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February, 1966 On a Chebyshev-Type Inequality for Sums of Independent Random Variables
S. M. Samuels
Ann. Math. Statist. 37(1): 248-259 (February, 1966). DOI: 10.1214/aoms/1177699614


Let $\mathscr{S}(\nu_1, \cdots, \nu_n)$ be the "class of all random variables", $S_n$, which are sums of $n$ independent, non-negative random variables, $X_1, \cdots, X_n$, with $EX_i = \nu_i, i = 1, \cdots, n$. We consider the problem of finding \begin{equation*}\tag{1.1}\inf_{S_n\varepsilon\mathscr{S}(\nu_1,\cdots, \nu_n)} P\{S_n < \lambda\}\end{equation*} where $\lambda$ is a positive constant. For $n = 1$, the infimum is $1 - \nu_1/\lambda$ from the well-known Markov inequality. The solution for $n = 2$ was given in [2]. We derive the solution for $n = 3$. From these results we conjecture what the solution is for arbitrary $n$. To lend support to the conjecture, we examine a sub-class of $\mathscr{S}(\nu_1, \cdots, \nu_n)$, namely those $S_n$'s for which the problem reduces to one of considering the number of successes in independent trials. We show that, within this subclass, the conjectured value does minimize $P\{S_n < \lambda\}$.


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S. M. Samuels. "On a Chebyshev-Type Inequality for Sums of Independent Random Variables." Ann. Math. Statist. 37 (1) 248 - 259, February, 1966.


Published: February, 1966
First available in Project Euclid: 27 April 2007

zbMATH: 0203.20003
MathSciNet: MR187270
Digital Object Identifier: 10.1214/aoms/1177699614

Rights: Copyright © 1966 Institute of Mathematical Statistics

Vol.37 • No. 1 • February, 1966
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