# Distance between two points with covariance [duplicate]

I would like to find the distance between two points location 1 and location 2. In 2D, location 1 is represented by a Gaussian distribution with mean $\mu_1$ and co-variance matrix $\Sigma_1$. Similarly location 2 is represented by a Gaussian distribution with mean $\mu_2$ and co-variance matrix $\Sigma_2$.

\begin{align} \mu_1 &= \begin{bmatrix} \mu_{x_1} \\ \mu_{y_1}\end{bmatrix} &{\rm and} &\qquad \Sigma_1 = \begin{bmatrix} \sigma_{x1} &\sigma_{x_1, y_1} \\ \sigma_{y1, x1} &\sigma_{y_1}\end{bmatrix} \\[10pt] \mu_2 &= \begin{bmatrix} \mu_{x_2} \\ \mu_{y_2}\end{bmatrix} &{\rm and} &\qquad \Sigma_2 = \begin{bmatrix}\sigma_{x_2} &\sigma_{x_2, y_2} \\ \sigma_{y_2, x_2} &\sigma_{y_2}\end{bmatrix} \end{align}

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:

\begin{align} \mu &= \mu_1-\mu_2 \\[5pt] \Sigma &= \Sigma_1 + \Sigma_2 \end{align}

But I am skeptical about this since $\mu_{x_1}$ and $\mu_{x_2}$ are correlated ($\sigma_{x_1y_1}$ is not equal to zero). But location 1 and location 2 distributions are not correlated. So do I have to worry about the correlation of $\mu_{x_1}$ and $\mu_{x_2}$?

• 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
• @gung OK, I've deleted my answer below and posted there; we can now close the present one as a duplicate. – Felipe G. Nievinski Dec 5 '15 at 18:46

## 1 Answer

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.