Sharp Inequalities of Bienaym\'e-Chebyshev and Gau\ss Type for Possibly Asymmetric Intervals around the Mean

Abstract

Gau\ss (1823) proved a sharp upper bound on the probability that a random variable falls outside a symmetric interval around zero when its distribution is unimodal with mode at zero. For the class of all distributions with mean at zero, Bienaym\'e (1853) and Chebyshev (1867) independently provided another, simpler sharp upper bound on this probability. For the same class of distributions, Cantelli (1928) obtained a strict upper bound for intervals that are a half line. We extend these results to arbitrary intervals for six classes of distributions, namely the general class of `distributions', the class of `symmetric distributions', of `concave distributions', of `unimodal distributions', of `unimodal distributions with coinciding mode and mean', and of `symmetric unimodal distributions'. For some of the known inequalities, such as the Gau\ss \, inequality, an alternative proof is given.