On a Chebyshev-Type Inequality for Sums of Independent Random Variables

Abstract

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 $P\{S_n < \lambda\}$.