Teoriâ veroâtnostej i ee primeneniâ, Tome 27 (1982) no. 2, pp. 339-341
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D. F. Vysočanskiǐ; Yu. I. Petunin. On a Gauss inequality for the unimodal distributions. Teoriâ veroâtnostej i ee primeneniâ, Tome 27 (1982) no. 2, pp. 339-341. http://geodesic.mathdoc.fr/item/TVP_1982_27_2_a12/
@article{TVP_1982_27_2_a12,
author = {D. F. Vyso\v{c}anskiǐ and Yu. I. Petunin},
title = {On a {Gauss} inequality for the unimodal distributions},
journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
pages = {339--341},
year = {1982},
volume = {27},
number = {2},
language = {ru},
url = {http://geodesic.mathdoc.fr/item/TVP_1982_27_2_a12/}
}
TY - JOUR
AU - D. F. Vysočanskiǐ
AU - Yu. I. Petunin
TI - On a Gauss inequality for the unimodal distributions
JO - Teoriâ veroâtnostej i ee primeneniâ
PY - 1982
SP - 339
EP - 341
VL - 27
IS - 2
UR - http://geodesic.mathdoc.fr/item/TVP_1982_27_2_a12/
LA - ru
ID - TVP_1982_27_2_a12
ER -
%0 Journal Article
%A D. F. Vysočanskiǐ
%A Yu. I. Petunin
%T On a Gauss inequality for the unimodal distributions
%J Teoriâ veroâtnostej i ee primeneniâ
%D 1982
%P 339-341
%V 27
%N 2
%U http://geodesic.mathdoc.fr/item/TVP_1982_27_2_a12/
%G ru
%F TVP_1982_27_2_a12
Let $\xi$ be a random variable with an unimodal distribution, $M$ be a mode of this distribution, $x_0\in(-\infty,\infty)$ and $\theta^2=\mathbf D\xi+(\mathbf E\xi-x_0)^2=\mathbf E(\xi-x_0)^2$. It is shown that for all $k\ge 2$$$ \mathbf P\{|\xi-x_0|\ge k\theta\}\le\frac{4}{9k^2}. $$ if the point $x_0$ separates the points $M$ and $\mathbf E\xi$ then the inequality is fulfilled for all $k\ge\sqrt 3$.
