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Bounds on Moments

  1. Nov 5, 2012 #1
    1. The problem statement, all variables and given/known data
    For any random variable X, prove that
    P{X[itex]\geq[/itex]0}[itex]\leq[/itex]inf[ E[ phi(t) : t [itex]\geq[/itex] 0] [itex]\leq[/itex] 1

    where phi(t) = E[exp(tX)] o<phi(t)[itex]\leq[/itex]∞


    2. Relevant equations



    3. The attempt at a solution
    I am not sure how to begin this. Any hints to get started would be greatly appreciated.
     
  2. jcsd
  3. Nov 6, 2012 #2

    haruspex

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    E[ phi(t)] doesn't mean anything. E[] requires a r.v., whereas phi(t) is just an ordinary function. So I guess you mean
    P{X[itex]\geq[/itex]0}[itex]\leq[/itex]inf[phi(t) : t [itex]\geq[/itex] 0] [itex]\leq[/itex] 1

    Write out E[phi()] as an integral and consider the positive and negative ranges of X separately.
     
  4. Nov 6, 2012 #3

    Ray Vickson

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    Broad hint: P{X ≥ 0} = E H(X), where H(x) = 0 for x < 0 and H(x) = 1 for x ≥ 0. So, you are really comparing expectations of two different functions of X.

    RGV
     
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