I need to calculate correlation $\rho_{XY}$ between two variable in range (0<x<y, 0<y<1), the range are related. I have no data, just the joint probability distribution. How do I calcultate this:
$$ \rho_{XY} = \frac{Cov(XY)}{\sqrt{Var_X}\sqrt{Var_y}} = \frac{E[XY] - E[X]E[Y]}{\sqrt{(E[X^2] - E[X]^2)(E[Y^2] - E[Y]^2 )}} $$
The joint function is:
$$ f_{XY}(x,y) = \frac{3x + 8y}{3.2} $$
I need numbers but the marginal of $x$ to find E[X] have the $y$ variable.
$$ E[X] = \int_0^y xf_x(x)dx = \frac{y^3 + 2y^2}{3.2} $$
But I can't find $\rho$ with that.