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On upper and lower bounds for probabilities of combinations of events

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  • Frolov, Andrei N.

Abstract

We derive new upper and lower bounds for probabilities that r or at least r out of n events occur. These bounds are optimal since they can turn to equalities. We describe a method of constructing of such bounds. It can be applied in case of measurable spaces and measures with sign as well. We also obtain bounds for conditional probabilities of combinations of events given σ-field. Averaging of both sides of inequalities for conditional probabilities can yield better bounds for unconditional probabilities.

Suggested Citation

  • Frolov, Andrei N., 2021. "On upper and lower bounds for probabilities of combinations of events," Statistics & Probability Letters, Elsevier, vol. 173(C).
    • Handle: RePEc:eee:stapro:v:173:y:2021:i:c:s0167715221000353
    DOI: 10.1016/j.spl.2021.109073
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    References listed on IDEAS

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    1. Masaaki Sibuya, 1991. "Bonferroni-type inequalities; Chebyshev-type inequalities for the distributions on [0, n]," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 43(2), pages 261-285, June.
    2. Endre Boros & András Prékopa, 1989. "Closed Form Two-Sided Bounds for Probabilities that At Least r and Exactly r Out of n Events Occur," Mathematics of Operations Research, INFORMS, vol. 14(2), pages 317-342, May.
    3. Frolov, Andrei N., 2017. "On inequalities for values of first jumps of distribution functions and Hölder’s inequality," Statistics & Probability Letters, Elsevier, vol. 126(C), pages 150-156.
    4. Frolov, Andrei N., 2012. "Bounds for probabilities of unions of events and the Borel–Cantelli lemma," Statistics & Probability Letters, Elsevier, vol. 82(12), pages 2189-2197.
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