On the Tchebyshev inequality ih the two-dimensional case
Teoriâ veroâtnostej i ee primeneniâ, Tome 16 (1971) no. 2, pp. 353-360
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Let $\xi_1$, $\xi_2$ be indepent random variables satisfying the conditions
$$
\mathbf P\{\xi_1\ge0\}=\mathbf P\{\xi_2\ge0\}=1\quad\mathbf M\xi_1=\mathbf M\xi_2=1.
$$
For positive $\Delta_1$ and $\Delta_2$, the inequality
\begin{gather*}
\mathbf P\{\Delta_1\min(\xi_1,\xi_2)+\Delta_2\max(\xi_1,\xi_2)\ge c\}\le
\\
\le\max[(\Delta_1+\Delta_2)^2,\Delta_2/(1-\Delta_1),2\Delta_2(1-\Delta_2)+\Delta_2^2]
\end{gather*}
is proved. Moreover, if $\xi_1$ and $\xi_2$ are equally distributed, then it is proved that
$$
\mathbf P\{\Delta_1\min(\xi_1,\xi_2)+\Delta_2\max(\xi_1,\xi_2)\ge c\}\le\max[(\Delta_1+\Delta_2)^2;2\Delta_2(1-\Delta_2)+\Delta_2^2].
$$
@article{TVP_1971_16_2_a13,
author = {L. V. Arharov},
title = {On the {Tchebyshev} inequality ih the two-dimensional case},
journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
pages = {353--360},
publisher = {mathdoc},
volume = {16},
number = {2},
year = {1971},
language = {ru},
url = {http://geodesic.mathdoc.fr/item/TVP_1971_16_2_a13/}
}
L. V. Arharov. On the Tchebyshev inequality ih the two-dimensional case. Teoriâ veroâtnostej i ee primeneniâ, Tome 16 (1971) no. 2, pp. 353-360. http://geodesic.mathdoc.fr/item/TVP_1971_16_2_a13/