Hostname: page-component-84b7d79bbc-g7rbq Total loading time: 0 Render date: 2024-07-29T16:27:09.086Z Has data issue: false hasContentIssue false
  • English
  • Français

Moments of coverage and coverage spaces

Published online by Cambridge University Press:  14 July 2016

Gedalia Ailam
Gedalia Ailam*
Affiliation:
Israel Institute for Biological Research, Ness-Ziona, Israel
Get access Rights & Permissions [Opens in a new window]

Extract

Moments of the measure of the intersection of a given set with the union of random sets may be called moments of coverage. These moments specified for the case of independent random sets in the Euclidean space, were treated by several authors. Kolmogorov (1933) and Robbins (1944), (1945), (1947) derived transformation formulas for the moments of coverage under some specified conditions. Robbins' formula was applied by Robbins (1945), Santaló (1947) and Garwood (1947) to computations of expectations and variances of coverage in the cases of spheres and intervals in 2-dimensional and in n-dimensional Euclidean space. The computations were carried out under the assumption of a simple distribution function for the centers of the covering figures (a homogeneous distribution in most of the cases and a normal one in some of them). Neyman and Bronowski (1945) computed by a different method the variance of coverage for some specific cases in the Euclidean plane.

Type
Short Communications
Information
Journal of Applied Probability , Volume 3 , Issue 2 , December 1966 , pp. 550 - 555
Copyright
Copyright © Applied Probability Trust 

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.)

References

[1] Garwood, F. (1947) The variance of overlap of geometrical figures with reference to a bombing problem. Biometrika 34, 1–17.CrossRefGoogle ScholarPubMed
[2] Halmos, P. R. (1956) Measure Theory. Van Nostrand, New York.Google Scholar
[3] Kolmogorov, A. (1933) Grundbegriffe der Wahrscheinlichkeitsrechnung. Springer Verlag, Berlin.CrossRefGoogle Scholar
[4] Neyman, J. and Bronowski, J. (1945) The variance of the measure of a two-dimensional random set. Ann. Math. Statist. 16, 330–341.Google Scholar
[5] Robbins, H. E. (1944) On the measure of a random set. I. Ann. Math. Statist. 15, 70–74.CrossRefGoogle Scholar
[6] Robbins, H. E. (1945) On the measure of a random set. II. Ann. Math. Statist. 16, 342–347.CrossRefGoogle Scholar
[7] Robbins, H. E. (1947) Acknowledgement of priority. Ann. Math. Statist. 18, 297.Google Scholar
[8] Santalò, L. A. (1947) On the first two moments of the measure of a random set. Ann. Math. Statist. 18, 37–49.Google Scholar