Square line picking is the selection of pairs of points (corresponding to endpoints of a line segment) randomly placed inside a square. random line segments can be picked in a unit square in the Wolfram Language using the function RandomPoint[Rectangle[], n, 2].
Picking two points at random from the interior of a unit square, the average distance between them is the case of hypercube line picking, i.e.,
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(OEIS A091505).
The exact probability function is given by
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(M. Trott, pers. comm., Mar. 11, 2004), and the corresponding distribution function by
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From this, the mean distance can be computed, as can the variance of lengths,
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The statistical median is given by the root of the quartic equation
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which is approximately .
The th raw moment is given for , 4, 6, ... as 1/3, 17/90, 29/210, 187/1575, 239/207, ... (OEIS A103304 and A103305).
If, instead of picking two points from the interior of a square, two points are chosen at random on different sides of the unit square, the average distance between two points picked in this manner is
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(OEIS A091506; Borwein and Bailey 2003, p. 25; Borwein et al. 2004, p. 66).