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Mathematics course of the University of Cambridge

Past exam of the mathematics course of the University of Cambridge

2022

Paper 320

At fixed intrinsic ratio $q$, part a gives
$$
Q^2=q^2+(1-q^2)\mu^2,
\qquad
\mu=\sqrt{\frac{Q^2-q^2}{1-q^2}}.
$$
Since $\mu$ is uniform, the <change-of-variables formula for a probability density> yields the conditional <probability density function>
$$
\boxed{
\mathcal P(Q\mid q)
=\frac{d\mu}{dQ}
=\frac{Q}{\sqrt{1-q^2}\sqrt{Q^2-q^2}},
\qquad q\leq Q\leq1}.
$$
The inverse-square-root singularity at $Q=q$ is integrable, and direct integration gives one.

Solution

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

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

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

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