On maximal tail probability of sums of nonnegative, independent and identically distributed random variables

We consider the problem of finding the optimal upper bound for the tail probability of a sum of $k$ nonnegative, independent and identically distributed random variables with given mean $x$. For $k=1$ the answer is given by Markov’s inequality and for $k=2$ the solution was found by Hoeffding and Shrikhande in 1955. We solve the problem for $k=3$ as well as for general $k$ and $x\le 1\mathrm{/}(2k-1)$ by showing that it follows from the fractional version of an extremal graph theory problem of Erd\H{o}s on matchings in hypergraphs.