Using disk point picking,
 |  |  |
(1)
|
 |  |  |
(2)
|
for ![r in [0,1]](/images/equations/DiskLinePicking/Inline7.svg)
,
, 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).
