If two Gaussian random variables are uncorrelated, they are statistically independent I read in a textbook that when two gaussian variables are uncorrelated, then they are statistically independent? How can I prove that?
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.
Two random variables, X,Y, are said to be uncorrelated if their co-variance, E(XY) − E(X)E(Y), is zero.

This mean :
E(XY) = E(X)E(Y)

then , they are statistically independent