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Gaussian Triangle Picking

المؤلف:  Eisenberg, B. and Sullivan, R.

المصدر:  "Random Triangles n Dimensions." Amer. Math. Monthly 103

الجزء والصفحة:  ...

8-2-2020

1144

Gaussian Triangle Picking

 

Finch (2010) gives an overview of known results for random Gaussian triangles.

Let the vertices of a triangle in  dimensions be normal (normal) variates. The probability that a Gaussian triangle in  dimensions is obtuse is

			(1)
			(2)
			(3)
			(4)
			(5)

where  is the gamma function,  is the hypergeometric function, and  is an incomplete beta function.

For even ,

(6)

(Eisenberg and Sullivan 1996).

The first few cases are explicitly

			(7)
			(8)
			(9)
			(10)

(OEIS A102519 and A102520). The even cases are therefore 3/4, 15/32, 159/512, 867/4096, ... (OEIS A102556 and A102557) and the odd cases are , where , 9/8, 27/20, 837/560, ... (OEIS A102558 and A102559).

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REFERENCES:

Eisenberg, B. and Sullivan, R. "Random Triangles  Dimensions." Amer. Math. Monthly 103, 308-318, 1996.

Finch, S. "Random Triangles." http://algo.inria.fr/csolve/rtg.pdf. Jan. 21, 2010.

Sloane, N. J. A. Sequences A102519, A102520, A102556, A102557, A102558, and A102559 in "The On-Line Encyclopedia of Integer Sequences."