We want to find the convex combination S of iid Bernoulli random variables that maximizes P(S ≥ t) for a given threshold t. Endre Csóka conjectured that such an S is an average if t ≥ p, where p is the success probability of the Bernoulli random variables. We prove this conjecture for a range of p and t

Let S and X be independent random variables, assuming values in the set of non-negative integers, and suppose further that both E(S) and E(X) are integers satisfying E(S) ≥ E(X). We establish a sufficient condition for the tail probability P(S ≥ E(S)) to be larger than the tail P(S + X ≥ E(S + X)), when the mean of S is equal to the mode.