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