Past exam of the mathematics course of the University of Cambridge / 2019 / ia / Paper 2 / 4F (Probability)

Independence of random variables and the two unit-rate exponential distributions giveThe linear transformationhas determinant $2$ and inverseThe inequalities $x,y\ge 0$ become $8v/3\le u\le 3v$. By the change-of-variables formula for probability densities,The variables $U$ and $V$ are not independent: the support of their joint density is a wedge rather than a Cartesian product of one-dimensional supports. For example, conditioning on a positive value of $V$ constrains $U$ to a finite interval depending on that value.