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Coverage problems and random convex hulls

Published online by Cambridge University Press:  14 July 2016

Nicholas P. Jewell  and
Joseph P. Romano
Nicholas P. Jewell*
Affiliation:
Princeton University
Joseph P. Romano*
Affiliation:
Princeton University
*
∗ Supported in part by a grant from the National Science Foundation.
∗∗ Part of this work is contained in this author's Junior Paper in the Department of Statistics, Princeton University.
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Abstract

Consider the placement of a finite number of arcs on the circle of circumference 2π where the midpoint and length of each arc follows an arbitrary bivariate distribution. In the case where each arc has lengthπ, the probability that the circle is completely covered is equal to the probability that the convex hull of a finite random sample of points, chosen according to a certain bivariate distribution in the plane contains the origin. In general, we show that evaluating the probability that the random convex hull contains a fixed disc is equivalent to solving the general coverage problem where the midpoint and length of each arc follows an arbitrary bivariate distribution. Exact formulae for the above probabilities are obtained and some examples are considered.

Keywords

GEOMETRICAL PROBABILITYCOVERAGE PROBABILITYCONVEX HULL

Information

Type
Research Papers
Information
Journal of Applied Probability , Volume 19 , Issue 3 , September 1982 , pp. 546 - 561
Copyright
Copyright © Applied Probability Trust 1982 

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Footnotes

Postal address for both authors: Department of Statistics, Princeton University, Fine Hall, P.O. Box 37, Princeton, NJ 08544, U.S.A.

References

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