Geodesic
On a Gauss inequality for the unimodal distributions
Teoriâ veroâtnostej i ee primeneniâ, Tome 27 (1982) no. 2, pp. 339-341

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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 $k\ge\sqrt 3$. 
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     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},
     publisher = {mathdoc},
     volume = {27},
     number = {2},
     year = {1982},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_1982_27_2_a12/}
}
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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/