Probability inequalities for continuous unimodal random variables with finite support

Abstract

A paramecer-free Bernstein-type upper bound is derived for the probability that the sum S of n i.i.d, unimodal random variables with finite support, X1 ,X2,…,Xn, exceeds its mean E(S) by the positive value nt. The bound for P{S - nμ ≥ nt} depends on the range of the summands, the sample size n, the positive number t, and the type of unimodality assumed for Xi. A two-sided Gauss-type probability inequality for sums of strongly unimodal random variables is also given. The new bounds are contrasted to Hoeffding's inequality for bounded random variables and to the Bienayme-Chebyshev inequality. Finally, the new inequalities are applied to a classic probability inequality example first published by Savage (1961).