if two random variables are jointly Gaussian, does that mean they are Gaussian individually? or vice-versa?

Question

I am new into the term of jointly Gaussian. I was wondering, suppose X and Y are two random variables having a Gaussian distribution. My question is whether they will be jointly Gaussian?

Or in the opposite sense, if (X,Y) is jointly Gaussian, does that mean X and Y both have Gaussian distribution?

What is the actual relation in these two case? Thank you.

Answer 1

If $X$ and $Y$ are jointly Gaussian then each one of them is Gaussian. The converse is false: Take $Y=X$ when $|X| \leq 1$ and $Y=-X$ when $|X| >1$ where $X$ has standard normal distribution. The joint distribution does not even have density in this case so it is not Gaussian.