Example X and Y are Gaussian R.V., but (X,Y) is not.

[Bridgeport]

Privetstvuu!

I am looking for an example such that X and Y are Gaussian random variables., but (X,Y) is not Gaussian vector.
Is is possible at all?

Spasibo!

[Mikhail Sokolov]

Yes, it is possible. See e.g. the book:
Stoyanov J. Counterexamples in probability. Wiley, 1997. (clause 10.1).

[Bridgeport]

Ura, Nashel!

Ogromnoe spasibo!