I have some problems with this exercise:
Let $X$ and $Y$ be real-valued random variables with a common density:
$f_{X,Y}=\left\{\begin{matrix} 2e^{-x-y} & 0<y<x \\ 0 & else \end{matrix}\right.$
a) Do the marginal distributions of $X$ and $Y$ each have a density? If so, calculate them.
b) Are $X$ and $Y$ uncorrelated?
So for a) Why shouldnt exist a marginal distributions?
I have calculated them: $f_x(x)=\int_0^x 2e^{-x-y}dy=2\left(\mathrm{e}^{-x}-\mathrm{e}^{-2x}\right)$
and for $f_y(y)=2\mathrm{e}^{-y}$
Have I done everything right?
To see if there are correlated I have to check:
$E[XY]=E[X]E[Y] \\ \int_0^x \int_0^{\infty}xy2e^{-x-y}dxdy=(\int_{-\infty}^{\infty} 2x\left(\mathrm{e}^{-x}-\mathrm{e}^{-2x}\right)dx)(\int_{-\infty}^{\infty} 2y\mathrm{e}^{-y}dy)$
The problem here that $E[X]$ diverges which makes me think that I have done something wrong. Can someone help me to make me understand where my error is?
The marginal densities are $$ f_X(x)=\int_0^x 2\,e^{-x-y}\,dy=2\,(e^{-x}-e^{-2x})\,,\quad f_Y(y)=\int_y^\infty 2\,e^{-x-y}\,dx = 2\,e^{-2y}\,. $$ which are both defined on $[0,\infty)\,.$
Your last formula looks totally wrong. It should be $$ \mathbb E[XY]=\int_0^\infty\int_0^x 2\,x\,y\,e^{-x-y}\,dy\,dx=\int_0^\infty 2\,x\,e^{-x}\underbrace{\int_0^xy\,e^{-y}\,dy}_{(*)}\,dx\,. $$ From $$ (*)=\int_0^xy\,e^{-y}\,dy=\int_0^xe^{-y}\,dy-y\,e^{-y}\Big|_0^x=1-e^{-x}-x\,e^{-x} $$ we get $$ \mathbb E[XY]=\int_0^\infty 2\,x\,e^{-x}(1-e^{-x}-x\,e^{-x})\,dx=1\,. $$ The random variables are not uncorrelated because $$ \mathbb E[X]=\int_0^\infty 2\,x\,(e^{-x}-e^{-2x})\,dx=\frac{3}{2}\,,\quad \mathbb E[Y]=\int_0^\infty 2\,y\,e^{-2y}\,dy=\frac{1}{2}\,. $$