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

LetSubtracting the measured distance modulus from the measured apparent magnitude givesFor the Hubble flow, defineThe Hubble law, with the Hubble constant parametrized by $\theta =5{\log }_{10}h$, gives $r_i\sim N(M_0-\theta ,H_i)$. After integrating over each intrinsic absolute magnitude and each unobserved true distance modulus, independence therefore gives the likelihood function