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Math Help - Multi variate normal

  1. #1
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    Multi variate normal

    Suppose X = (x1, x2, x3)' is multivariate normal

    ((3 2 2), ((4 -2 -2) (-2 2 1) (-2 1 2)))

    (for the 3*3 matrix, the first row is 4 -2 -2 and second -2 2 1 etc...)

    What is the joint distribution function Y1 = X1 + X2 and Y2 = X1+X3?

    multivariates are relatively new to me, could I ask for how this might be calculated?

    Many thanks
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  2. #2
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    Quote Originally Posted by sharpe View Post

    What is the joint distribution function Y1 = X1 + X2
    Hi there sharpe, this is a good question.

    What you need to know is that if X_1 and X_2 are normal then the linear combination Y_1 = X_1+X_2 is also normal.

    So you need to find the expectation and variance of this joint distribution.

    Consider these

    E(aX_1+bX_2) =aE(X_1)+bE(X_2)

    V(aX_1) = a^2V(X_1)

    V(aX_1+bX_2) = a^2V(X_1)+b^2V(X_2)+2ab\times COV(X_,X_2)

    It follows in your case that a=b=1 then

    E(Y_1) = E(X_1)+E(X_2)

    and

    V(Y_1) = V(X_1)+V(X_2)
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  3. #3
    MHF Contributor matheagle's Avatar
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    (Y_1,Y_2)=\left(\matrix{1&1&0,\cr <br />
                                             1&0&1\cr}\right)X
    where that's supposed to be a 2 by 3 matrix
    Last edited by matheagle; August 8th 2010 at 03:46 PM.
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  4. #4
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    Thanks that is helpful.

    For some reason I could not see what the second matrix represented, but with your comments the penny dropped and it shows the variances and covariances. It was then pretty trivial

    Thanks a lot!
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