I'm having problems figuring out the limits for this. The exponential is e^-(x + y) apologies about the lack of clarity on it.
Hello,
You must have $\displaystyle \iint_{\mathbb{R}^2} f_{X,Y}(x,y) ~dxdy=1$
That is to say, by writing the boundaries:
$\displaystyle \int_0^\infty \int_0^y cxe^{-(x+y)} ~dxdy=1$
maybe reversing the order of integration will help :
$\displaystyle =\int_0^\infty \int_x^\infty cxe^{-(x+y)} ~dydx=c\int_0^\infty x \int_x^\infty e^{-(x+y)} ~dydx=1$
and that's all
I'd go one further and also pull out the $\displaystyle e^{-x}$, as I did below.
$\displaystyle 1=c\int_0^\infty x e^{-x}e^{-x} dx=c\int_0^\infty x e^{-2x}dx$
Which can either be solved by parts or by recognizing the constant in the gamma density
with $\displaystyle \alpha=2$ and what I call $\displaystyle \beta =.5$.