The expected number in a spherical shell is $4\pi r^2\rho_0e^{-r/r_0}dr$. Since
$$
\int_0^\infty4\pi r^2\rho_0e^{-r/r_0}dr=8\pi\rho_0r_0^3,
$$
the normalized radial density is the <Gamma distribution>
$$
\boxed{p(r)=\frac{r^2}{2r_0^3}e^{-r/r_0}},\qquad r>0.
$$

Solution

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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2021/iii/paper-219/1/a/solution