Covariance of centred random vairables

Question

Let $X$ and $Y$ be two random variables, with $\mathbb{E}(X)=\mathbb{E}(Y)=0$.

Is it true that $\mathbb{E}(XY)=0$ ? In other word, is it true that cov$(X,Y)=0$ ?

Answer 1

It is not true.

Consider $$P(-1,-1)=P(1,1)=\frac12$$

Then $E(X)=0=E(Y)$ but $E(XY)=1$.