Consider two random variables $X$ and $Y$. If X and Y are independent random variables, then it can be shown that: $$E(XY) = E(X)E(Y).$$
Let $X$ be the random variable that takes each of the values $-1\!\!\!$, $0$, and $1$ with probability $1/3$. Let $Y$ be the random variable with value $Y = X^2$.
Prove that $X$ and $Y$ are not independent.
Prove that $E(XY) = E(X)E(Y)$.
I understand that $E(XY) = E(X^3)$ since $Y = X^2$ so that makes each side of the equation equal to zero.
But I am not sure how to go about proving that $X$ and $Y$ are not independent.