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

Integrating out the second coordinate gives the probability density function of the marginal distributionThus the ratio has the normalized target probability density function. The factor $s$ from the Jacobian determinant is essential; merely transforming the rectangular coordinates as if they remained uniform would give a different probability density function. This is the basis of the ratio-of-uniforms method.