{
  "id": "0906.2270",
  "version": "v1",
  "published": "2009-06-12T07:36:25.000Z",
  "updated": "2009-06-12T07:36:25.000Z",
  "title": "On the Expectations of Maxima of Sets of Independent Random Variables",
  "authors": [
    "D. V. Tokarev",
    "K. A. Borovkov"
  ],
  "comment": "13 pages",
  "categories": [
    "math.PR"
  ],
  "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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2009-06-12T07:36:25.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60E15",
      "60K10"
    ],
    "keywords": [
      "independent random variables",
      "expectations",
      "component parallel system",
      "optimal initial composition",
      "jointly independent copies"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 13,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0906.2270T"
    }
  }
}