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Random triangles in convex regions

Published online by Cambridge University Press:  14 July 2016

Norbert Henze
Norbert Henze*
Affiliation:
University of Hannover
*
∗ Postal address: Institut für Mathematische Stochastik, Universität Hannover, Welfengarten 1, D-3000 Hannover 1, Federal Republic of Germany.
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Abstract

Consider a random triangle formed by vertices which are independent and distributed uniformly inside a given bounded convex region K. Then a well-known result of W. Blaschke states that, among all regions of fixed area, the area AK of this random triangle is stochastically bounded by Acircle (Ac) and A triangle (At). The distribution of the latter was obtained by Alagar (1977). We use the classical Crofton technique to derive an integral representation for the distribution function Fc of Ac from which Fc is calculated using numerical quadrature. The maximum deviation between Fc and the distribution function of At is less than 0.04. Additionally, we obtain an explicit form for the distribution of Aparallelogram which may be represented as a mixture of functions involving Dirichlet-distributed random vectors.

Keywords

RANDOM TRIANGLEGEOMETRIC PROBABILITY
Type
Research Papers
Information
Journal of Applied Probability , Volume 20 , Issue 1 , March 1983 , pp. 111 - 125
Copyright
Copyright © Applied Probability Trust 1983 

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