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

An apparent magnitude is the sum of absolute magnitude and distance modulus. Thus

\overset{m}{ˉ}_{LMC} \sim N (M_{0}^{*} + μ_{LMC}, \frac{σ _{*}^{2}}{N _{LMC}}) .

The independent unbiased distance-modulus estimate is

N (μ_{LMC}, σ_{LMC}^{2})

, so closure of independent normal distributions under linear combinations gives

S_{1} \sim N (M_{0}^{*}, V_{1}), V_{1} = \frac{σ _{*}^{2}}{N _{LMC}} + σ_{LMC}^{2} .

Hence

E S_{1} = M_{0}^{*}

and

Var (S_{1}) = V_{1}

.

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