The Annals of Probability

On the Distribution of the Number of Successes in Independent Trials

Leon Jay Gleser

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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.

Article information

Ann. Probab., Volume 3, Number 1 (1975), 182-188.

First available in Project Euclid: 19 April 2007

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Zentralblatt MATH identifier


Primary: 60C05: Combinatorial probability
Secondary: 60E05: Distributions: general theory 62E15: Exact distribution theory 26A86

Independent Bernoulli trials number of successes inequalities on cumulative distribution function inequalities on expected values majorization Schur conditon


Gleser, Leon Jay. On the Distribution of the Number of Successes in Independent Trials. Ann. Probab. 3 (1975), no. 1, 182--188. doi:10.1214/aop/1176996461.

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