On the monotonicity of tail probabilities

Abstract

. Let S and X be independent random variables, assuming values in the set of non-negative integers, and suppose further that both E ( S ) and E ( X ) are integers satisfying E ( S ) › E ( X ) . We establish a sufficient condition for the tail probability P ( S › E ( S )) to be larger than the tail P ( S + X › E ( S + X )) , when the mean of S is equal to the mode.