What is equation of the line passing through two points sampled from bivariate Gaussians? Suppose we are given two bivariate Gaussian distributions.  To be concrete:
$\mu_1=(1,1)$
$\mu_2=(-1,-1)$
and the two Gaussians have the same covariance matrix,
$\Sigma=\left( \begin{array}{ll} 1 & 0 \\ 0 & \delta^2 \end{array} \right)$
Let $X_1$ and $X_2$ be a single sample drawn from the first and second Gaussians, respectively.
How to characterize the line passing through $X_1$ and $X_2$? What is the joint distribution of the slope and the intercept of that line?
More broadly, how to characterize the line that maximally separates those two points, i.e., the line that has one point to its left, the other point to its right, and whose distance to the points is maximum? This question is motivated by SVM
 A: Thanks for an interesting question. I have only an idea, I didnt really computed all integrals.
First, consider that $X_2=[y_1, y_2]$ is not random, but a fixed point. Then, you can actually find a density of the angle pretty easily.
WLOG let $\mu_1=(0,0)$ (and $\mu_2=[-2,-2]$, but we have a fixed $X_2$). Denote $\phi(x)$ density of standard normal distribution. Then, density of the slope will be $f_{y_1, y_2}(\alpha)= \phi (y_1 - y_2\cdot cotg(\alpha) )$. The reason is because there is a correspondence one to one between the angle and the point on the line, and this corresponence may be clear from the picture.
Now, all you need to do is integrate this function wrt the distribution of $y_1, y_2$ that is the distribution of the point $X_2$. E.g. if $\delta=0$ then $y_2=-2$ a.s. and we have an integral
$$f(\alpha)=\int_{R}f_{y_1, y_2}(\alpha)*\phi(y_1) dy_1.$$
This may be ugly I dont know.
For $\delta>0$ I only suppose that first you will need to integrate it wrt second axis, and then you will obtain the previous case. I.e. There will be 3 integrals which is kinds ugly :(
Not sure if this helps, but GL
