On the expectations of maxima of sets of independent random variables

Abstract

Let X 1 , … , X k and Y 1 , … , Y m be jointly independent copies of random variables X and Y , respectively. For a fixed total number n of random variables, we aim at maximising M ( k , m ) ≔ E max { X 1 , … , X k , Y 1 , … , Y m } in k = n − m ≥ 0 , which corresponds to maximising the expected lifetime of an n -component parallel system whose components can be chosen from two different types. We show that the lattice { M ( k , m ) : k , m ≥ 0 } is concave, give sufficient conditions on X and Y for M ( n , 0 ) to be always or ultimately maximal and derive a bound on the number of sign changes in the sequence M ( n , 0 ) − M ( 0 , n ) , n ≥ 1 . The results are applied to a mixed population of Bienayme–Galton–Watson processes, with the objective to derive the optimal initial composition to maximise the expected time to extinction.