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 $\int_q^1\mathcal P(Q\mid q)dQ=1$.

Solution

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

 Solution

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