distribution of distance between two points whose coordinates are normal random variables

Question

let there be two random variables $(X_1,Y_1)$ and $(X_2,Y_2)$, where $X_1\sim N(m_1,s)$, $X_2\sim N(m2,s)$, $Y_1\sim N(n,t)$, $Y_2\sim N(n,t)$. What is the distribution of $\|(X_1,Y_1)-(X_2,Y_2)\|$?

Answer 1

The distance is chi-square distributed.

Intuitively, look at the distance between the two normal random variables as an error (for example, distance between randomly sampled point and the mean). We know that this error is distributed chi square.

Here's a more rigorous description:

https://stats.stackexchange.com/questions/9220/what-is-the-distribution-of-the-euclidean-distance-between-two-normally-distribu