Consider a line segment of length 1, and pick a point at random between . 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 points picked at random on a unit line segment determine a triangle is given by 1/4.
The probability and distribution functions for the ratio of small to large pieces are given by
(1)
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(2)
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for . The raw moments are therefore
(3)
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where is the digamma function. The first few are therefore
(4)
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(5)
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(6)
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(7)
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(OEIS A115388 and A115389). The central moments are therefore
(8)
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where is a Pochhammer symbol. The first few are therefore
(9)
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(10)
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(11)
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This therefore gives mean, variance, skewness, and kurtosis excess of
(12)
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(13)
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(14)
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(15)
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The mean can be computed directly from
(16)
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(17)
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(18)
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The probability and distribution functions for the ratio of large to small pieces are given by
(19)
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(20)
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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. Taking to be the smallest possible piece in which is bone cen be cut, the mean is then given by
(21)
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