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BONFERRONI-TYPE INEQUALITIES; CHEBYSHEV-TYPE

INEQUALITIES FOR THE DISTRIBUTIONS ON [0,*n*]

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MASAAKI SIBUYA

*Department of Mathematics, Keio University, Kohoku-ku, Yokohama 223, Japan*
(Received March 18, 1989; revised March 26, 1990)

**Abstract.**
An elementary ``majorant-minorant method'' to
construct the
most stringent Bonferroni-type inequalities is
presented. These are essentially Chebyshev-type inequalities for
discrete probability distributions on the set {0, 1,...., *n*},
where *n* is the number of concerned events, and polynomials with
specific properties on the set lead to the inequalities. All the known
results are proved easily by this method. Further, the inequalities in
terms of all the lower moments are completely solved by the method. As
examples, the most stringent new inequalities of degrees three and
four are obtained. Simpler expressions of Margaritescu's
inequality (1987, *Stud. Cerc. Mat.*, **39**, 246-251),
improving Galambos' inequality, are given.

*Key words and phrases*:
Binary random variable, Galambos'
inequality,
Kwerel's inequality, moment problem.

**Source**
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