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Approximations to densities in geometric probability

Published online by Cambridge University Press:  14 July 2016

H. Solomon  and
M. A. Stephens
H. Solomon*
Affiliation:
Stanford University
M. A. Stephens*
Affiliation:
Simon Fraser University
*
∗Postal address: Department of Statistics, Stanford University, Stanford, CA 94305, U.S.A.
∗∗Postal address: Department of Mathematics, Simon Fraser University, Burnaby, B.C., Canada V5A 1S6.
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Abstract

Many random variables arising in problems of geometric probability have intractable densities, and it is very difficult to find probabilities or percentage points based on these densities. A simple approximation, a generalization of the chi-square distribution, is suggested, to approximate such densities; the approximation uses the first three moments. These may be theoretically derived, or may be obtained from Monte Carlo sampling.

The approximation is illustrated on random variables (the area, the perimeter, and the number of sides) associated with random polygons arising from two processes in the plane. Where it can be checked theoretically, the approximation gives good results. It is compared also with Pearson curve fits to the densities.

Keywords

GEOMETRIC PROBABILITYPOISSON PROCESS IN THE PLANERANDOM POLYGONSVORONOI POLYGONS
Type
Research Papers
Information
Journal of Applied Probability , Volume 17 , Issue 1 , March 1980 , pp. 145 - 153
Copyright
Copyright © Applied Probability Trust 1980 

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Footnotes

Research supported by the National Research Council of Canada, by the U.S. Office of Naval Research, Contract No. N00014–76–C–0475, and by the U.S. Army Research Office, Grant DAAG 29–77–G–0031.

References

Baddeley, A. (1977) A fourth note on recent research in geometrical probability. Adv. Appl. Prob. 9, 824–860.Google Scholar
Crain, I. K. (1972) Monte Carlo simulation of the random Voronoi polygons: preliminary results. Search, 3, 220: Australian and New Zealand A.A.S. Google Scholar
Crain, I. K. and Miles, R. E. (1976) Monte Carlo estimates of the distributions of the random polygons determined by random lines in the plane. J. Statist. Comput. Simulation 4, 293–325.Google Scholar
Matheron, G. (1975) Random Sets and Integral Geometry. Wiley, New York.Google Scholar
Miles, R. E. (1973) The various aggregates of random polygons determined by random lines in a plane. Adv. Math. 10, 256–290.Google Scholar
Pearson, E. S. and Hartley, H. O. (1972) Biometrika Tables for Statisticians, Vol. 2. Cambridge University Press.Google Scholar
Santaló, L. A. (1976) Integral Geometry and Geometric Probability. Encyclopaedia of Mathematics, Vol. 1. Addison-Wesley, Reading, Mass. Google Scholar
Solomon, H. (1978) Geometric Probability, Monograph 28, Regional Conference Series in Applied Mathematics, S.I.A.M., Philadelphia.Google Scholar
Solomon, H. and Stephens, M. A. (1978) Approximations to density functions using Pearson curves, J. Amer. Statist. Assoc. 73, 150–160.Google Scholar