Let X and Y be two random variables.
If we know that both Y|X and X|Y are normal, then can we conclude that
(1) X and Y are normal respectively;
(2) X and Y are bivariate normal?
Thanks!
Edited:
The motivation behind this question is that
If the random vector [X Y]' follows a bivariate normal distribution,
then I think we know that X, Y, X|Y, and Y|X are all normal.
So my question is whether X|Y and Y|X being normal also implies that [X Y]' is bivariate normal.