Past exam of the mathematics course of the University of Cambridge / 2002 / iii / Paper 40 / 1 / a / i / Solution

Write $H={\int }_{\mathbb{R}}h(x)dx$, and assume $0<H<\infty$; a uniform distribution on the region requires positive area. Set $x=V/U$, $s=U$, so the inverse transformation is $U=s$, $V=sx$. Its Jacobian determinant has absolute value $s$ for $s>0$. The region becomes $x\in \mathbb{R}$, $0<s\le \sqrt{h(x)}$, and its area isThe original joint probability density function is therefore $2/H$ on $C_h$. The change of variables formula givesThe boundary $U=0$ has probability zero, so division by $U$ causes no ambiguity for the resulting random variable.