Distance between two points with covariance

Question

I would like to find the distance between two points locaion1 and location2. In 2D, location1 is represented by a Gaussian distribution with mean m1 and co-variance matrix P1. Similarly location2 is represented by a Gaussian distribution with mean m2 and co-variance matrix P2.

Here m1= [m_x1; m_y1] and P1 = [sigma_x1 sigma_x1_y1; sigma_y1_x1 sigma_y1].

m2= [m_x2; m_y2] and P2 = [sigma_x2 sigma_x2_y2; sigma_y2_x2 sigma_y2]

By looking at other questions, I believe the distance between these two locations will be a Gaussian distribution as well. In that case, distance would be a Gaussian would be m = m1-m2 and P = P1 + P2.

But I am skeptical about this since m_x1 and m_x2 are correlated ( sigma_x1_y1 not equal to zero). But location1 and location2 distributions are not correlated. So do I have to worry about the correlation of m_x1 and m_x2?

Answer 1

The distance between two points from multivariate Gaussian distributions with the same covariance is the Mahalanobis distance. It is more complicated when the covariance matrices are different. The distance between the two means is a positive constant. If you are looking at the distance between the sample mean vectors this is a random variable but not a normal random variable. The distance is nonnegative. To be specific if the bivariate normals have independent components with the same variance, the distance has a Rayleigh distribution.