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

• Main Authors: Ion, Roxana A, Klaassen, Chris A. J, Heuvel, Edwin R. van den
• Format: Journal Article
• Language: English
• Published: 18-08-2022
• Subjects: Mathematics - Probability; Mathematics - Statistics Theory; Statistics - Theory
• DOI: 10.48550/arxiv.2208.08813

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.