Disk Line Picking

Disk line picking concerns the statistical distribution of the distance between two points chosen independently and uniformly in a disk. Using disk point picking,

			(1)
			(2)

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

			(3)
			(4)
			(5)
			(6)

(OEIS A093070; Uspensky 1937, p. 258; Solomon 1978, p. 36).

This is a special case of ball line picking with , so the full probability density function for a disk of radius is

(7)

for , and is zero outside this interval (Solomon 1978, p. 129; Mathai 1999, p. 204).

The raw moments of the distribution of line lengths are given by

			(8)
			(9)

where is the gamma function and is a real number with . The expected value of is given by , giving

(10)

(Solomon 1978, p. 36; Pure et al. ). The first few moments are then

			(11)
			(12)
			(13)
			(14)
			(15)
			(16)

(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).