# What is the distribution of the euclidean distance between two random points in 2d space?

I would like to know the distribution of z as the euclidean distance between 2 points which are not centred in the origin. If I assume 2 points in the 2D plane $A = (X_a,Y_a)$ and $B = (X_b,Y_b)$, where the $X_a \sim \mathcal N(\mu_a,s^2)$, $X_b\sim\mathcal N(\mu_b,s^2)$, $Y_a\sim\mathcal N(\nu_a,s^2)$, $Y_b\sim\mathcal N(\nu_b,s^2)$, then the Euclidean distance between $A$ and $B$, would be $z= \sqrt{(X_a-X_b)^2 + (Y_a-Y_b)^2}$. Now: $X=X_a-X_b$ and $Y=Y_a-Y_b$ are themselves random variables with means $(\mu_b-\mu_a)$ and $(\nu_b-\nu_a)$ and variance $2s^2$, so the problem that I have is determining the pdf of $z=\sqrt{X^2 +Y^2}$, knowing that $X$ and $Y$ are 2 uncorrelated Gaussian random variables with non-zero means and the same variance, $2s^2$.

The Rician distribution applies when $z$ is the distance from the origin to a bivariate random variable. This has been proven only when the random variables ($A$ and $B$) are circular bivariate random variables (A proof can be found in L. C. Andres and R. L. Phillips, Mathematical Techniques for Engineers and Scientists, 2003, Ch. 13, Sec. 13.8.2, p. 680). I would like to know the pdf/cdf of $z$ as a distance between two points (none of them being centred in the origin) when they are not circular. Is there a known parametric distribution for $z$? What would this distribution look like if it is a generalized form of the Rician distribution?

Thank you!

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– Andy W Mar 27 '12 at 13:24
The answer is here: en.wikipedia.org/wiki/Chi_distribution – whuber Mar 27 '12 at 13:32
Cardinal: "determining the pdf of z=sqrt(X^2 +Y^2), knowing that X and Y are 2 uncorrelated Gaussian RVs with NON-ZERO MEANS and the same variance, 2s^2" defines the chi distribution, doesn't it (up to scaling by 2s^2)? When the means are unequal it should be a non-central chi distribution. – whuber Mar 27 '12 at 13:49
Thanks for that, Cardinal. Your reading matches my interpretation of the OP's previous question, but this question explicitly refers to "uncorrelated" RVs (at the end of the first paragraph). Ana Maria: please clarify this and help us understand how the present question differs from your previous one on this topic. – whuber Mar 27 '12 at 14:05
Hi Ana. I'm sorry. What part, precisely, remains unclear? I (or perhaps someone else) can then post a short proof to remove all doubt. Cheers. – cardinal Mar 27 '12 at 14:55