Given that $X\sim N(0,1)$, $Z\sim Unif(\{\pm 1\})$, and $Y=XZ$. $Z$ is independent of $X$. After calculation we found that $Y\sim N(0,1)$. Show that X and Y are uncorrelated but not independent.
I can only assume that since $Y=XZ$ then, $Y$ must be dependent of $X$.
If $X$ and $Y$ are indepedent then so are $X^{2}$ and $Y^{2}$. But $Y^{2}=X^{2}$ since $Z^{2}=1$. This makes $X^{2}$ independent of itself which is possible only when it is a constant. This contradicts the fact that $X \sim N(0,1)$.
By independence $EXY=EX^{2}Z=EX^{2}EZ=(1)(0)=0$ and $EXE(XY)=(0)E(XY)=0$ so cavariance of $X$ and $Y$ is $0$.