$E(XY)=E(X)E(Y)\neq 0$ implies independance

Question

I have a little question. I know that $E(XY)=E(X)E(Y)$ does not imply independence. There are a lot of examples with $E(XY)=E(X)E(Y)=0$.

But if $E(XY)=E(X)E(Y)\neq 0$, is it implying that $X$ and $Y$ are independent ? If not, have you an example where $X$ and $Y$ are dependent ?

Thanks a lot !

Answer 1

Since $E(XY)-E(X)E(Y)$ doesn't change if you add a constant to one or both of $X,\,Y$, any dependent $X,\,Y$ with $E(XY)=E(X)E(Y)=0$ can be shifted (thereby preserving their dependence) to give $E(XY)=E(X)E(Y)\ne0$.

Answer 2

Let $X \sim N(0,1)$. then $1+X^{2}$ and $1+X$ are not independent but $E(1+X^{2})(1+X)=E(1+X^{2})E(1+X)=2$.