Consider two random variables $x$ and $Y,$ independent and following a standard

normal distribution, i$.$e$.,xN(0,1)$ and $YN(0,1).$ Consider a random variable

$,$ independent of $x$ and $Y,$ such that $P(=1)=P(=-1)=\frac{1}{2}.$ Let $Z=x+$

$Y.$

What is the distribution of $Z?$

Is $(x,Y,Z)$ a Gaussian vector?