Title: On some conjectures of Samuels and Feige,
Abstract: Let $\mu_1 \ge \dotsc \ge \mu_n > 0$ and $\mu_1 + \dotsm + \mu_n = 1$. Let
$X_1, \dotsc, X_n$ be independent non-negative random variables with $EX_1 =
\dotsc = EX_n = 1$, and let $Z = \sum_{i=1}^n \mu_i X_i$. 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.