Show that there exist a random vector \((U, V)\) such that \(U\) and \(V\) are one-dimensional Gaussian random variables but \((U, V)\) is not Gaussian.

Try \(f(u, v)=g(u) g(v)(1-\sin u \sin v)\) where \(g(u)=(2 \pi)^{-1 / 2} e^{-u^{2} / 2}\).