Use the deterministic evidence-synthesis relation
$$
\pi_2=\pi_1+\lambda,
$$
with support
$$
\mathcal T=\{(\pi_1,\lambda):\pi_1\geq0,
\lambda\geq0, \pi_1+\lambda\leq1\}.
$$
A uniform prior on this triangle has density $2\mathbf1_{\mathcal T}$. Conditional on the prevalences, model the independent surveys by
$$
Y_{S1}\sim\operatorname{Binomial}(n_{S1},\pi_1),
\qquad
Y_{S2}\sim\operatorname{Binomial}(n_{S2},\pi_2),
$$
with the two counts <conditionally independent> and $\pi_2=\pi_1+\lambda$.

Solution

Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 207 / 1 / e / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2019/iii/paper-207/1/e/solution