Integrating the constant density $2$ across horizontal slices of the triangular support gives
$$
f_{\pi_1}(x)=2(1-x),\qquad0<x<1,
$$
so $\pi_1\sim\operatorname{Beta}(1,2)$. For $z=\pi_2=\pi_1+\lambda$, the line segment at fixed $z$ has length $z$, and the transformation has unit Jacobian. Hence
$$
f_{\pi_2}(z)=2z,\qquad0<z<1,
$$
so
$$
\boxed{\pi_1\sim\operatorname{Beta}(1,2),
\qquad\pi_2\sim\operatorname{Beta}(2,1).}
$$

Solution

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

 Solution

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