On the expectations of maxima of sets of independent random variables

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 grater or equal to 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 grater or equal to 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 grater or equal to 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