Marginally normally distributed and uncorrelated, but not jointly distributed.

Question

Let $X$ be a standard Normal r.v., and $\epsilon$ a r.v. with $P(\epsilon=-1)=P(\epsilon=1)=1/2$

If $X_1=X$ and $X_2=\epsilon X$, then both are $N(0,1)$ and uncorrelated.

However, $(X_1,X_2)$ is not normally distributed... Why is that?

For every linear combination of $aX_1+bX_2$, with $a,b \in \mathbb{R}$, I think we get a normal r.v.

Answer 1

Try the linear combination $Y=X_1-X_2$, i.e. with $a=1, b=-1$

This is not normally distributed since $\mathbb P (Y=0)=\frac12$ while normal distributions are continuous so the probability of any specific value should be $0$