= Solution
For an intrinsic <probability density function> $\mathcal P(q)$, the <law of total probability> averages the conditional density from part c. An object observed with ratio $Q$ can only have $q\leq Q$, so
$$
\boxed{
\mathcal P(Q)
=Q\int_0^Q
\frac{\mathcal P(q)}{\sqrt{1-q^2}\sqrt{Q^2-q^2}}\,dq,
\qquad 0<Q<1}.
$$
This is an <Abel transform> of $\mathcal P(q)/\sqrt{1-q^2}$. Its normalization follows by reversing the order of integration and using $\int_q^1\mathcal P(Q\mid q)dQ=1$.
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