Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 219 / 1

A spherical shell contributes a factor $4\pi r^2$ to the number of stars. Since ${\int }_0^{\infty }{r}^2{e}^{-r/r_0}dr=2r_0^3$, the normalized distance density isThus $C=(2r_0^3{)}^{-1}$, $\alpha =2$, and $\beta =1$. This is a gamma distribution with shape three and scale $r_0$.

Ignoring terms independent of $r_0$, the log-likelihood isIts score vanishes atThe expected Fisher information isBecause a shape-three gamma variable has mean $3r_0$ and variance $3r_0^2$,The estimator is unbiased and attains the Cramér-Rao lower bound ${\mathcal{I}}_N(r_0{)}^{-1}$.

The inverse-square law gives $f_s=L_0/(4\pi r_s^2)$, so a star is observed exactly whenWriting $a=R/r_0$, integration of the shape-three gamma density givesTherefore the fully normalized truncated distribution is

At distance $r_s$, detection requires $L_s\ge 4\pi r_s^2{f}_{\min }$. HenceThe probability is $1/2$ when its normal quantile is zero, namely at