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On the Classical Bonferroni Inequalities and the Corresponding Galambos Inequalities

A. M. Walker
Journal of Applied Probability
Vol. 18, No. 3 (Sep., 1981), pp. 757-763
DOI: 10.2307/3213333
Stable URL: http://www.jstor.org/stable/3213333
Page Count: 7
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Since scans are not currently available to screen readers, please contact JSTOR User Support for access. We'll provide a PDF copy for your screen reader.
On the Classical Bonferroni Inequalities and the Corresponding Galambos Inequalities
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Abstract

Let (A1, A2,⋯, An) be a set of n events on a probability space. Let S0 = 1, Sr, 1 ≤ r ≤ n be the sum of the probabilities of all $\binom{n}{r}$ intersections of r events, and Mn the number of events in the set which occur. The classical Bonferroni inequalities provide upper and lower bounds for the probabilities P(Mn = m), 0 ≤ m ≤ n, and P(Mn ≥ m), 1 ≤ m ≤ n, equal to partial sums of series of the form σm ncrS r which give the exact probabilities. These inequalities have recently been extended by J. Galambos to give sharper bounds. Here we present straightforward proofs of the Bonferroni inequalities, using indicator functions, and show how they lead naturally to new simple proofs of the Galambos inequalities.

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