Let $X \sim \text{Gaussian}(0, 1)$ be a D211: Standard gaussian random real number such that
| (i) | \begin{equation} Y : = X^2 \end{equation} |
Then
| (1) | \begin{equation} \mathbb{E} X Y = \mathbb{E} X^3 = 0 \end{equation} |
| (2) | \begin{equation} \mathbb{P}(X > 1, Y^2 < 1) = 0 \neq \mathbb{P}(X > 1) \mathbb{P}(Y^2 < 1) \end{equation} |