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The distance between two random points in plane regions

Published online by Cambridge University Press:  01 July 2016

T. K. Sheng
T. K. Sheng*
Affiliation:
The University of Newcastle
*
∗Postal address: Faculty of Mathematics, The University of Newcastle, NSW 2308, Australia.
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Abstract

Let T be a triangle. P be a parallelogram, E be an ellipse, A, B be concentric circles, C, D be concentric dartboard regions, R, S be rectangles of the same orientation, U, V be two finite unions and/or differences of convex regions in the Euclidean plane. Given a function f on [0,∞), let E[/(r), U, V] denote the mean value of f(|u–v|), where |u–v| is the distance between u∊U and v∊V. Using Borel’s overlap technique, a specific distance weight function and a specific equivalence relation, we obtain formulae expressing E[f(r), U, V] in terms of triple integrals, expressing E(rn, U, V), E[f(r), A, V] and E[f(r), R, V] in terms of double integrals, expressing E[f(r), A, B], E[f(r), R, S], E[f(r), T, T], E[f(r), P, P], E(rn, C, D) and E(rn, R, V) in terms of single integrals, and expressing E(rn, R, S), E(rn, P, P), E(rn, T, T), E(rn, E, E) in terms of elementary functions, where n is an integer ≧−1. Many other related results are also given.

Keywords

GEOMETRICAL PROBABILITYINTEGRAL GEOMETRYMOMENTS OF DISTANCEDISTANCE WEIGHT FUNCTIONEQUIVALENT PAIRSCUSHIONSWEDGESCIRCLESDARTBOARD REGIONSELLIPSESPARALLELOGRAMSPOLYGONSRECTANGLESTRIANGLES
Type
Research Article
Information
Advances in Applied Probability , Volume 17 , Issue 4 , December 1985 , pp. 748 - 773
Copyright
Copyright © Applied Probability Trust 1985 

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