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 $k=n-m\ge 0$, 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 $\{M(k,m):k,m\ge 0\}$ 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 $M(n,0)-M(0,n)$, $n\ge 1$. 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.

13 pages