probability - Understanding range of probability density function Consider:$$f_{X,Y}\left(x,y\right)\:=\:\begin{pmatrix}8xy &amp; 0\le x\le y\le 1 \\0 &amp; otherwise\end{pmatrix}$$
I want to calculate $f_X\left(x\right)$, $f_Y\left(y\right)$, the expectations of them, and $E\left[XY\right]$.  

But, i don't how to determine the range of the integrals for each $f_X\left(x\right)$, $f_Y\left(y\right)$. What is the rule? (for example, for $f_X\left(x\right)$, i need to do $\int _0^1f_{X,Y}\left(x,y\right)dy\:$ or $\int _x^1f_{X,Y}\left(x,y\right)dy\:$?)
tnx!
$f_X(x)=\int_x^1f_{X,Y}(x,y)dy$. Why? If you integrate from $0$ to $1$ you are saying that $x=0$ which is not well deduced.
The definition, in this case, of the marginal distribution $F_X(x)$ is 
$$F_X(x) = \int_{-\infty}^\infty f(x,y)\,dy.$$

From the domain of definition of your pdf, this simplifies to the integral $\int_0^1 f(x,y)\,dy$.  Do you see why?