I'm trying to determine whether the following continuous random variables are independent or not given their joint probability density.
$f_{X,Y}(x,y) = 2(x+y)$ for $ 0\le y\le x\le 1$
I calculated the marginal densities as follows:
$f_x(x) = \int_0^1{2(x+y)} dy$ = $ 2x+1$
Using the same method for $f_y(y)$ I got $f_y(y) = 2y+1$
Since $f_{X,Y}(x,y)\neq f_X(x)f_Y(y) $ they are not independent.
My concern is that my calculation of the marginal densities is incorrect, could someone verify whether my method is correct? Or if this is the best way to check if the two are independent.