Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 219 / 1 / a

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Solution

Let

C_{k} = σ_{int}^{2} + σ_{m, k}^{2} + σ_{C, k}^{2}, H_{i} = σ_{int}^{2} + σ_{m, i}^{2} .

Subtracting the measured distance modulus from the measured apparent magnitude gives

q_{k} = m_{k} - μ_{C, k} \sim N (M_{0}, C_{k}) .

For the Hubble flow, define

r_{i} = m_{i} - 25 - 5 lo g_{10} (\frac{c z _{i}}{100 km s ^{- 1}}) .

The Hubble law, with the Hubble constant parametrized by

θ = 5 lo g_{10} h

, gives

r_{i} \sim N (M_{0} - θ, H_{i})

. After integrating over each intrinsic absolute magnitude and each unobserved true distance modulus, independence therefore gives the likelihood function

L (M_{0}, θ) = k = 1 \prod K \frac{e ^{- (q_{k} - M_{0})^{2} / (2 C_{k})}}{2 π C _{k}} i = 1 \prod N \frac{e ^{- (r_{i} - M_{0} + θ)^{2} / (2 H_{i})}}{2 π H _{i}} .

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