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Variance-Covariance Matrix

  1. Apr 15, 2009 #1
    1. The problem statement, all variables and given/known data

    Let [tex]\Sigma = [/tex]
    ( var(X1) cov(X1, X2) )
    ( cov (X2. X1) var(X2) )

    Show that [tex]Var (a_1 X_1 + a_2 X_2) = a^T \Sigma a[/tex]

    where [tex]a^T = [a_1 a_2][/tex] is the transpose of the of the column vector a

    2. Relevant equations



    3. The attempt at a solution

    I got this far:

    [tex]Var (a_1 X_1 + a_2 X_2) = a_1^2 Var(X_1) + a_2^2 Var(X_2) + 2a_1 a_2 Cov (X_1, X_2) = a_1^2 Var(X_1) + a_2^2 Var(X_2) + a_1 a_2 Cov (X_1, X_2) + a_1 a_2 Cov (X_2, X_1)[/tex]

    Thats all I got so far, any hints
     
  2. jcsd
  3. Apr 16, 2009 #2
    Aren't you done? Isn't that what [tex]a^T \Sigma a[/tex] is?
     
  4. Apr 17, 2009 #3
    Thought there was more to it than that.

    There's another part of the question that says: Using [tex]Var (a_1 X_1 + a_2 X_2)[/tex] show that for every choice of a1 and a2 that [tex]a^T \Sigma a \geq 0[/tex]

    Can I assume that [tex]\Sigma[/tex] is always positive?
     
  5. Apr 17, 2009 #4
    [tex]Var (a_1 X_1 + a_2 X_2)\ge0[/tex] always, since it's variance! And you just showed it equals [tex]a^T \Sigma a[/tex]
     
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