Past exam of the mathematics course of the University of Cambridge / 2006 / ia / Paper 2 / 10F (Probability)

Independence gives the joint probability density function $f_{X,Y}(x,y)=4e^{-(x^2+y^2)}/\pi$ on $x,y>0$. The inverse transformation isIts Jacobian matrix has absolute determinant $1/[2(1+v^2)]$. One way to see the determinant is to use polar coordinates: $u=r^2$, $v=\tan \theta$, with $0<\theta <\pi /2$. Then $dxdy=rdrd\theta =dudv/[2(1+v^2)]$. The change-of-variables formula for probability densities givesand zero outside that support. Integrating over the complementary variable givesThese are respectively a unit-rate exponential distribution and a unit-scale half-Cauchy distribution. Their product equals the joint probability density function, so $U$ and $V$ are independent. This radial-ratio independence of two half-normal variables results from the radial symmetry of the positive-quadrant joint probability density function.