An equivalent conjecture to Feige's Conjecture

Abstract

Let X1, ..., Xn be arbitrary non-negative independent random variables with respective expected values $\mu_{i}$ at most one. We sketch but do not prove an equivalent conjecture to Feige's Conjecture $\mathbb{P} \left( \sum_{i=1}^{n} X_{i} < \mu + 1 \right) \geq \exp \left(-1 \right)$, where $\mu$ is the expected value of the sum of the random variables. We show by a simple example how this inequality finds use in mathematical finance.