Suppose X,Y are random variables and their joint pdf is given by: f(x,y)=2g(x)g(y) where x*y>0, and zero otherwise.
g(x) and g(y) are pdfs of standard normal distribution.
I was first able to prove that the marginal distribution of X and Y is standard normal.
However, I am having trouble with the question: Is the random vector (X,Y) bivariate normal?
Is it correct to say that since the support of bivariate normal RV is the entire R^2 plane, but since this random vector does not have a positive density at points wheres XY<0, then it cannot be bivariate normal?
I also tried to find a linear combination of X and Y and prove that it is not normally distribution, but to no avail.