Jointly Gaussian uncorrelated random variables are independent [closed]

Question

Let $X,Y$ be jointly normally distributed and uncorrelated. Why are they independent?

Answer 1

In short, they are independent because the bivariate normal density, in case they are uncorrelated, i.e. $\rho =0$, reduces to a product of two normal densities the support of each one ranges from $(-\infty, \infty)$. If the joint distribution can be written as a product of nonnegative functions, we know that the RVs are independent. Moreover, we know, and can show, that each marginal density is normal on its own.

That is easy to see in the bivariate density below:

$$f(x,y)= \frac{1}{2 \pi \sigma_1 \sigma_2 \left( 1-\rho^2 \right)^{1/2}} \exp\{-q/2 \}, \quad -\infty<x<\infty,\quad -\infty<y<\infty $$

where $$q= \frac{1}{1-\rho^2} \left[ \left( \frac{x-\mu_1}{\sigma_1} \right)^2-2\rho \left(\frac{x-\mu_1}{\sigma_1} \right) \left(\frac{y-\mu_2}{\sigma_2} \right)+\left(\frac{y-\mu_2}{\sigma_2} \right)^2 \right]$$

Put $\rho=0$. There is also a nice proof involving mfgs. Is that what you were looking for?