In the problem I am given $f(x,y)=2,\ 0 < x < y,\ 0 < y <1$.
I'm trying to find the correlation $\rho$ which I know is equal to
$$\rho = \frac{Cov(x,y)}{\sqrt{Var(x)Var(y)}}$$
where $Cov(x,y) = E(XY) - E(X)E(Y)$
$$Var (X) = E(X^2) - [E(X)]^2$$ and likewise with $Var(Y)$ just with $Y$.
I've tried solving this all out and the book says $\rho = 1/2$.
So here's what I got (which is wrong), and if anyone can, please help me see where I'm going wrong so I can get the right answer please!
$$EX = \iint\limits_A xf(x,y) \ dy \ dx = \frac{1}{3} \\ EX^2 = \frac{1}{6} \ \ EY = \frac{2}{3}\ \ EY^2 = \frac{1}{3}\ \ EXY = 1 \\ Cov(x,y) = \frac{7}{9}\ \ Var(X) = \frac{1}{18}\ \ Var(Y) = \frac{-1}{9} $$
(which I don't think can be right)
My $\rho$ did not equal $1/2$.
Please help!!(Feel free to edit the problem to make it more presentable. I did the best I knew how, which is not much, sorry.)
Here are some corrections to your derivatons: $EY^2=\frac{1}{2}$, $EXY=\frac{1}{4}$, $Cov(X,Y)=\frac{1}{36}$, $Var(y)=\frac{1}{18}$, hence $\rho=\frac{1}{2}$ is correct. The other values you calculated correctly.