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?
 A: 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.
A: 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.   
A: The covariance matrix of uncorrelated random variables $X$ and $Y$ is a diagonal matrix. If the two random variables are jointly Gaussian, we can write the joint PDF as
$$\frac{1}{\sqrt{(2\pi)^2\cdot|K|}}\cdot e^{-\frac{(x-\mu)^TK^{-1}(x-\mu)}{2}}$$
where K is the covariance matrix of X and Y,
$$K = \begin{bmatrix}
 \sigma_x^2 & 0 \\ 
 0 & \sigma_y^2 
\end{bmatrix}.$$
If we use this $K$ in above equation, we can split the joint pdf into marginal pdf of $X$ and $Y$, hence independence is proved.
A: 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
