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Empirical Analysis of Probabilistic Bounds

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Logistics, Supply Chain and Financial Predictive Analytics

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Abstract

Empirical analysis is done to the sharp bounds presented in [12] for the probability of union of arbitrary events following monotonic distribution. Given any number of binomial moments, the closed form sharp bounds for the probability of union of events are presented in [12]. In this paper, we analyze the bounds and the probability distribution generated from the optimal basis for different monotonic functions with different monotonicity.

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References

  1. Boole G (1854) Laws of thought. Dover, Newyork

    Google Scholar 

  2. Boros E, Prekopa A (1989) Closed form two-sided bounds for probabilities that at least r and exactly r out of n events occur. Math Oper Res 14:317–342

    Article  Google Scholar 

  3. Boros E, Prekopa A (1989) Probabilistic bounds and algorithms for the maximum satisfiability problem. Ann Oper Res 21:109–126

    Article  Google Scholar 

  4. Boros E, Scozzari A, Tardella F, Veneziani P (2014) Polynomially computable bounds for the probability of the union of events. Math Oper Res 39:1311–1329

    Article  Google Scholar 

  5. Bukszar J, Madi-Nagy G, Szantai T (2012) Computing bounds for the probability of the union of events by different methods. Ann Oper Res 201:63–81

    Article  Google Scholar 

  6. Chung KL, Erdos P (1952) On the application of the Borel-Cantelli lemma. Trans Am Math Soc 72:179–186

    Article  Google Scholar 

  7. Dawson DA, Sankoff D (1967) An inequality for probability. Proc Am Math Soc 18:504–507

    Article  Google Scholar 

  8. Frechet M (1940/43) Les Probabilities Associees a un systeme d’Evenement Compatibles et Dependants, Actualites Scientifique et Industrielles, Nos. 859,942, Paris

    Google Scholar 

  9. Gao L, Prekopa A (2001) Lower and Upper bounds for the probability of at least r and exactly r out of n events that occur, Rutcor Research report

    Google Scholar 

  10. Hunter D (1976) Bounds for the probability of a union. J Appl Probab 13:597–603

    Article  Google Scholar 

  11. Kumaran V, Prekopa A (2005) Bounds on probability of a finite union. In: Mohan SR, Neogy SK (eds) Operations research with economic and industrial applications: emerging trends. Anamaya Publishers, New Delhi, India, pp 77–84

    Google Scholar 

  12. Kumaran V, Swarnalatha R (2017) Bounds for the probability of union of events following monotonic distribution. Discrete Appl Math 223:98–119

    Article  Google Scholar 

  13. Kwerel SM (1975) Most stringent bounds on aggregated probabilities of partially specified dependent probability systems. J Am Stat Assoc 70:472–479

    Google Scholar 

  14. Prekopa A (1988) Boole-Bonferroni inequalities and linear programming. Oper Res 36:145–162

    Article  Google Scholar 

  15. Prekopa A (1995) Stochastic programming. Kluwer Academic Publishers, Netherlands

    Book  Google Scholar 

  16. Prekopa A, Gao L (2005) Bounding the probability of the union of events by aggregation and disaggregation in linear programs. Discrete Appl Math 145:444–454

    Article  Google Scholar 

  17. Prekopa A, Ninh A, Alexe G (2016) On the relationship between the discrete and continuous bounding moment problems and their numerical solutions. Ann Oper Res 238:521–575

    Article  Google Scholar 

  18. Prekopa A, Subasi M, Subasi E (2008) Sharp bounds for the probability of the union of events under unimodal condition. Eur J Pure Appl Math 1:60–81

    Google Scholar 

  19. Sathe YS, Pradhan M, Shah SP (1980) Inequalities for the probability of the occurrence of at least \(m\) out of \(n\) events. J Appl Probab 17:1127–1132

    Article  Google Scholar 

  20. Subasi E, Subasi M, Prekopa A (2009) Discrete moment problem with distributions known to be unimodal. Math Inequalities Appl 1:587–610

    Article  Google Scholar 

  21. Swarnalatha R, Kumaran V (2017) Bounds for the probability of the union of events with unimodality. Ann Oper Res. https://doi.org/10.1007/s10479-017-2629-6

  22. Unuvar M, Ozguven EE, Prekopa A (2015) Optimal capacity design under \(k\)-out-of-\(n\) and consecutive \(k\)-out-of-\(n\) type probabilistic constraints. Ann Oper Res 226:643–657

    Article  Google Scholar 

  23. Veneziani P (2002) New Bonferroni-type inequalities, Rutcor Research report

    Google Scholar 

  24. Yoda K, Prekopa A (2016) Improved bounds on the probability of the union of events some of whose intersections are empty. Oper Res Lett 44(1):39–43

    Article  Google Scholar 

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Acknowledgements

The first author thanks MHRD (Government of India) and National Institute of Technology, Tiruchirappalli, India for financial support.

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Correspondence to R. Swarnalatha .

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Swarnalatha, R., Kumaran, V. (2019). Empirical Analysis of Probabilistic Bounds. In: Deep, K., Jain, M., Salhi, S. (eds) Logistics, Supply Chain and Financial Predictive Analytics. Asset Analytics. Springer, Singapore. https://doi.org/10.1007/978-981-13-0872-7_11

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