I'm trying to get more familiar with $E(XY) = E(X)E(Y)$, and all examples that I have come across seems to satisfy this expression. My question then is, if there exist an example of two random variables $X$ and $Y$ for which $E(XY) \neq E(X)E(Y)$?
I am looking for an example using a joint probability distribution table.
Thank you.
Suppose that $\operatorname EX=0$, $\operatorname EX^2=1$ and $X=Y$. Then $$ 1=\operatorname EX^2=\operatorname E[XY]\ne\operatorname EX\operatorname EY=\operatorname EX\operatorname EX=0. $$