Hi

If $\displaystyle X$ and $\displaystyle Y$ has joint density function

$\displaystyle f_{X,Y}(x,y)=\begin{cases} \frac{2|x|+y}{\sqrt{2\pi}}e^{-\frac{(2|x|+y)^2}{2}}, y \geq -|x| \\ 0 , y < -|x| \end{cases} $

,how do I prove X and Y are standard normal variables?

Also show that they are not independent, but uncorrelated.

For the first part, can I calculate the marginal density functions and they would be the density function of a standard normal?

Or should I use moment generating functions?

Thanks