The common distance modulus of calibrator galaxy $g$ cancels between its supernova and mean Cepheid apparent magnitudes. Their independent intrinsic scatters therefore give
$$
\widehat S_2^g\sim N(\Delta M,V_2),
\qquad
\Delta M=M_0^{\mathrm{SN}}-M_0^*,
\qquad
V_2=\sigma_{\mathrm{SN}}^2+\frac{\sigma_*^2}{N_{\mathrm{Ceph}}}.
$$
Thus $\mathbb E\widehat S_2^g=\Delta M$ and $\operatorname{Var}(\widehat S_2^g)=V_2$.


Solution

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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2025/iii/paper-219/1/b/solution