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

By invariance of maximum likelihood,Because $\hat{\theta }-\theta \sim N(0,{\sigma }_{\theta }^2)$, the ratio $\hat{h}/h$ is log-normal. Its exact variance isAs $N_{\mathrm{SN}},G,N_{\mathrm{LMC}}\to \infty$, all sampling terms vanish but $V_1$ retains ${\sigma }_{\mathrm{LMC}}^2$. The dominant remaining uncertainty is therefore the parallax calibration of the LMC distance modulus.