Past exam of the probability course of the University of Oxford / 2015 / 2 / b / ii

Integrating the joint probability density across $u<v<2u$ gives the marginal densityFor every $u>0$, the conditional probability density is consequentlyHenceThis also follows from two independent exponential variables conditioned on their sum: given $X+Y=u$, the variable $Y$ is uniform on $(0,u)$, and $V=u+Y$. Conditioning on $U=u$ here is defined through the conditional probability density, rather than division by $\mathbb{P}(U=u)$, which is zero. The endpoint $u=0$ lies outside the positive marginal support and needs no specified conditional law.