Distance between two points with covariance

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?

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The (unsigned) distance between two points is a nonnegative quantity that may be difficult to model as a Gaussian random variable, and this is supported by the fact that the square of the distance is (at least in the independent case) a non-central chi-square random variable, whose square root is not a Gaussian random variable. –  Dilip Sarwate Jun 11 '12 at 2:39