Picking two independent sets of points and from a unit uniform distribution and placing them at coordinates gives points uniformly distributed over the unit square.
The distribution of distances from a randomly selected point in the unit square to its center is illustrated above.
The expected distance to the square's center is
(1)
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(2)
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(3)
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(4)
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(Finch 2003, p. 479; OEIS A103712), where is the universal parabolic constant. The expected distance to a fixed vertex is given by
(5)
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(6)
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which is exactly twice .
The expected distances from the closest and farthest vertices are given by
(7)
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(8)
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Pick points at randomly in a unit square and take the convex hull . Let be the expected area of , the expected perimeter, and the expected number of vertices of . Then
(9)
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(10)
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(11)
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(12)
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(13)
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(14)
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(OEIS A096428 and A096429), where is the multiplicative inverse of Gauss's constant, is the gamma function, and is the Euler-Mascheroni constant (Rényi and Sulanke 1963, 1964; Finch 2003, pp. 480-481).
In addition,
(15)
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(16)
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where
(17)
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(18)
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(19)
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and
(20)
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(21)
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(Groeneboom 1988; Cabo and Groeneboom 1994; Keane 2000; Finch 2003, p. 481).