g} - Δ M ) ^{2}}{2 V _{2}} - \sum_{i = 1}^{N_{SN}} \frac{( s _{3 i} - M ) ^{2}}{2 σ _{SN}^{2}}] .

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