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**Zentralblatt MATH**

**Publications of (and about) Paul Erdös**

**Zbl.No: ** 129.31402

**Autor: ** Erdös, Pál; Neveu, J.; Rényi, Alfréd

**Title: ** An elementary inequality between the probabilities of events (In English)

**Source: ** Math. Scand. 13, 90-104 (1963).

**Review: ** The reviewer (Zbl 064.13005) has proved that for any n events A_{1},A_{2},...,A_{n} such that Pr(A_{i}) = \omega_{1} for i = 1,2,...,n and Pr(A_{i} \cap A_{j}) = \omega_{2} for i \ne j we have the inequality \omega_{2} \geq \omega_{1}^{2}-{\omega_{1}(1-\omega_{1}) \over n-1}+{(n\omega_{1}-[n\omega_{1}]) (1-n\omega_{1}+[n \omega_{1}]) \over n(n-1)} (1) with [n\omega_{1}] denoting the integral part of n\omega_{1}, and that this inequality is an equality for some collection of events A_{1}, A_{2},...,A_{n} whatever \omega_{1} and n.

Here the authors consider the closely related more general problem of the determination, for any natural n and a in (0,1), of the constant \epsilon_{n} (\alpha) defined as the least real number \epsilon such that for any collection of events A_{1},A_{2},...,A_{n} subject to the only condition (2) Pr(A_{i} \cap A_{j}) \leq \alpha^{2} for i \ne j we have the inequality **sum** Pr(A_{i}) \leq n\alpha+\epsilon. With \nu denoting the largest integer such that \nu(\nu-1) \leq n(n-1)\alpha^{2} the constant sought for is found to be given by

\epsilon_{n}(\alpha) = ^{1}/_{2} (1-\alpha)+(n \alpha-\nu) ((n-1)\alpha-\nu)/2 \nu. The second term in this formula vanishes if n\alpha or (n-1)\alpha is an integer; otherwise for n ––> oo it is of the order of 1/n. An explicit extremal collection of events A_{1},A_{2},...,A_{n} is constructed in the case of \alpha = ^{1}/_{2} and n \equiv 3 (mod 4) by the use of the method of quadratic residues.

**Reviewer: ** S.Zubrzycki

**Classif.: ** * 60C05 Combinatorial probability

60E05 General theory of probability distributions

**Index Words: ** probability theory

© European Mathematical Society & FIZ Karlsruhe & Springer-Verlag