If the marginal distributions of X1 and X2 are both normal, it does not necessarily imply that the vector (X1, X2) is distributed as bivariate normal. To see this, consider the bivariate density function 1 1 f (21, 12) = exp 47 \\/1 - p2 2(1 - p2) (21 -2px1x2 + 23) , + exp1- 2(1 - p2) for -co