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

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 $r_0$.