Open Access
Translator Disclaimer
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


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.


Download Citation

Leon Jay Gleser. "On the Distribution of the Number of Successes in Independent Trials." Ann. Probab. 3 (1) 182 - 188, February, 1975.


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

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

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


Vol.3 • No. 1 • February, 1975
Back to Top