Let $A$ be a $3x3$ matrix.
Each element in A is $ a_{ij}=\begin{cases} 1, & w.p\ 1/2\\ 0, & w.p\ 1/2 \end{cases}$
The elements are independent.
Let $X$ be the number of rows with 3 zeros in them.
Let $Y$ be the number of columns with 3 zeros in them.
What is the Correlation of $X$ and $Y$, e.g. $\rho_{X,Y}$ ?
I know the answer to this should be $\rho_{X,Y}=\frac{3}{7}$
I have tried this:
$\rho_{X,Y}=\frac{\operatorname{E}(XY)-\operatorname{E}(X) \operatorname{E}(Y)}{ \sqrt{(\operatorname{Var}(X))} \sqrt{(\operatorname{Var}(Y))} }$
I computed $\operatorname{E}(X) , \operatorname{E}(Y) , \operatorname{Var}(X) , \operatorname{Var}(Y) $ and got $\frac{9}{8}$ for all of them.
edit:
The expected value and variances i computed at first was wrong.
Those (I hope, are the correct ones) $\operatorname{E}(X)=\frac{3}{8}$ , $\operatorname{E}(Y)=\frac{3}{8}$ and $ \operatorname{Var}(X)=\frac{21}{64} , \operatorname{Var}(Y)=\frac{21}{64} $
How can one evaluate $\operatorname{E}(XY)$ ?
Or there is some other way to do it?