I have some questions concerning gaussian variables.
Suppose I have two gaussians RV $g_1$ and $g_2$ such that $\text{corr}(g_1, g_2)=\rho$ .
$g_1$ and $g_2$ are not necessarly jointly gaussian, my questions are the following:
-Do we have an idea about the joint distribution of $(g_1,g_2)$ ? Can we write that $g_1 = \rho g_2 + \sqrt{1-\rho^2} Z$ with $Z$ independant of $g_2$ or are we allowed to write it only if $g_1$ and $g_2$ are jointly gaussian?
Lot of thanks