On the Distribution of the Number of Successes in Independent Trials

Let S be the number of successes in n independent Bernoulli trials, where pⱼ is the probability of success on the jth trial. Let p = (p₁, p₂, ⋯, pₙ), and for any integer c, 0 c n, let H(c p) = P c\. Let p⁽¹⁾ be one possible choice of p for which E(S) = λ. For any n × n doubly stochastic matrix Π, let p⁽²⁾ = p⁽¹⁾Π. Then in the present paper it is shown that H(c p⁽¹⁾) H(c p⁽²⁾) for 0 c λ - 2, and H(c p⁽¹⁾) H(c p⁽²⁾) for λ + 2 c n. These results provide a refinement of inequalities for H(c p) obtained by Hoeffding [3]. Their derivation is achieved by applying consequences of the partial ordering of majorization.