Journal of Applied Probability

Size distributions in random triangles

D. J. Daley, Sven Ebert, and R. J. Swift

Abstract

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).

Article information

Source
J. Appl. Probab., Volume 51A (2014), 283-295.

Dates
First available in Project Euclid: 2 December 2014

Permanent link to this document
https://projecteuclid.org/euclid.jap/1417528481

Digital Object Identifier
doi:10.1239/jap/1417528481

Mathematical Reviews number (MathSciNet)
MR3317364

Zentralblatt MATH identifier
1314.60043

Subjects
Primary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65]
Secondary: 52A22: Random convex sets and integral geometry [See also 53C65, 60D05] 53C65: Integral geometry [See also 52A22, 60D05]; differential forms, currents, etc. [See mainly 58Axx]

Keywords
Random directions

Citation

Daley, D. J.; Ebert, Sven; Swift, R. J. Size distributions in random triangles. J. Appl. Probab. 51A (2014), 283--295. doi:10.1239/jap/1417528481. https://projecteuclid.org/euclid.jap/1417528481


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References

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