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Some remarks on lower bounds of Chebyshev’s type for half-lines. (English) Zbl 1053.60012
Summary: We prove that for any r.v. $$X$$ such that $$E\{X\}=0$$, $$E\{X^2\}=1$$ and $$E\{X^\}=\mu$$, and for any $$\varepsilon\geq 0$$ $P(X\geq\varepsilon) \geq \frac{K_0} {\mu}-\frac{K_1} {\sqrt\mu} \varepsilon+ \frac{K_1} {\mu\sqrt\mu} \varepsilon,$ where absolute constants $$K_0=2\sqrt 3-3\approx 0.464$$, $$K_1=1.397$$, and $$K_2= 0.0231$$. The constant $$K_0$$ is sharp for $$\mu\geq\frac{3}{\sqrt 3+1} \approx 1.09$$. Some other bounds and examples are given.
##### MSC:
 6e+16 Inequalities; stochastic orderings 6e+06 Probability distributions: general theory
##### Keywords:
inequality of Chebyshev’s type
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