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The random volume of interpenetrating spheres in space

Published online by Cambridge University Press:  14 July 2016

P. A. P. Moran
P. A. P. Moran*
Affiliation:
The Australian National University, Canberra
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Abstract

The distribution of the volume occupied by random spheres in a cube is considered, both when the number of spheres is fixed and when their centres form a Poisson field. The mean and variance are obtained and in the latter case the distribution is proved to converge to normality. The probability of complete coverage is also obtained heuristically.

Keywords

GEOMETRICAL PROBABILITYCOVERAGEINTERPENETRATING SPHERES

Information

Type
Short Communications
Information
Journal of Applied Probability , Volume 10 , Issue 2 , June 1973 , pp. 483 - 490
Copyright
Copyright © Applied Probability Trust 1973 

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References

[1] Bernstein, S. (1926–7) Sur l'extension du théorème limité du calcul des probabilités aux sommes des quantités dépendents. Math. Ann. 97, 1–59.Google Scholar
[2] Melnyk, T. W. and Rowlinson, J. S. (1971) The statistics of the volumes covered by systems of penetrating spheres. Jour. Comput. Physics 7, 385–393.CrossRefGoogle Scholar
[3] Miles, R. E. (1972) The random division of space. Proceedings of Symposium on Statistical and Probabilistic Problems in Metallurgy, Seattle, August 1971 (Ed. W.L. Nicholson). Adv. Appl. Prob. (1972) Special Supplement, 243–266.Google Scholar
[4] Moran, P. A. P. and Fazekas De St Groth, S. N. E. E. (1962) Random circles on a sphere. Biometrika 49, 389–396.Google Scholar
[5] Santaló, L. A. (1948) Sobre la distribucion de pianos en el espacio. Riv. Union. Mat. Argentina 13, 120–124.Google Scholar
[6] Widom, B. and Rowlinson, J. S. (1970) New model for the study of liquid-vapour phase transitions. J. Chem. Phys. 52, 1670–1684.Google Scholar
[7] Takács, L. (1958) On the probability distribution of the measure of the union of random sets placed in a Euclidean space. Ann. Univ. Sci. Budapest Eötvös Sect. Math. 1, 89–95.Google Scholar
[8] Santaló, L. A. (1947) On the first two moments of the measure of a random set. Ann. Math. Statist. 18, 37–49.Google Scholar