Bi variate normal distribution question

Question

$X \sim N(0,1)$ and $c>0$ such that $ \begin{equation} \nonumber Y =\left\{ \begin{array}{l l} X & \quad |X|<c \\ & \quad \\ -X & \quad |X| \geq c \end{array} \right. \end{equation}$

so $Y$ is normal distributed beacuse $P(Y\leq y) = P(X\leq y)+P(X > y)= \int_{-\infty}^{y}\frac{1}{\sqrt{2 \pi}} e^{-\frac{u^2}{2}}du+\int_{y}^{\infty}\frac{1}{\sqrt{2 \pi}} e^{-\frac{u^2}{2}}du=\int_{-\infty}^{\infty}\frac{1}{\sqrt{2 \pi}} e^{-\frac{u^2}{2}}du $

but I dont understand how can I continiue from here to show that $(X,Y)$ is not normal distributed

Answer 1

To show not normaL: I suggest you work it out for all cases. First case $P(X\le x,Y\le y)=0$ if $x\lt -c$ and $y\lt c$, since $Y=-X$.