Abstract
We consider the problem of finding the sharp bounds for the probability of the union of n events via linear programming. The probability of occurrences is supposed to be unimodal with known mode. Probability bounds are found for the union of n events when the first m \(\left( 2 \le m \le n-1 \right) \) out of n binomial moments are known. Inverse is found to the class of matrices corresponding to the dual feasible bases and it is exploited to find the sharp bounds.
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References
Boole, G. (1854). Laws of thought. New York: Dover.
Boros, E., & Prekopa, A. (1989). Closed form two-sided bounds for probabilities that at least r and exactly r out of n events occur. Mathematics of Operations Research, 14, 317–342.
Boros, E., & Prekopa, A. (1989). Probabilistic bounds and algorithms for the maximum satisfiability problem. Annals of Operations Reaserch, 21, 109–126.
Boros, E., Scozzari, A., Tardella, F., & Veneziani, P. (2014). Polynomially computable bounds for the probability of the union of events. Mathematics of Operations Research, 39, 1311–1329.
Bukszar, J., Madi-Nagy, G., & Szantai, T. (2012). Computing bounds for the probability of the union of events by different methods. Annals of Operations Reaserch, 201, 63–81.
Chung, K. L., & Erdos, P. (1952). On the application of the Borel-Cantelli lemma. Transactions of the American Mathematical Society, 72, 179–186.
Dawson, D. A., & Sankoff, D. (1967). An inequality for probability. Proc. Am. Math. Soc., 18, 504–507.
Frechet, M. (1940/43). Les probabilities associees a un systeme d’Evenement Compatibles et Dependants, Actualites Scientifique et Industrielles, Nos. 859,942, Paris.
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.
Hunter, D. (1976). Bounds for the probability of a union. Journal of Applied Probability, 13, 597–603.
Kumaran, V., & Prekopa, A. (2005). Bounds on Probability of a Finite Union. In S. R. Mohan and S. K. Neogy (Eds.), Operations Research with Economic and Industrial Applications: Emerging trends (pp. 77–84). New Delhi, India: Anamaya Publishers.
Kumaran, V., & Swarnalatha, R. (2017). Bounds for the probability of union of events following monotonic distribution. Discrete Applied Mathematics, 223, 98–119.
Kwerel, S. M. (1975). Most stringent bounds on aggregated probabilities of partially specified dependent probability systems. Journal of the American Statistical Association, 70, 472–479.
Prekopa, A. (1988). Boole-Bonferroni inequalities and linear programming. Operation Research, 36, 145–162.
Prekopa, A. (1995). Stochastic programming. Netherlands: Kluwer.
Prekopa, A. (1999). The use of discrete moment bounds in probabilistic constrained stochastic programming models. Annals of Operations Reaserch, 85, 21–38.
Prekopa, A., & Gao, L. (2005). Bounding the probability of the union of events by aggregation and disaggregation in linear programs. Discrete Applied Mathematics, 145, 444–454.
Prekopa, A., Ninh, A., & Alexe, G. (2016). On the relationship between the discrete and continuous bounding moment problems and their numerical solutions. Annals of Operations Reaserch, 238, 521–575.
Prekopa, A., Subasi, M., & Subasi, E. (2008). Sharp bounds for the probability of the union of events under unimodal condition. European Journal of Pure and Applied Mathematics, 1, 60–81.
Sathe, Y. S., Pradhan, M., & Shah, S. P. (1980). Inequalities for the probability of the occurrence of at least \(m\) out of \(n\) events. J. Appl. Probab., 17, 1127–1132.
Subasi, E., Subasi, M., & Prekopa, A. (2009). Discrete moment problem with distributions known to be unimodal. Mathematical Inequalities and Applications, 1, 587–610.
Subasi, M. M., Subasi, E., Binmahfoudh, A., & Prekopa, A. (2017). New bounds for the probability that at least k-out-of-n events occur with unimodal distributions. Discrete Applied Mathematics, 226, 138–157.
Unuvar, M., Ozguven, E. E., & Prekopa, A. (2015). Optimal capacity design under \(k\)-out-of-\(n\) and consecutive \(k\)-out-of-\(n\) type probabilistic constraints. Annals of Operations Reaserch, 226, 643–657.
Veneziani, P. (2002). New Bonferroni-type inequalities, Rutcor Research report.
Acknowledgements
The first author thanks MHRD (Government of India) and National Institute of Technology, Tiruchirappalli, India for financial support. Both the authors thank Prof. Andras Prekopa, who is although no longer with us, for motivating them to study Discrete Moment Problem by his inspiring work. Also the authors thank the referees for their encouragement and valuable suggestions which made this paper to take a better shape.
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Swarnalatha, R., Kumaran, V. Bounds for the probability of the union of events with unimodality. Ann Oper Res (2017). https://doi.org/10.1007/s10479-017-2629-6
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DOI: https://doi.org/10.1007/s10479-017-2629-6