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I have a problem where I'm interested in the distance between a fixed point and a bivariate normal distribution (where the two random variables are correlated). How does one find the distribution of that euclidean distance? The mahalanobis distance may seem appropriate, but it's scaled. I'd like to have a distance in terms of the right units.

anonymouspoet
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    You appear to ask two slightly different questions: do you want the distribution of the Euclidean distance or of the Mahalanobis distance? In either case, the solution is given by a non-central chi distribution. – whuber Apr 25 '21 at 17:43
  • @whuber i'm interested in the distribution of the euclidean distance. is there a way to analytically come to that solution or a paper that contains that proof? – anonymouspoet Apr 29 '21 at 13:19
  • Follow the link I gave. – whuber Apr 29 '21 at 13:31
  • thanks for that suggestion. what if the fixed point i mentioned in the original post is not at the origin. is there a way to scale that? – anonymouspoet Apr 29 '21 at 13:40
  • It makes no difference: simply make that point the new origin. The distribution is still Normal, just with a shifted mean. – whuber Apr 29 '21 at 13:46
  • That makes sense. Back to the non-central chi distribution. I don't see how that would give me the euclidean distance of X and Y - which would be sqrt(X^2 + Y^2). Based on this formulation of distance, it would require finding the distribution of the sum of squares of X and Y (which are correlated). – anonymouspoet Apr 29 '21 at 14:51
  • When the variances differ, you're right: you need a generalization of the chi distribution. That's sufficiently messy that you might consider numerical integration, simulation, or approximations unless you need to perform precise mathematical analysis of the distribution. – whuber Apr 29 '21 at 14:55

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