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On Bounds for Probabilities of Combinations of Events, the Jordan Formula, and the Bonferroni Inequalities

  • A. N. FrolovEmail author
Mathematics

Abstract

This paper presents a method for deriving optimal lower and upper bounds for probabilities and conditional probabilities (given a σ-field) for various combinations of events. The optimality is understood as the possibility that inequalities become equalities for some sets of events. New generalizations of the Jordan formula and the Bonferroni inequalities are obtained. The corresponding conditional versions of these results are also considered.

Keywords

Bonferroni inequalities Jordan formula probabilities of combinations of events probabilities of occurrence of several events 

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Notes

Acknowledgments

The author is grateful to anonymous reviewers, whose comments favored the improvement of the text of this paper.

References

  1. 1.
    A. N. Frolov, “On inequalities for probabilities wherein at least r from n events occur,” Vestn. St. Petersburg Univ.: Math. 50, 287–296 (2017).  https://doi.org/10.3103/S1063454117030074 MathSciNetCrossRefGoogle Scholar
  2. 2.
    A. N. Frolov, “On inequalities for probabilities of joint occurrence of several events,” Vestn. St. Petersburg Univ.: Math. 51, 286–295 (2018).  https://doi.org/10.3103/S1063454118030032 MathSciNetCrossRefGoogle Scholar
  3. 3.
    A. N. Frolov, “Bounds for probabilities of unions of events and the Borel-Cantelli lemma,” Stat. Probab. Lett. 82, 2189–2197 (2012).MathSciNetCrossRefGoogle Scholar
  4. 4.
    A. N. Frolov, “On lower and upper bounds for probabilities of unions and the Borel-Cantelli lemma,” Stud. Sci. Math. Hung. 52, 102–128 (2015).MathSciNetzbMATHGoogle Scholar
  5. 5.
    W. Feller, An Introduction to Probability Theory and Its Applications (Wiley, New York, 1957; Mir, Moscow, 1967), Vol. 1, in Ser.: Wiley Series in Probability and Mathematical Statistics.zbMATHGoogle Scholar
  6. 6.
    J. Galambos, “Bonferroni inequalities,” Ann. Probab. 5, 577–581 (1977).MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    J. Galambos, I. Simonelli, Bonferroni-Type Inequalities with Applications (Springer-Verlag, New York, 1996).zbMATHGoogle Scholar
  8. 8.
    F. M. Hoppe and E. Seneta, “A Bonferroni-type identity and permutation bounds,” Int. Stat. Rew. 58, 253–261 (1990).CrossRefzbMATHGoogle Scholar
  9. 9.
    S. Kounias and J. Marin, “Best linear Bonferroni bounds,” SIAM J. Appl. Math. 30, 307–323 (1976).MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    S. Kounias and K. Sotirakoglou, “Upper and lower bounds for the probability that r events occur,” J. Math. Programming. Oper. Res. 27, 63–78 (1993).MathSciNetzbMATHGoogle Scholar
  11. 11.
    E. Margaritescu, “Improved Bonferroni inequalities,” Rev. Roum. Math. Pures Appl. 33, 509–515 (1988).MathSciNetzbMATHGoogle Scholar
  12. 12.
    T. F. Móri and G. J. Székely, “A note on the background of several Bonferroni-Galambos-type inequalities,” J. Appl. Probab. 22, 836–843 (1985).MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    A. Prékopa, “Boole-Bonferroni inequalities and linear programming,” Oper. Res. 36, 145–162 (1988).MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Recsei E and E. Seneta, “Bonferroni-type inequalities,” Adv. Appl. Probab. 19, 508–511 (1987).MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    M. Sobel and V. R. R. Uppuluri, “On Bonferroni-type inequalities of the same degree for probabilities of unions and intersections,” Ann. Math. Stat. 43, 1549–1558 (1972).CrossRefzbMATHGoogle Scholar
  16. 16.
    A. M. Walker, “On the classical Bonferroni inequalities and the corresponding Galambos inequalities,” J. Appl. Probab. 18, 757–763 (1981).MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    K. L. Chung and P. Erdős, “On the application of the Borel-Cantelli lemma,” Trans. Am. Math. Soc. 72, 179–186 (1952).MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    D. A. Dawson and D. Sankoff, “An inequality for probabilities,” Proc. Am. Math. Soc. 18, 504–507 (1967).MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    E. G. Kounias, “Bounds for the probability of a union, with applications,” Ann. Math. Stat. 39, 2154–2158 (1968).MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    S. M. Kwerel, “Bounds on the probability of the union and intersection of m events,” Adv. Appl. Probab. 7, 431–448 (1975).MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    E. Boros and A. Prékopa, “Closed form two-sided bounds for probabilities that at least r and exactly r out of n events occurs,” Math. Oper. Res. 14, 317–342 (1989).MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    D. de Caen, “A lower bound on the probability of a union,” Discrete Math. 169, 217–220 (1997).MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    H. Kuai, F. Alajaji, and G. Takahara, “A lower bound on the probability of a finite union of events,” Discrete Math. 215, 147–158 (2000).MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    A. N. Frolov, “On inequalities for probabilities of unions of events and the Borel-Cantelli lemma,” Vestn. St. Petersburg Univ.: Math. 47, 68–75 (2014).  https://doi.org/10.3103/S1063454114020034 MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    A. N. Frolov, “On estimation of probabilities of unions of events with applications to the Borel-Cantelli lemma,” Vestn. St. Petersburg Univ.: Math. 48, 175–180 (2015).  https://doi.org/10.3103/S1063454115030036 MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    A. N. Frolov, “On inequalities for conditional probabilities of unions of events and the conditional Borel-Cantelli lemma,” Vestn. St. Petersburg Univ.: Math. 49, 379–388 (2016).  https://doi.org/10.3103/S1063454116040063 MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    A. N. Frolov, “On inequalities for values of first jumps of distribution functions and Hölder’s inequality,” Stat. Probab. Lett. 126, 150–156 (2017).CrossRefzbMATHGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2019

Authors and Affiliations

  1. 1.St. Petersburg State UniversitySt. PetersburgRussia

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