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Covariance With Random Vector

  1. Mar 3, 2015 #1

    ElijahRockers

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    1. The problem statement, all variables and given/known data
    Let ##X## be a random variable such that ##\mu_X = 0## and ##K_{XX} = I##.
    Find ##Cov(a^T X, b^T X)## for ##a = (1, 1, 0, 0)## and ##b = (0, 1, 1, 0)##.

    3. The attempt at a solution
    I guess I am assuming that ##X## is a 4 element random vector. I can't know values of the random variables, but I know their mean, and I think from ##K_{XX} = I## that
    ##E[X_i X_j] = 0, i≠ j##
    ##E[X_i X_j] = 1, i= j##

    So..

    ##a^T X = X_1 + X_2 = A##
    ##b^T X = X_2 + X_3 = B##
    ##Cov(A,B) = E[AB]-E[A]E[ B]##

    ##E[A]## and ##E[ B]## are 0, so

    ##Cov(A,B) = E[AB] = E[X_1 X_2 + X_1 X_3 + X_2 X_2 + X_2 X_3]##

    From ##K_{XX}##, ##E[AB] = E[X_2 X_2] = 1 = Cov(A,B)##

    Not sure if this is correct or not.
     
  2. jcsd
  3. Mar 3, 2015 #2

    Ray Vickson

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    It is correct if your interpretation of ##K_{XX}## is correct (which I cannot speak to because the notation is unfamiliar to me).
     
  4. Mar 3, 2015 #3

    ElijahRockers

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    ##K_{XX}## is the covariance matrix of ##X##, where ##K_{XX_{i,j}} = E[X_i X_j] - E[X_i]E[X_j]## is each element in the matrix... I believe.
     
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