$$f_{X,Y}(x,y)=\begin{cases}\tfrac 8 3(xy) & \textsf{whereas } 0<x<1 , x<y<2x \\ 0 & \textsf{elsewhere}\end{cases}$$
I need to calculate covariance. I managed to calculate that $E(X)$ is $4/5$ and $E(XY)$ is $1$. However I cannot get number for $E(Y)$ because of its boundry. Any suggestions on how to solve for $E(Y)$?
Since the function appears to be a joint pdf, the interval $0<x< 1$ and $x<y<2x$ means that
$$\mathsf E(X) = \int\limits_0^1\int\limits_x^{2x} \tfrac 8 3 xy\cdot x \operatorname d y\operatorname d x ~=~\frac 4 5\quad\color{green}\checkmark$$
$$\mathsf E(XY) = \int\limits_0^1\int\limits_x^{2x} \tfrac 8 3 xy\cdot xy \operatorname d y\operatorname d x ~\neq ~ 1$$
$$\mathsf E(Y) = \int\limits_0^1\int\limits_x^{2x} \tfrac 8 3 xy\cdot y \operatorname d y\operatorname d x$$