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On the expectations of maxima of sets of independent random variables

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  • Tokarev, Daniel V.
  • Borovkov, Konstantin A.

Abstract

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 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.

Suggested Citation

  • Tokarev, Daniel V. & Borovkov, Konstantin A., 2009. "On the expectations of maxima of sets of independent random variables," Statistics & Probability Letters, Elsevier, vol. 79(23), pages 2381-2388, December.
    • Handle: RePEc:eee:stapro:v:79:y:2009:i:23:p:2381-2388
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    References listed on IDEAS

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    1. Navarro, Jorge & Rychlik, Tomasz, 2007. "Reliability and expectation bounds for coherent systems with exchangeable components," Journal of Multivariate Analysis, Elsevier, vol. 98(1), pages 102-113, January.
    2. S. E. Jacobsen & S. Arunkumar, 1973. "Investment in Series and Parallel Systems to Maximize Expected Life," Management Science, INFORMS, vol. 19(9), pages 1023-1028, May.
    3. Bairamov, Ismihan & Arnold, Barry C., 2008. "On the residual lifelengths of the remaining components in an n-k+1 out of n system," Statistics & Probability Letters, Elsevier, vol. 78(8), pages 945-952, June.
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