this is the kind of questions where from just looking at you realize that the answer might not be straightforward:
$$f_{X,Y}(x,y)\begin{cases} e^{-y}e^{-xy}y^2 & y>0 ,x\geq0\\ 0 & else \end{cases} $$
And the questions is: What can you conclude about X,Y? (Are they independent? $Cov(X,Y)>0$ ? $COV(X,Y)<0$? are they coordinated? etc.
Well, as a newbie, I started by calculating $ f_X(x)$. It didn't work out. It was too complicated and I got stuck. I then turned to $f_Y(y)$ which gave me $f_Y(y) = ye^{-y}$, but I couldn't figure out what to do with that data. So I then though that this is probably deeper than just a technically solvable problem, there's probably a concept behind it.
Could you please enlighten me?
Thanks!
The random variables $X$ and $Y$ are independent iff the joint density is the product of the marginal densities. You have found the density of $Y$. For the density of $X$, note that $$ f_{X}(x)=\int_0^\infty f(x,y)\, dy=\frac{2}{(x+1)^3}\int_{0}^\infty \frac{(x+1)^3}{2}y^{2}e^{-y(x+1)}\, dy=\frac{2}{(x+1)^3}\quad (x\geq0) $$ where the integral equals one since it is the integral of a gamma density with shape parameter $3$ and rate parameter $x+1$.
It follows that $X$ and $Y$ are not independent.