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Random coverage of a circle with applications to a shadowing problem

Published online by Cambridge University Press:  14 July 2016

M. Yadin  and
S. Zacks
M. Yadin*
Affiliation:
Technion — Israel Institute of Technology
S. Zacks*
Affiliation:
State University of New York at Binghampton
*
∗Postal address: Faculty of Industrial and Management Engineering, Technion — Israel Institute of Technology, Haifa, Israel.
∗∗Postal address: Department of Mathematical Sciences, State University of New York, Binghampton, NY 13901, U.S.A.
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Abstract

The coverage problem on the circle is considered from the shadowing process point of view. A random number of shadow arcs are distributed on a circle. The length of each arc is a random variable which depends on the random diameter of a shadowing disk and its random location. Formulae are derived for the numerical determination of the moments of the measure of vacancy of arcs on the circle, for a special example. An approximation to the distribution of the measure of vacancy is also provided.

Keywords

RANDOM ARCSVACANCY PROBABILITIESCOVERAGEPOISSON FIELDSSHADOWING PROCESSES
Type
Research Papers
Information
Journal of Applied Probability , Volume 19 , Issue 3 , September 1982 , pp. 562 - 577
Copyright
Copyright © Applied Probability Trust 1982 

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Footnotes

Partially supported by ONR Contract N00014-80-C-0325 (NR 042–276) at Virginia Polytechnic Institute and State University; and Contract ONR N00014-81-K-0407 (NR 042–276) at State University of New York at Binghamton.

References

[1] Chernoff, H. and Daly, J. F. (1957) The distribution of shadows. J. Math. Mech. 6, 567–584.Google Scholar
[2] Domb, C. (1947) The problem of random intervals on a line. Proc. Camb. Phil. Soc. 43, 329–341.CrossRefGoogle Scholar
[3] Johnson, N. L. and Kotz, S. (1970) Distributions in Statistics: Continuous Univariate Distributions. Houghton Mifflin, New York.Google Scholar
[4] Robbins, H. E. (1944) On the measure of random sets. Ann. Math. Statist. 15, 70–74.CrossRefGoogle Scholar
[5] Robbins, H. E. (1945) On the measure of random sets, II. Ann. Math. Statist. 16, 342–347.CrossRefGoogle Scholar
[6] Siegel, A. F. (1978) Random space filling and moments of coverage in geometric probability, J. Appl. Prob. 15, 340–355.CrossRefGoogle Scholar
[7] Siegel, A. F. (1978) Random arcs on the circle. J. Appl. Prob. 15, 774–789.CrossRefGoogle Scholar
[8] Solomon, H. (1978) Geometric Probability. SIAM, Philadelphia, PA.CrossRefGoogle Scholar
[9] 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, Sect. Math. 1, 89–95.Google Scholar