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.

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    $\begingroup$ 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. $\endgroup$
    – whuber
    Apr 25, 2021 at 17:43
  • $\begingroup$ @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? $\endgroup$ Apr 29, 2021 at 13:19
  • $\begingroup$ Follow the link I gave. $\endgroup$
    – whuber
    Apr 29, 2021 at 13:31
  • $\begingroup$ 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? $\endgroup$ Apr 29, 2021 at 13:40
  • $\begingroup$ It makes no difference: simply make that point the new origin. The distribution is still Normal, just with a shifted mean. $\endgroup$
    – whuber
    Apr 29, 2021 at 13:46


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