Let $ X $ and $Y$ have the joint PDF $f(x,y)=4e^{-2(x+y)}$ on $x>0$ and $y>0$ and zero otherwise. Find the distribution of $W=X+Y$
I make the substitutions $W=X+Y$ and $U=X$ to obtain that the Jacobian is simply $1$.
So then I calculate the joint distribution of $U$ and $W$ to be $f(u,w) = 4e^{-2w}$ on $u>0$ and $w>0$
But when I go to find the marginal distribution function of $W$, $f_W(w)$, I get stuck with a divergent integral. What's going wrong here? Is my support for $u$ and $w$ incorrect? I just don't understand it. Please, somebody explain what's going wrong.
You probably did not get the support of $u$ correctly. The density $f_{U,W}$ is zero outside of the set $\{(u,w): 0 \le u \le w\}$. $$f_W(w) = \int_0^w f_{U,W}(u,w) \mathop{du} = \int_0^w 4e^{-2w} \mathop{du} = 4we^{-2w}.$$ This is the gamma distribution with parameters $\alpha=2$ and $\beta=2$.