# 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