If two Gaussian random variables are uncorrelated, they are statistically independent

Question

I read in a textbook that when two gaussian variables are uncorrelated, then they are statistically independent? How can I prove that?

Answer 1

If $X$ and $Y$ are jointly gaussian, and uncorrelated, you can show that $$f_{XY}(x,y) = f_X(x)f_Y(y);$$ this assures independence.

Answer 2

Here is a counterexample.

Here is the definition of "'jointly' normally distributed". This article states, but I'm not sure it proves, that jointly normally distributed random variables are independent if they are uncorrelated.