2008/08/12 by Ludolf Erwin Meester, Ludolf E. Meester, Meester, Ludolf E.
Decision Sciences · Mathematics · #60E15 #FOS: Mathematics #Probability (math.PR) #Probability and Risk Models #math.PR #msc:60E15
paper · pdf · doi:10.48550/arxiv.0808.1669
arxiv created 2008/08/12 · openalex publication_date 2008/08/12 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Two old conjectures from problem sections, one of which from SIAM Review, concern the question of finding distributions that maximize P(Sn <= t), where Sn is the sum of i.i.d. random variables X1, ..., Xn on the interval [0,1], satisfying E[X1]=m. In this paper a Lagrange multiplier technique is applied to this problem, yielding necessary conditions for distributions to be extremal, for arbitrary n. For n=2, a complete solution is derived from them: extremal distributions are discrete and have one of the following supports, depending on m and t: 0,t, t-1,1, t/2,1, or 0,t,1. These results suffice to refute both conjectures. However, acquired insight naturally leads to a revised conjecture: that extremal distributions always have at most three support points and belong to a (for each n, specified) finite collection of two and three point distributions.