A Markovian Gau\ss \, Inequality for Asymmetric Deviations from the Mode of Symmetric Unimodal Distributions

Abstract

For a random variable with a unimodal distribution and finite second moment Gau\ss \, (1823) proved a sharp bound on the probability of the random variable to be outside a symmetric interval around its mode. An alternative proof for it is given based on Khintchine's representation of unimodal random variables. Analogously, a sharp inequality is proved for the slightly broader class of unimodal distributions with finite first absolute moment, which might be called a Markovian Gau\ss\ inequality. For symmetric unimodal distributions with finite second moment Semenikhin (2019) generalized Gau\ss's inequality to arbitrary intervals. For the class of symmetric unimodal distributions with finite first absolute moment we construct a Markovian version of it. Related inequalities of Volkov (1969) and Sellke and Sellke (1997) will be discussed as well.