Line Point Picking

Line point picking is the selection of a uniformly distributed point on a unit line segment . This point divides the line into line segments of length and . If a set of points are thus picked at random, the resulting distribution of lengths has a uniform distribution on . Similarly, separating the two pieces after each break, the larger piece has uniform distribution on (with mean 3/4), and the smaller piece has uniform distribution on (with mean 1/4).

The probability that the line segments resulting from cutting at two independently and uniformly sampled points on a unit line segment determine a triangle is given by 1/4.

The probability density function and distribution function for the ratio of small to large pieces are given by

			(1)
			(2)

for . The raw moments are therefore

(3)

where and is the digamma function. The first few are therefore

			(4)
			(5)
			(6)
			(7)

(OEIS A115388 and A115389). The central moments are therefore

(8)

where and is a Pochhammer symbol. The first few are therefore

			(9)
			(10)
			(11)

This therefore gives mean, variance, skewness, and kurtosis excess of

			(12)
			(13)
			(14)
			(15)

The mean can be computed directly from

			(16)
			(17)
			(18)

The probability density function and distribution function for the ratio of large to small pieces are given by

			(19)
			(20)

for . Paradoxical though it may be, this distribution has infinite mean and other moments. The reason for this is that a theoretical bone can be cut extremely close to one end, thus giving huge ratio of largest to smallest pieces, whereas there is some limit for a real physical bone. Conditioning the cut to be uniformly distributed on , where , ensures that each piece has length at least . The conditional mean is then given by

(21)