On the monotonicity of tail probabilities
Let S and X be independent random
variables, assuming values in the set of non-negative integers, and suppose further
that both expectations ES and EX are integers satisfying ES>EX.
We establish a sufficient condition for the tail probability P(S> ES)
to be larger than the tail probability P(S+X> E(S+X)), when the mean of S is equal to the mode.
2010 AMS Mathematics Subject Classification: Primary 60G50; Secondary 60E15.
Keywords and phrases: tail comparisons, sums of independent random variables,
(negative) binomial distribution, Poisson distribution, Simmons' inequality.