$X$ and $Y$ are independent exponential random variables with parameter λ. We want to find joint distribution of $(U,V)$ where $U=X+Y$ and $V=X/Y.$ Then find marginal dist of U and V.
I was thinking of using change of variables to solve this but i'm having trouble at several steps. This is what I have so far:
$f_{(x,y)} (x,y)= λ^2e^{-λ(x+y)}$
$u=x+y$, $v=x/y$
$x=\frac{uv}{v+1}$, $y=\frac{u}{v+1}$
Region K: $(0,\infty)^2$, Region L: $\{(u,v): 0<u<\infty, ?<v<? \}$ (how do i find bounds?)
$J_k(x,y) = \frac{-x-y}{y^2}$ (is this right?)
Then $f_{u,v}(u,v) = f_{(x,y)} (\frac{uv}{v+1},\frac{u}{v+1})*\frac{1}{|J_k(\frac{uv}{v+1},\frac{u}{v+1})|}=$ $\frac{λ^2 e^{-λu}u}{(v+1)^2}$ (is this right?)