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Covariance involving Expectation

  1. Nov 1, 2013 #1
    1. The problem statement, all variables and given/known data
    Suppose ##X,Y## are random variables and ##\varepsilon = Y - E(Y|X)##. Show that ##Cov(\varepsilon , E(Y|X)) = 0##.


    2. Relevant equations
    ##E(\varepsilon) = E(\varepsilon | X) = 0##
    ##E(Y^2) < \infty##


    3. The attempt at a solution
    ##Cov(\varepsilon , E(Y|X)) = E((\varepsilon - E(\varepsilon)(E(Y|X) - E(E(Y|X))))
    = E(\varepsilon (E(Y|X) - E(Y)))
    = E(\varepsilon E(Y|X)) - E(\varepsilon E(Y))##

    From there, I didn't know what to do next so I tried it again using the formula for covariance:

    ##Cov(\varepsilon , E(Y|X)) = E(\varepsilon E(Y|X)) - E(\varepsilon )E(E(Y|X))
    = E(\varepsilon E(Y|X))##

    Which means ##E(\varepsilon E(Y|X)) - E(\varepsilon E(Y) = E(\varepsilon E(Y|X))## or in other words, ##E(\varepsilon E(Y)) = 0##. I'm not sure what to do now.
     
  2. jcsd
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