Estimating Sums of Independent Random Variables

The paper deals with a problem proposed by Uriel Feige in 2005: if X1, . . . , Xn is a set of independent nonnegative random variables with expectations equal to 1, is it true that P ( ∑n i=1 Xi < n + 1) > 1 e ? He proved that P ( ∑n i=1Xi < n + 1) > 1 13 . In this paper we prove that infimum of the P ( ∑n i=1Xi < n + 1) can be achieved when all random variables have only two possible values, and one of them is 0. We also give a partial solution to the case when all random variables are equally distributed. We prove that the inequality holds when n goes to infinity and provide numerical evidence that the probability decreases when n increases.