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Math Help - Jointly normal distribution

  1. #1
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    Jointly normal distribution

    Let X and Y be jointly normal, with the mean vectors and covariance matrix below:

    <br />
\mu =\left(\begin{array}{cc}1\\2\end{array}\right)   \Sigma = \left(\begin{array}{cc}2&.5\\.5&1\end{array}\right  )

    Let Z_{1} = X + Y \mbox{ and } Z_{2} = X - 2Y

    Find the mean vector and covariance matrix of Z_{1} \mbox{ and }Z{2}

    I figured out the following:

    \mu_{Z_{1}} = \mu_X + \mu_Y = 3,

     \mu_{Z_{2}} = \mu_X - 2\mu_Y = -3,

    \sigma_{Z_{1}}^{2} = \sigma_{X}^{2} + \sigma_{Y}^{2} + 2\rho\sigma_{X}\sigma_{Y} = 3 + \sqrt{2},

     \sigma_{Z_{2}}^{2} = \sigma_{X}^{2} + 4\sigma_{Y}^{2} - 4\rho\sigma_{X}\sigma_{Y} = 6 - 2\sqrt{2}

    But how do I figure out Cov(Z_{1}, Z_{2})?
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  2. #2
    Moo
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    Hello,

    Well, the elements of the covariance matrix are the covariances...
    So we have cov(X,Y)=cov(Y,X)=.5

    And finding cov(Z1,Z2) is then easy, using the fact that the covariance is a bilinear form (cov(ax,y)=a*cov(x,y) , cov(x+y,z)=cov(x,z)+cov(y,z))



    Otherwise, you can note that :
    \begin{pmatrix}Z_1 \\ Z_2\end{pmatrix}=\begin{pmatrix} 1&1 \\ 1&-2\end{pmatrix} \begin{pmatrix}X\\Y\end{pmatrix}

    Let's denote A=\begin{pmatrix} 1&1 \\ 1&-2\end{pmatrix}


    And we know that A\begin{pmatrix}X\\Y\end{pmatrix} will follow a normal distribution \mathcal{N}(A\mu,A\Sigma A')

    See a proof in the attached pdf
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