On the Number of Successes in Independent Trials

Abstract

Let S be the number of successes in n independent Bernoulli trials,where p_j is the probability of success on the jth 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 whichE(S) = λ. For any n \times n doubly stochastic matrix Π,let \mathbf{p}^{(2)} = \mathbf{p}^{(1)}Π. Then in the presentpaper it is shown that H(c \mid \mathbf{p}^{(1)}) \leqq H(c \mid\mathbf{p}^{(2)}) for 0 \leqq c \leqq \lbrackλ- 2\rbrack,and H(c \mid \mathbf{p}^{(1)}) \geqq H(c \mid \mathbf{p}^{(2)})for \lbrackλ+ 2\rbrack \leqq c \leqq n. These results providea refinement of inequalities for H(c \mid \mathbf{p}) obtainedby Hoeffding [3]. Their derivation is achieved by applying consequencesof the partial ordering of majorization.