Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 320 / 1 / e / Solution

For the uniform distribution $\mathcal{P}(q)=1$, substitute $q=Q\sin \theta$ into part d:where $K$ is the complete elliptic integral of the first kind. Both $Q$ and $K(Q)$ increase on $(0,1)$; indeed $\mathcal{P}(Q)\sim \pi Q/2$ near zero and diverges logarithmically as $Q\uparrow 1$. Thus the apparent distribution is strongly skewed toward higher $Q$: projection makes many intrinsically flattened systems look round.