Square Point Picking

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)
			(2)
			(3)
			(4)

(Finch 2003, p. 479; OEIS A103712), where  is the universal parabolic constant. The expected distance to a fixed vertex is given by

			(5)
			(6)

which is exactly twice .

The expected distances from the closest and farthest vertices are given by

			(7)
			(8)

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)
			(10)
			(11)
			(12)
			(13)
			(14)

(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)
			(16)

where

			(17)
			(18)
			(19)

and

			(20)
			(21)

(Groeneboom 1988; Cabo and Groeneboom 1994; Keane 2000; Finch 2003, p. 481).

REFERENCES:

Bailey, D. H.; Borwein, J. M.; and Crandall, R. E. "Box Integrals." Preprint. Apr. 3, 2006.

Cabo, A. J. and Groeneboom, P. "Limit Theorems for Functionals of Convex Hulls." Probab. Th. Related Fields 100, 31-55, 1994.

Finch, S. R. Mathematical Constants. Cambridge, England: Cambridge University Press, pp. 480-481, 2003.

Groeneboom, P. "Limit Theorems for Complex Hulls." Probab. Th. Related Fields 79, 327-368, 1988.

Heuter, I. "Limit Theorems for the Convex Hull of Random Points in Higher Dimensions." Trans. Amer. Math. Soc. 351, 4337-4363, 1999.

Keane, J. "Convex Hull Integrals and the 'Ubiquitous Constant.' " Unpublished note, 2000.

Rényi, A. and Sulanke, R. "Über die konvexe Hülle von  zufällig gewählten Punkten, I." Z. Wahrscheinlichkeits 2, 75-84, 1963.

Rényi, A. and Sulanke, R. "Über die konvexe Hülle von  zufällig gewählten Punkten, II." Z. Wahrscheinlichkeits 3, 138-147, 1964.

Sloane, N. J. A. Sequences A096428, A096429, and A103712 in "The On-Line Encyclopedia of Integer Sequences."