Solution
= Solution
Integrating the <joint probability density> over the auxiliary coordinate, as in (i), gives the <marginal distribution>
$$
\boxed{\pi_X(x)=\int_0^1\pi(x,u)\,du=p(x).}
$$
In particular, augmenting with $U$ preserves the target <probability distribution> of $X$. The other <marginal distribution>, though not needed to identify $X$, is $\pi_U(u)=M\int_\Omega q(x)\mathbf1_{\{w(x)\geq Mu\}}\,dx$ for $0<u<1$. It need not be a <uniform distribution> on $(0,1)$.