Square Point Picking

Square point picking generates points uniformly distributed over the unit square by independently selecting values and from a uniform distribution on and placing them at coordinates .

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)

For an integer , pick points independently and uniformly 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).

The variance of the perimeter satisfies

			(15)
			(16)

where

			(17)
			(18)
			(19)

and

			(20)
			(21)

(Groeneboom 1988; Cabo and Groeneboom 1994, pp. 39 and 43; Keane 2000; Finch 2003, p. 481).