Implicitly defined (and easily approximated) universal constants 1.1 < an < 1.6, n = 2,3, ... , are found so that if X1, X2, ... are i.i.d. non-negative random variables and if the Tn is the set of stop rules for X1, ..., Xn, then E (max {X1, ..., Xn}) ≤ ansup {EXt : t ε Tn}, and the bound an is best possible. Similar universal constants 0 < bn < 1/4 are found so that if the (Xi) are i.i.d. random variables taking values only in [a,b], then E (max {X1, ..., Xn}) ≤ ansup {EXt : t ε Tn} + bn (b - a ), where again the bound bn is the nest possible. In both situations, extremal distributions for which equality is attained (or nearly attained) are given in implicit form.



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