Sharp Inequalities of Bienaymé-Chebyshev and GaußType 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 ofsymmetric distributions', of concave distributions', ofunimodal distributions', of unimodal distributions with coinciding mode and mean', and ofsymmetric unimodal distributions'. For some of the known inequalities, such as the Gau\ss \, inequality, an alternative proof is given.