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Math Help - conditional expectation

  1. #1
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    conditional expectation

    If E(X^2)< \infty, then E((X-E(X|G))^2) \leq E((X-E(X))^2)
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  2. #2
    MHF Contributor matheagle's Avatar
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    Start with E((X-E(X))^2) and add and subtract \mu_g=E(X|G)

    E((X-E(X))^2)=E((X-\mu_g+\mu_g-E(X)^2)

    =E((X-\mu_g)^2)+2E((X-\mu_g)(\mu_g-\mu))+E(\mu_g-\mu)^2 where \mu=EX

    =E((X-\mu_g)^2)+E(\mu_g-\mu)^2\ge E((X-\mu_g)^2)

    Since

     E((X-\mu_g)(\mu_g-\mu))=E(E((X-\mu_g)(\mu_g-\mu)|G))

     =E ((\mu_g-\mu)(E(X-\mu_g) |G))=E((\mu_g-\mu)(\mu_g-\mu_g))=0
    Last edited by matheagle; November 22nd 2009 at 12:59 AM.
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