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Math Help - Describe the range of a random vector with multivariate normal distribution

  1. #1
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    Describe the range of a random vector with multivariate normal distribution

    Let  X = \begin{bmatrix} X_1 \\ X_2 \\ X_3 \end{bmatrix} have covariance matrix  \Sigma =  \begin{bmatrix}3 & 1 & 1 \\ 1 & 4 & -7 \\ 1 & -7 & 15 \end{bmatrix}.

    Note that  \Sigma \cdot \begin{bmatrix} 1 \\ -2 \\ -1 \end{bmatrix} = 0, so  \Sigma have rank 2. If  \mu _X = \begin{bmatrix} 10 \\ 20 \\ 30 \end{bmatrix} and  X has a multivariate normal distribution, describe the range of X.

    Solution so far:

    Now, since  \Sigma \cdot \begin{bmatrix} 1 \\ -2 \\ -1 \end{bmatrix} = 0, I have:

     \begin{bmatrix} 1 & -2 & -1 \end{bmatrix} \Sigma \cdot \begin{bmatrix} 1 \\ -2 \\ -1 \end{bmatrix} = 0

     \begin{bmatrix} 1 & -2 & -1 \end{bmatrix} Cov(X,X) \begin{bmatrix} 1 \\ -2 \\ -1 \end{bmatrix} = 0

      Cov( \begin{bmatrix} 1 & -2 & -1 \end{bmatrix} \begin{bmatrix} X_1 \\ X_2 \\ X_3 \end{bmatrix} )  = 0

     Var( X_1-2X_2-X_3) = 0

    So I need to find out when does that happens, but I'm kind of stuck at the moment, any help please? Thanks!
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  2. #2
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    Re: Describe the range of a random vector with multivariate normal distribution

    Hey tttcomrader.

    What part of R^3 describes this linear equation involving X1, X2, and X3? (Hint: It will be on a plane in 3 dimensional space with two independent parameters).
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  3. #3
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    Re: Describe the range of a random vector with multivariate normal distribution

    So since  Var(X_1-2X_2-X_3)=0 , then we must have  X_1-2X_2-X_3 = k for some constant k.

    That also means that  E[X_1-2X_2-X_3] = k , which is  10-40-30 = -60 , so  X_1-2X_2-X_3 = -60 ?
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    Re: Describe the range of a random vector with multivariate normal distribution

    Looks pretty good.
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