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On some conjectures of Samuels and Feige
Roland Paulin · 2017-03-15 · via math.PR updates on arXiv.org

Let $μ_1 \ge \dotsc \ge μ_n > 0$ and $μ_1 + \dotsm + μ_n = 1$. Let $X_1, \dotsc, X_n$ be independent non-negative random variables with $EX_1 = \dotsc = EX_n = 1$, and let $Z = \sum_{i=1}^n μ_i X_i$. Let $M = \max_{1 \le i \le n} μ_i = μ_1$, and let $δ> 0$ and $T = 1 + δ$. Both Samuels and Feige formulated conjectures bounding the probability $P(Z < T)$ from above. We prove that Samuels' conjecture implies a conjecture of Feige.

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