On the expectations of maxima of sets of independent random variables

Let X1,…,Xk and Y1,…,Ym 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)≔Emax{X1,…,Xk,Y1,…,Ym} 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.