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Bounds on the probability of the union and intersection of m events

Published online by Cambridge University Press:  01 July 2016

Seymour M. Kwerel
Seymour M. Kwerel*
Affiliation:
Baruch College, City University of New York
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Abstract

For dependent probability systems of m events partially specified only by the quantities S1, the sum of the probabilities of the m individual events; S2, the sum of the probabilities of each of the (m) pairs of events and S3 the sum of the probabilities of each of the (m 3) combinations of three events; this paper develops the most stringent upper and lower bounds on P1, the probability of the union of the m events; and on P[m], the probability of the simultaneous occurrence of the m events.

Keywords

PARTIALLY SPECIFIED DEPENDENT SYSTEMSMOST STRINGENT BOUNDS
Type
Research Article
Information
Advances in Applied Probability , Volume 7 , Issue 2 , June 1975 , pp. 431 - 448
Copyright
Copyright © Applied Probability Trust 1975 

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References

[1] Chung, K. L. and Erdös, P. (1952) On the application of the Borel-Cantelli lemma. Trans. Amer. Math. Soc. 72, 179–186.Google Scholar
[2] Dawson, D. A. and Sankoff, D. (1967) An inequality for probabilities. Proc. Amer. Math. Soc. 18, 504–507.Google Scholar
[3] Feller, W. (1957) An Introduction to Probability Theory and Its Applications, Volume I. Second edition. Wiley, New York.Google Scholar
[4] Fréchet, M. (1940) Les Probabilités Associées à un Système D'Evénements Compatibles et Dépendants, Premiere Partie. Hermann & Cie., Paris.Google Scholar
[5] Galambos, J. (1973) A general Poisson limit theorem of probability theory. Duke Math. J. 40, 581–586.CrossRefGoogle Scholar
[6] Hadley, G. (1962) Linear Programming. Addison-Wesley, Palo Alto, California.Google Scholar
[7] Kwerel, S. M. (1968) Information retrieval for media planning. Management Sci. 15, 137–160.CrossRefGoogle Scholar
[8] Kwerel, S. M. (1974) Most stringent bounds on aggregated probabilities of partially specified dependent probability systems. J. Amer. Statist. Assoc. 70, To appear.CrossRefGoogle Scholar
[9] Sobel, M. and Uppuluri, V. R. R. (1972) On Bonferroni-type inequalities of the same degree for the probability of unions and intersections. Ann. Math. Statist. 43, 1549–1558.Google Scholar