{"id":"palomar/PALOMAR-2026-09-07-000011","url":"https://github.com/geoconuk/lean-misc-math/tree/cb359c08d6515e97c4b7ec1ccdd0f56b0ddbe722","json":"https://data.palomar-registry.org/entries/PALOMAR-2026-09-07-000011-v1.json","repo":"https://github.com/geoconuk/lean-misc-math","tags":["math.PR","math.ST","msc:60E15","msc:62E15","msc:60C05","probability","statistics","binomial","extremal-bounds"],"score":0,"title":"Hoeffding's extrema of the number of successes at a fixed mean","trust":"high","domain":"math.PR","source":"palomar","authors":["George A. Constantinides"],"reviews":0,"subject":"math.PR","summary":"For n independent trials with success probabilities p_1, ..., p_n in [0,1] -- Poisson trials -- and S the number of successes, three results about how the law of S varies over all trial vectors with a given mean lam, with pbar = lam/n the common success probability of the homogeneous trials of the same mean. First, for g convex on the integer grid, E[g S] is maximised by the binomial, and under strict grid convexity equality holds exactly at the constant vector (Hoeffding's Theorem 3). Second, for arbitrary g with no convexity assumed, the maximum and the minimum of E[g S] over the…","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"verified":true,"discussed":0,"published":"2026-09-07","reactions":{},"formalized":true,"formalizes":[{"title":"On the distribution of the number of successes in independent trials","identifier":"doi:10.1214/aoms/1177728178","relationship":"formalizes"}]}