For the <uniform distribution> $\mathcal P(q)=1$, substitute $q=Q\sin\theta$ into part d:
$$
\mathcal P(Q)
=Q\int_0^{\pi/2}
\frac{d\theta}{\sqrt{1-Q^2\sin^2\theta}}
=\boxed{QK(Q)},
$$
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 $Q\uparrow1$. Thus the apparent distribution is strongly skewed toward higher $Q$: projection makes many intrinsically flattened systems look round.

Solution

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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2022/iii/paper-320/1/e/solution