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Size distributions in random triangles

Part of: General convexity Geometric probability and stochastic geometry Global differential geometry

Published online by Cambridge University Press:  30 March 2016

D. J. Daley ,
Sven Ebert  and
R. J. Swift
D. J. Daley
Affiliation:
Department of Mathematics and Statistics, The University of Melbourne, Parkville, VIC 3052, Australia. Email address: dndaley@gmail.com.
Sven Ebert
Affiliation:
Institut für Stochastik, Karlsruhe Institut fur Technologie, 76128 Karlsruhe, Germany. Email address: ebert@stoch.uni-karlsruhe.de.
R. J. Swift
Affiliation:
Department of Mathematics, California State Polytechnic University, Pomona, CA 91768, USA. Email address: rjswift@csupomona.edu.
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Abstract

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The random triangles discussed in this paper are defined by having the directions of their sides independent and uniformly distributed on (0, π). To fix the scale, one side chosen arbitrarily is assigned unit length; let a and b denote the lengths of the other sides. We find the density functions of a / b, max{a, b}, min{a, b}, and of the area of the triangle, the first three explicitly and the last as an elliptic integral. The first two density functions, with supports in (0, ∞) and (½, ∞), respectively, are unusual in having an infinite spike at 1 which is interior to their ranges (the triangle is then isosceles).

Keywords

Random directions

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry
Secondary: 52A22: Random convex sets and integral geometry 53C65: Integral geometry
Type
Part 7. Stochastic geometry
Information
Journal of Applied Probability , Volume 51 , Issue A: Celebrating 50 Years of The Applied Probability Trust , December 2014 , pp. 283 - 295
Copyright
Copyright © Applied Probability Trust 2014 

References

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Kendall, M. G., and Moran, P. A. P. (1963). Geometric Probability. Hafner Publishing, New York.Google Scholar
Mathai, A. M. (1999). An Introduction to Geometrical Probability: Distributional Aspects with Applications. Gordon and Breach, Amsterdam.Google Scholar