Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 4 / 27L / b / i / Solution

Write $r=(x^2+y^2+z^2{)}^{1/2}$, so $f(x,y,z)=r^3$. For $t\ge 0$,whose volume isThus the pushforward intensity of $[0,t]$ isThe pushforward is locally finite and diffuse, so the radial volume transform of a homogeneous Poisson point process and the mapping theorem show that