Let $X,Y, \epsilon$ be independent random variables such that $X,Y\sim \mathcal{N}(0,1)$ and $$\mathbb{P}\{\epsilon=1\}=\frac{1}{2}=\mathbb{P}\{\epsilon=-1\}.$$
Which of the vectors $(\epsilon|X|,\epsilon|Y|),\, (X,\epsilon X),\, (X,\epsilon X+Y)$ are gaussian random vectors (GRV)?
My definition of GRV is the following:
A GRV is a two-dimensional random vector $(X_1,X_2)$ such that for all $\alpha,\beta\in\mathbb{R}$, $\alpha X_1+\beta X_2$ is normally distributed.
I would assume that the first one is a GRV, while the two others aren't. How would one show this? Blindly applying the definition hasn't worked for me.