An upper bound for the probability of a union
- 1 September 1976
- journal article
- Published by Cambridge University Press (CUP) in Journal of Applied Probability
- Vol. 13 (3) , 597-603
- https://doi.org/10.2307/3212481
Abstract
The problem of bounding P(∪ Ai) given P(Ai) and P(AiAj) for i ≠ j = 1, …, k goes back to Boole (1854) and Bonferroni (1936). In this paper a new family of upper bounds is derived using results in graph theory. This family contains the bound of Kounias (1968), and the smallest upper bound in the family for a given application is easily derivable via the minimal spanning tree algorithm of Kruskal (1956). The properties of the algorithm and of the multivariate normal and t distributions are shown to provide considerable simplifications when approximating tail probabilities of maxima from these distributions.Keywords
This publication has 7 references indexed in Scilit:
- A theorem on treesPublished by Cambridge University Press (CUP) ,2009
- Best Linear Bonferroni BoundsSIAM Journal on Applied Mathematics, 1976
- Bounds for the Probability of a Union, with ApplicationsThe Annals of Mathematical Statistics, 1968
- Best Possible Inequalities for the Probability of a Logical Function of EventsThe American Mathematical Monthly, 1965
- The One-Sided Barrier Problem for Gaussian NoiseBell System Technical Journal, 1962
- On the shortest spanning subtree of a graph and the traveling salesman problemProceedings of the American Mathematical Society, 1956
- A BIVARIATE GENERALIZATION OF STUDENT'S t-DISTRIBUTION, WITH TABLES FOR CERTAIN SPECIAL CASESBiometrika, 1954