1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: 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

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    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

    User Avatar
    Science Advisor
    Homework Helper

    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
     
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook