Hoeffding's extrema of the number of successes at a fixed mean
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…
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