Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 219 / 1

The expected number in a spherical shell is $4\pi r^2{\rho }_0{e}^{-r/r_0}dr$. Sincethe normalized radial density is the Gamma distribution

In the stated units, $\alpha =1/r$. The change of variables formula givesThus, with ${\alpha }_0=1/r_0$,

Writing $r_s=1/{\alpha }_s$, the log likelihood isHenceThe score variance and negative expected Hessian give Fisher informationSince $r_s$ is Gamma with mean $3r_0$ and variance $3r_0^2$,The estimator is unbiased and attains the Cramér-Rao lower bound.

The inverse-square law gives $f=L_0/(4\pi r^2)$, so observation is equivalent toFor $u=R/r_0$, integrating the Gamma density givesTherefore

At distance $r$, inclusion requires $L\ge 4\pi r^2{f}_{\min }$. ThusIt equals $1/2$ when