= Solution
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.
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