On the Expectations of Maxima of Sets of Independent Random Variables

Details On the Expectations of Maxima of Sets of Independent Random Variables On the Expectations of Maxima of Sets of Independent Random Variables

2009-06-12

arxiv.org/abs/0906.2270v1

Mathematics

Probability

13 pages

Scientific paper

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.

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