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 $\mu_{x_2}$?
