If $(X,Y)$ have the following joint distribution:
$$f_{X,Y}(x,y) = \begin{cases} 2 f_X(x)f_Y(y) & \text{if }xy>0 \\[6pt] 0 & \text{otherwise} \end{cases} $$
where $f_X(·)$ and $f_Y(·)$ are standard univariate normal densities
How to show $X, Y \sim N(0,1)$?
How to check if $(X,Y)$ jointly normal?
Please provide an informative step by step answer, I need to understand it. Thanks in advance!