Question

problem#1

Answer 1

1) U = X/Y

Let V=X

Y = V/U

The jacobian is given by

$\large J=\begin{vmatrix} \frac{\partial x}{\partial u} & \frac{\partial x}{\partial v} \\ \frac{\partial y}{\partial u} & \frac{\partial y}{\partial v} \end{vmatrix} =\begin{vmatrix} 0 & 1\\ \frac{-v}{u^2}& 1/u \end{vmatrix}=\frac{v}{u^2}$

The joint p.d.f of U and V is given by

$\large f_{u,v}(u,v) = f_{X,Y}(\frac{u}{v},u).\frac{v}{u^2}$