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Distance distributions associated with poisson processes of geometric figures

Published online by Cambridge University Press:  14 July 2016

Mark Berman
Mark Berman*
Affiliation:
C. S. I. R. O. Division of Mathematics and Statistics, Newtown, New South Wales
*
*Now at Imperial College London.
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Abstract

Consider a process of identically-shaped (but not necessarily equal-sized) figures (e.g. points, clusters of points, lines, spheres) embedded at random in n-dimensional space. A simple technique is derived for finding the distribution of the distance from a fixed point, chosen independently of the process of figures, to the k th nearest figure. The technique also shows that the distribution is independent of the distribution of the orientations of the figures. It is noted that the distribution obtained above (for equal-sized figures) is identical to the distribution of the distance from a fixed figure to the k th nearest of a random process of points.

Keywords

POISSON PROCESSESNEAREST NEIGHBOURDISTANCE METHODSGEOMETRICAL PROBABILITY
Type
Short Communications
Information
Journal of Applied Probability , Volume 14 , Issue 1 , March 1977 , pp. 195 - 199
Copyright
Copyright © Applied Probability Trust 1977 

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References

[1] Coleman, R. (1974) The distance from a given point to the nearest end of one member of a random process of linear segments. In Stochastic Geometry, ed. Harding, E. F. and Kendall, D. G., Wiley, New York, 192–201.Google Scholar
[2] Eberhardt, L. L. (1967) Some developments in distance sampling. Biometrics 23, 207–216.CrossRefGoogle ScholarPubMed
[3] Giger, H. and Hadwiger, H. (1968) Über Treffzahlwahrscheinlichkeiten im Eikörperfeld. Z. Wahrscheinlichkeitsth. 10, 329–334.CrossRefGoogle Scholar
[4] Miles, R. E. (1974) A synopsis of ‘Poisson flats in Euclidean spaces’. In Stochastic Geometry, ed. Harding, E. F. and Kendall, D. G., Wiley, New York, 202–227.Google Scholar
[5] Ogston, A. G. (1958) The spaces in a uniform random suspension of fibres. Trans. Faraday. Soc. 54, 1754–1757.CrossRefGoogle Scholar
[6] Parker, P. and Cowan, R. (1976) Some properties of line segment processes. J. Appl. Prob. 13, 96–107.CrossRefGoogle Scholar
[7] Thompson, H. R. (1956) Distribution of distance to n th neighbour in a population of randomly distributed individuals. Ecology 37, 391–394.CrossRefGoogle Scholar