Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 219 / 2 / d

Table of contents
- Solution d

Solution

The stated conditional independences give

p (d ∣ x_{i}, m_{i}) = \prod_{s = 1}^{N_{sat}} \frac{1}{2 π σ _{s}^{2}} exp [- \frac{( d _{s} - x _{s i} ) ^{2}}{2 σ _{s}^{2}}] .

Thus

E [m ∣ d] = \sum_{i = 1}^{K} m_{i} w_{i}, w_{i} = \frac{\prod _{s} p ( d _{s} ∣ x _{s i} )}{\sum _{j} \prod _{s} p ( d _{s} ∣ x _{s j} )} .

The dependence among satellite properties and host mass remains encoded in each jointly simulated row.

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