Assume $X \sim \mathcal{N}(0,1)$ and $Y \sim \mathcal{N}(0,1)$. For $E[XY]$, we have the following: \begin{equation} E[XY] = \int XY exp(-X^2 - Y^2) dXdY = \int X exp(-X^2)dX \int Y exp(-Y^2)dY = 0, \end{equation}
then $E[XY] = 0$. Is this true (there is no condition that $X$ and $Y$ are independent)?