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Bonferroni-type inequalities
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- Journal:
- Advances in Applied Probability / Volume 19 / Issue 2 / June 1987
- Published online by Cambridge University Press:
- 01 July 2016, pp. 508-511
- Print publication:
- June 1987
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On the classical Bonferroni inequalities and the corresponding Galambos inequalities
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- Journal:
- Journal of Applied Probability / Volume 18 / Issue 3 / September 1981
- Published online by Cambridge University Press:
- 14 July 2016, pp. 757-763
- Print publication:
- September 1981
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Let (A1A2, · ··, An) be a set of n events on a probability space. Let
be the sum of the probabilities of all 
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), 
and 
equal to partial sums of series of the form 
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.
be the sum of the probabilities of all 
intersections of r events, and 
and 
equal to partial sums of series of the form 
which give the exact probabilities. These inequalities have recently been extended by J. Galambos to give sharper bounds.