Solution
= Solution
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
$$
\boxed{\pi_1\sim\operatorname{Beta}(1,2),
\qquad\pi_2\sim\operatorname{Beta}(2,1).}
$$