  A remarkable conjecture of Feige (2006) asserts that for any collection of
$n$ independent non-negative random variables $X_1, X_2, \dots, X_n$, each with
expectation at most $1$, $$ \mathbb{P}(X < \mathbb{E}[X] + 1) \geq \frac{1}{e},
$$ where $X = \sum_{i=1}^n X_i$. In this paper, we investigate this conjecture
for the class of discrete log-concave probability distributions and we prove a
strengthened version. More specifically, we show that the conjectured bound
$1/e$ holds when $X_i$'s are independent discrete log-concave with arbitrary
expectation.
