Two correlated marginally Gaussian RV, but not Jointly Gaussian We know if two RVs are marginally Gaussian and uncorrelated, they can be not jointly Gaussian. How about two **correlated RVs** which are **marginally Gaussian**? Are they Jointly Gaussian or no, there are occasions when they are not Jointly Gaussian?

Suppose $U,V$ are i.i.d. $N(0,1).$ Let $(X,Y)=(U,U)$ if $U>0$ and let $(X,Y)=(-|U|,-|V|)$ if $U\le0.$

Or let $Z\sim N(0,1)$ and let $(X,Y)=(Z,Z)$ if $|Z|>1$ and let $(X,Y)=(Z,2-Z)$ if $|Z|\le 1.$

In both cases, $X$ and $Y$ are marginally gaussian, correlated, but not jointly gaussian.