Let
$$
C_k=\sigma_{\mathrm{int}}^2+\sigma_{m,k}^2+\sigma_{C,k}^2,
\qquad
H_i=\sigma_{\mathrm{int}}^2+\sigma_{m,i}^2.
$$
Subtracting the measured <distance modulus> from the measured <apparent magnitude> gives
$$
q_k=\widehat m_k-\widehat\mu_{C,k}\sim N(M_0,C_k).
$$
For the <Hubble flow>, define
$$
r_i=\widehat m_i-25-5\log_{10}\!\left(\frac{cz_i}{100\ {\rm km\,s^{-1}}}\right).
$$
The <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>
$$
\boxed{
L(M_0,\theta)=
\prod_{k=1}^K\frac{e^{-(q_k-M_0)^2/(2C_k)}}{\sqrt{2\pi C_k}}
\prod_{i=1}^N\frac{e^{-(r_i-M_0+\theta)^2/(2H_i)}}{\sqrt{2\pi H_i}}.}
$$

Solution

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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2019/iii/paper-219/1/a/solution