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 function
for a disk of radius
is

(7)

(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 . 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).

Sloane, N. J. A. Sequences A014847, A093070, A093526,
A093527, A093528,
and A093529 in "The On-Line Encyclopedia
of Integer Sequences."Mathai, A. M. An Introduction to
Geometrical Probability: Distributional Aspects with Applications. Amsterdam,
Netherlands: Gordon and Breach, 1999.Pure, R.; Durran, S.; Tong, F.;
Pan, J. "Distance Distribution Between Two Random Points in Arbitrary Polygons."
To appear in Math. Meth. Appl. Sci.Solomon, H. Geometric
Probability. Philadelphia, PA: SIAM, 1978.Uspensky, J. V.
Ch. 12, Problem 5 in Introduction
to Mathematical Probability. New York: McGraw-Hill, pp. 257-258, 1937.