## Journal of Applied Probability

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

### 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