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Gaussian Triangle Picking

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

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

P_n=(3Gamma(n))/([Gamma(1/2n)]^2)int_0^(1/3)(x^((n-2)/2))/((1+x)^n)dx
(1)
=(3Gamma(n))/([Gamma(1/2n)]^22^(n-1))int_0^(pi/3)sin^(n-1)thetadtheta
(2)
=(6Gamma(n)_2F_1(1/2n,n;1+1/2n;-1/3))/(3^(n/2)n[Gamma(1/2n)]^2)
(3)
=(3B(-1/3;1/2n,1-n)Gamma(n))/([Gamma(1/2n)]^2)
(4)
=(3B(3/4;1/2n,1/2)Gamma(n))/([Gamma(1/2n)]^2),
(5)

where Gamma(n) is the gamma function, _2F_1(a,b;c;x) is the hypergeometric function, and B(z;a,b) is an incomplete beta function.

For even n=2k,

P_(2k)=3sum_(j=k)^(2k-1)(2k-1; j)(1/4)^j(3/4)^(2k-1-j)
(6)

(Eisenberg and Sullivan 1996).

The first few cases are explicitly

P_2=3/4=0.75
(7)
P_3=1-(3sqrt(3))/(4pi)=0.586503...
(8)
P_4=(15)/(32)=0.46875
(9)
P_5=1-(9sqrt(3))/(8pi)=0.37975499...
(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 1-rsqrt(3)/pi, where r=3/4, 9/8, 27/20, 837/560, ... (OEIS A102558 and A102559).

See also

Disk Triangle Picking

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References

Eisenberg, B. and Sullivan, R. "Random Triangles n 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."

Referenced on Wolfram|Alpha

Gaussian Triangle Picking

Cite this as:

Weisstein, Eric W. "Gaussian Triangle Picking." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/GaussianTrianglePicking.html

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