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Math Help - variance cov matrix

  1. #1
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    variance cov matrix

    One question:

    If we have the vector x=[x_1,x_2,x_3] ;  E(x_1)=8 \ E(x_2)=0, \  E(x_3)=2:

    where

    var(x_1)=4, var(x_2)=1, var(x_3)=4

    and

    cov(x_1,x_2)=-1,cov(x_1,x_3)=2, cov(x_2,x_3)=-2

    what is the variance of the linear combination:

    z=2x_2 + x_3

    *****

    I get that

    var(z)=4var(x_2)+var(x_3)+2cov(2x_2,x_3)

    which is equal to


    var(z)=4(1)+4+2(2)(-2)=0

    Am I doing everything right? How can we get that variance? I can't calculate correlation coefficients with a variance of 0, so I am stucked here. I would really appreaciate any sort of help
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  2. #2
    MHF Contributor matheagle's Avatar
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    yes that variance is zero
    also note that the correlation between X2 and X3 is -1.
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  3. #3
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    thnk you!

    So I am supposed to calculate the correlation between Z and another linear combination. What am I supposed to do if var(z)=0? Just say that it is impossible to calculate?
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  4. #4
    MHF Contributor matheagle's Avatar
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    what is this other linear combination?
    and note anyone can make this absurd.
    we need to follow the cauchy-schwartz inequality
    that's why I calculated your correlation
    you can make the covariance between X2 and X3 equal -100 which makes this impossible
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  5. #5
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    the other linear combinations are

    w=x_1+x_2+x_3
    y=x_1-x_3

    yahh, I guess that it is a problem in the question. Thank you!
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