# Marginal normality and joint normality

Let $X$ and $Y$ be two independent standard normally distributed random variables $N(0,1)$ .If we define a new random variable $Z$ such that : $$Z = \begin{cases}X & \text{if} &XY > 0\\ -X & \text{if} & XY < 0\end{cases}$$ Prove that the joint distribution of $Z$ and $Y$ is not bivariate normal by trying to show that $Z$ and $Y$ always have the same sign .

In fact I could prove that $Z$ and $Y$ have the same sign , since I proved that $Z$ is standard normally distributed $N(0,1)$ and by definition of $Z$ , $Z > 0$ if either $X<0$ and $Y<0$ or $X>0$ and $Y>0$ .

But I did not get the idea of that proof , why the joint distribution of $Y$ and $Z$ will not be bivariate normal as long as $Y$ and $Z$ have the same sign ? I know that the normality does not imply joint normality , I just want to know the relation between the sign of the random variables and the joint bivariate normal distribution ?

• This is a specific question so I don't think this should be closed. Commented Jan 25, 2016 at 15:24