Suppose the random variables $X$ and $Y$ have a joint pdf $f(x,y)=6y$, $0<y<x<1$

Question

Find $Cov(X,Y)$

My work:

I found that $E(X) = .75$ and $E(Y)=.50$ $E(XY) = \int_0^1{\int_0^1} xyf(x,y) dydx = {\int_0^1}\int_0^16xy^2 dy dx = \int_0^1 2x dx = 1$

so $Cov(X,Y) = E(XY) - E(X)\cdot E(Y) = 1 - .75\cdot.5= \frac58$

But my book says that the correct answer is $\cfrac{1}{40}$

Where did I go wrong? I would appreciate any tips.

Answer 1

$EXY=\int_0^{1}\int_0^{x} xy (6y)dydx$. This evaluates to $\frac 2 5$ so $cov(X,Y)=\frac 2 5-(0.75)(0.5)=0.025$

Answer 2

$$ E(XY) = \int_0^1 \int_0^1 6xy^2 \mathbb{1}_{0 < y<x<1}dx dy = \int_0^1 \int_y^1 6xy^2dx dy= \frac{2}{5} $$