Using disk point picking,
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for , , choose two points at random in a unit disk and find the distribution of distances between the two points. Without loss of generality, take the first point as and the second point as . Then
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(OEIS A093070; Uspensky 1937, p. 258; Solomon 1978, p. 36).
This is a special case of ball line picking with , so the full probability function for a disk of radius is
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(Solomon 1978, p. 129; Mathai 1999, p. 204).
The raw moments of the distribution of line lengths are given by
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where is the gamma function and . The expected value of is given by , giving
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(Solomon 1978, p. 36; Pure et al. ). The first few moments are then
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(OEIS A093526 and A093527 and OEIS A093528 and A093529). The moments that are integers occur at , 2, 6, 15, 20, 28, 42, 45, 66, ... (OEIS A014847), which rather amazingly are exactly the values of such that , where is a Catalan number (E. Weisstein, Mar. 30, 2004).