there's a something that I though about that Disturbing me for quite some time now:
Let $X$ , $Y $ be a random variables. I call random variables independent if for every possible value of $x_{i}$ that $X$ might get and $y_{j}$ that $Y$ might get:
$p(X=x_{i} \cap Y=y_{j})=P(X=x_{i})\cdot P(Y=y_{j})$.
In addition two random variables are Non-correlated if $ E[X\cdot Y]=E[X]\cdot E[Y]$,
when $E$ represents the expected value.
I know that if $X$ and $Y$ are independent so they are Non-correlated.
when $X$ and $Y$ are dependent, I can think of only cases which $E[X\cdot Y]=0$ because one of the expected value of $X$ or $Y$ is o.
my QUESTION is: Are they any dependent random varaibles and non-correlated so that $E[X]\neq0$ and $E[Y]\neq0$, and if not, how does one prove that?
Thank you for the help.