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February, 1975 On the Distribution of the Number of Successes in Independent Trials
Leon Jay Gleser
Ann. Probab. 3(1): 182-188 (February, 1975). DOI: 10.1214/aop/1176996461

Abstract

Let $S$ be the number of successes in $n$ independent Bernoulli trials, where $p_j$ is the probability of success on the $j$th trial. Let $\mathbf{p} = (p_1, p_2, \cdots, p_n)$, and for any integer $c, 0 \leqq c \leqq n$, let $H(c \mid \mathbf{p}) = P\{S \leqq c\}$. Let $\mathbf{p}^{(1)}$ be one possible choice of $\mathbf{p}$ for which $E(S) = \lambda$. For any $n \times n$ doubly stochastic matrix $\Pi$, let $\mathbf{p}^{(2)} = \mathbf{p}^{(1)}\Pi$. Then in the present paper it is shown that $H(c \mid \mathbf{p}^{(1)}) \leqq H(c \mid \mathbf{p}^{(2)})$ for $0 \leqq c \leqq \lbrack\lambda - 2\rbrack$, and $H(c \mid \mathbf{p}^{(1)}) \geqq H(c \mid \mathbf{p}^{(2)})$ for $\lbrack\lambda + 2\rbrack \leqq c \leqq n$. These results provide a refinement of inequalities for $H(c \mid \mathbf{p})$ obtained by Hoeffding [3]. Their derivation is achieved by applying consequences of the partial ordering of majorization.

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Leon Jay Gleser. "On the Distribution of the Number of Successes in Independent Trials." Ann. Probab. 3 (1) 182 - 188, February, 1975. https://doi.org/10.1214/aop/1176996461

Information

Published: February, 1975
First available in Project Euclid: 19 April 2007

zbMATH: 0301.60010
MathSciNet: MR365651
Digital Object Identifier: 10.1214/aop/1176996461

Subjects:
Primary: 60C05
Secondary: 26A86 , 60E05 , 62E15

Keywords: independent Bernoulli trials , inequalities on cumulative distribution function , inequalities on expected values , majorization , number of successes , Schur conditon

Rights: Copyright © 1975 Institute of Mathematical Statistics

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Vol.3 • No. 1 • February, 1975
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