Hostname: page-component-586b7cd67f-t7czq Total loading time: 0 Render date: 2024-11-24T01:35:59.189Z Has data issue: false hasContentIssue false

Coverage problems and random convex hulls

Published online by Cambridge University Press:  14 July 2016

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.

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.

Type
Research Papers
Copyright
Copyright © Applied Probability Trust 1982 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

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

References

Eddy, W. F. (1980) The distribution of the convex hull of a Gaussian sample. J. Appl. Prob. 17, 686695.CrossRefGoogle Scholar
Efron, B. (1965) The convex hull of a random set of points. Biometrika 52, 331343.CrossRefGoogle Scholar
Mardia, K. V. (1972) Statistics of Directional Data. Academic Press, New York.Google Scholar
Rogers, L. C. G. (1978) The probability that two samples in the plane will have disjoint convex hulls. J. Appl. Prob. 15, 790802.CrossRefGoogle Scholar
Siegel, A. F. and Holst, L. (1982) Covering the circle with random arcs. J. Appl. Prob. 19, 373381.CrossRefGoogle Scholar
Solomon, H. (1978) Geometric Probability. SIAM, Philadelphia.CrossRefGoogle Scholar
Stevens, W. L. (1939) Solution to a geometrical problem in probability. Ann. Eugenics 9, 315320.CrossRefGoogle Scholar
Whitworth, W. A. (1897) DCC Exercises on Choice and Chance. Deighton Bell and Co., Cambridge. (Republished in 1959 by Hafner, New York.) Google Scholar