Past exam of the options II course of the University of Oxford / 2010 / M / M4 / a / i / Solution

Away from the null event $X=0$, the inverse transformation is $x=u$, $y=uv$. Its Jacobian determinant is $u$, so its absolute value is $|u|$. Independence therefore gives the joint densityA value at $u=0$ may be assigned arbitrarily without changing the distribution.