I'd like to check if my results are correct. Could you guys please verify them?
$X,Y \sim \; Exp(1), X \perp Y$
Find the joint density of $U = X, V = X/Y$:
\begin{align} V = X/Y & \rightarrow Y = U/V\\ U = X & \rightarrow X = U \\ |Det(J)| &= U/V^{2} \end{align}where J is the jacobian matrix.
As long as they are iid,
$f_{Y}(U/V)f_{X}(U)|Det(J)| = e^{-U/V} e^{-U} \frac{U}{V^{2}}$