Let $X$ and $Y$ have joint pdf $f(x,y) = 4e^{-2(x+y)}$ for $0 < x < \infty$, $0 < y < \infty$, and zero otherwise.
(a) Find the CDF of $W = X + Y$
(b) Find the joint pdf of $U = X/Y$ and $V=X$
(c) Find the marginal pdf of $U$
Could someone show me the statistics behind setting up the integration? I can do the computation myself. So for instance, for (b). I will at least need the Jacobian, $\begin{vmatrix} U_x & U_y\\ V_x & V_y \end{vmatrix} = \dfrac{-X}{Y^2}$
Then subbing, I get $f(u,v)=4e^{2(\frac{v}{u}+v)}$
And for the marginal, I am not continuing until I am sure (b) is right otherwise I will waste my time doing unnecessary computation.
(a) $\int_{0}^{w} \int_{0}^{w-x} 4e^{-2(x+y)}dydx$
Thanks