Example of $X,Y$ normal ditributed s.t. $\mathbb E[XY]=\mathbb E[Y]\mathbb E[Y]$ but $X$ and $Y$ are not independent.

Question

I always thought that if $X$ and $Y$ were normal distributed then $\mathbb E[XY]=\mathbb E[X]\mathbb E[Y]$ implies $X$ and $Y$ are not independent. But my teacher said it was wrong. Could someone gives a counter-example ? I really don't see why is it wrong.

Answer 1

Let $X$ be a standard normal random variable. Let $B$ take values $\pm 1$, each with probability $1/2$, and independent of $X$. Let $Y = B X$. Then $Y$ has a standard normal distribution, and $$\mathbb E[XY] = \mathbb E[B X^2] = \mathbb E[B] \mathbb E[X^2] = 0 = \mathbb E[X] \mathbb E[Y]$$ but $X$ and $Y$ are not independent.