Ball line picking is the problem of finding the statistical distribution of the lengths 
of line segments determined
by two points chosen independently and uniformly within
an 
-ball 
of radius 
, where 
is a positive integer.
The figure plots the dimensionless density 
against 
for 
, 2, ..., 8. The probability
density function of these lengths is given by
 |
(1)
|
for 
,
and is zero outside this interval. Here
 |
(2)
|
and
 |
(3)
|
is a regularized beta function, where 
is an incomplete
beta function and 
is a beta function (Tu and Fischbach 2001, p. 8).
The first few are
 |  |  |
(4)
|
 |  |  |
(5)
|
 |  |  |
(6)
|
 |  |  |
(7)
|
The mean line segment lengths for 
and the first few dimensions 
are given by
 |  |  |
(8)
|
 |  |  |
(9)
|
 |  |  |
(10)
|
 |  |  |
(11)
|
(OEIS A093530 and A093531 and OEIS A093532 and A093533), corresponding to line line picking, disk line picking, (3-D) ball line picking, and so on.
For arbitrary radius 
, these means scale
as 
.
The second raw moment is
 |
(12)
|
which gives 
,
, 
, and 
for 
, 2, 3, and 4, respectively (Tu and Fischbach 2001, p. 28).
The standard deviation is therefore
 |
(13)
|
-Dimensional Sphere and Its Applications to Physics."
7 May 2001.