On some conjectures of Samuels and Feige

Abstract

Let ${\mu }_1\ge \dots \ge {\mu }_n>0$ and ${\mu }_1+\cdots +{\mu }_n=1$. Let $X_1,\dots ,X_n$ be independent non-negative random variables with $E{X}_1=\dots =E{X}_n=1$, and let $Z={\sum }_{i=1}^n{\mu }_i{X}_i$. Let $M={\max }_{1\le i\le n}{\mu }_i={\mu }_1$, and let $\delta >0$ and $T=1+\delta$. Both Samuels and Feige formulated conjectures bounding the probability $P(Z<T)$ from above. We prove that Samuels' conjecture implies a conjecture of Feige.

4 pages