On the Probability a Weighted Bernoulli Sum Exceeds Its Mean

Abstract

Let $w_1,\dots ,w_m$ be positive real weights whose sum is $1$, and let $v_1,\dots ,v_m$ be i.i.d. Bernoulli$(p)$ random variables. If we let $X={\sum }_{i=1}^m{w}_i{v}_i$, then we conjecture that for all $0\le p\le 1/3$ we have \[\mathbb{P}\big[X\geq \mathbb{E}[X]\big]\geq p.\] In this short note, we observe a connection of this conjecture with a version of the Manickam-Miklós-Singhi conjecture, which allows one to prove it for sufficiently small values of $p$.