Suppose that one selects two random points x,y in a sphere of radius R. Is there a closed-form expression for the probability density P(d_x,y), i.e. the probability that x and y have Euclidean distance d in the sphere? Note that I'm referring to points within a sphere, not the probability density over the surface area.
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In mathematics it is conventional to use "sphere" to refer to the surface and "ball" to refer to the enclosed region. You mean the points are selected from the latter, yes? – Jan 04 '16 at 18:50
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I found an existing question about the CDF. One of the answers there mentions the PDF. Or you could use the method in the accepted answer to find the CDF and then differentiate. – epimorphic Jan 04 '16 at 18:55
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Thanks - I can see why "sphere" would cause confusion. – Max Jan 04 '16 at 18:59
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Where $d$ is the distance and $R$ is the radius.
$$f(d)=3\frac{d^2}{R^3}-\frac{9}{4}\frac{d^3}{R^4}+\frac{3}{16}\frac{d^5}{R^6}$$
See for instance Tu and Fischbach, 2011 who also provide results for this problem in spherical objects of general dimension and density.
CommonerG
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Thank you. Has this problem also been solved for an ellipsoid with axes R1,R2,R3? – Max Jan 04 '16 at 19:16
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Yes. Take a look at the introduction of the referenced paper. 3 dimensional ellipsoids are mentioned. – CommonerG Jan 04 '16 at 19:17