Generalization of Gauss inequality for unimodal distributions. (Russian) Zbl 0575.60021

Vysochanskij, D. F.; Petunin, Yu. I.

Teor. Veroyatn. Mat. Stat. 31, 26-31 (1984).

Let \(\xi\) be a unimodal random variable with a mode \(m\) and let \(\tau^2 := \mathbb{E}(\xi -m)^2\) and \(\theta^2 := \mathbb{E}(\xi-x_0)^2\) where \(x_0\) is an arbitrary real number. The Gauss inequality gives the following estimate (1) \(\mathbb{P}[| \xi -m| \geq k\tau] \leq 4/(9k^2)\), for all \(k>0\). As a consequence of the above one can get that \[ \mathbb{P}[|\xi-x_0| \geq k\theta] \leq 4(1+r^2)/9(k-| r|)^2, \tag{2} \] where \(r:=(x_ 0-m)/\theta\) and \(k>| r|\). Usually the quantity \(| r|\) is difficult to estimate and therefore (2) seems to be of no interest. In the paper it has been shown that the left-hand probability in (2) always can be estimated by the right-hand side of (1). Thus the authors have generalized the Gauss inequality and also improved the inequality (2).

Reviewer: Z.Jurek

MSC:

60E15	Inequalities; stochastic orderings
60E05	Probability distributions: general theory

Keywords:

unimodal distribution; Gauss inequality; generalized the Gauss inequality