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Math Help - Covariance matrix for two independent variables

  1. #1
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    Covariance matrix for two independent variables

    Hi,

    Let Y1 and Y2 be independent random variables with Y1 ~ Norm(1, 3) and Y2 ~ Norm(2, 5). If W1 = Y1 + Y2 and W2 = 4Y1 - Y2 what is the joint distribution of W1 and W2?

    I see how the mean vector is simply the variables' means, and I see the diagonal of the covariance matrix is just the variance, but according to the solution the off diagonal, which is the covariance between W1 and W2, right?, should be 2. I dont see how this can be computed without the correlation coefficient, p?

    Overall the covariance matrix is [23, 2; 2, 53].

    Any help much appreciated. MD
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  2. #2
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    Re: Covariance matrix for two independent variables

    Quote Originally Posted by Mathsdog View Post
    Hi,

    Let Y1 and Y2 be independent random variables with Y1 ~ Norm(1, 3) and Y2 ~ Norm(2, 5). If W1 = Y1 + Y2 and W2 = 4Y1 - Y2 what is the joint distribution of W1 and W2?

    I see how the mean vector is simply the variables' means, and I see the diagonal of the covariance matrix is just the variance, but according to the solution the off diagonal, which is the covariance between W1 and W2, right?, should be 2. I dont see how this can be computed without the correlation coefficient, p?

    Overall the covariance matrix is [23, 2; 2, 53].

    Any help much appreciated. MD
    Have you expanded the covariance of W_1 and W_2?

    {\rm{CoV}}(W_1,W_2)={\rm{E}\left[ (W_1-\overline{W_1}\ )(W_2-\overline{W_2}\ )\right]= ...

    CB
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  3. #3
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    Re: Covariance matrix for two independent variables

    Hi, Thanks for the reply. I am familiar with the calculation of the covariance matrix from the data. I think its also pretty clear how a correlation can be calcultated too. In this question only the mean and the variance for the independent variables, and by implication the mean and the variance for the variables W1 and W2, Norm(5, 23) and Norm(2, 53) respectively.

    However, I don't see how the non-diagonal elements of the covariance matrix are calculated from these facts.

    Basically, how did 2 get arrived at as the value for the non-diagonal elements?

    Apologies if I have misunderstood you CB.
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  4. #4
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    Re: Covariance matrix for two independent variables

    Quote Originally Posted by Mathsdog View Post
    Hi, Thanks for the reply. I am familiar with the calculation of the covariance matrix from the data. I think its also pretty clear how a correlation can be calcultated too. In this question only the mean and the variance for the independent variables, and by implication the mean and the variance for the variables W1 and W2, Norm(5, 23) and Norm(2, 53) respectively.

    However, I don't see how the non-diagonal elements of the covariance matrix are calculated from these facts.

    Basically, how did 2 get arrived at as the value for the non-diagonal elements?

    Apologies if I have misunderstood you CB.
    You have been told the values of:

    \rm{E}(Y_1) , \rm{E} (Y_2) , \rm{E}( (Y_1-\overline{Y_1} )^2 ) , \rm{E}( (Y_2-\overline{Y_2} )^2 )

    Now if you try doing as I suggested and expand the covariance of W_1 and W_2 in terms of Y_1 and Y_2 you will find that what you have been given are sufficient to find a numerical value for the covariance.

    CB
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  5. #5
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    Re: Covariance matrix for two independent variables

    Thanks so much. I did as you said, and with a bit of umming and scratching my head got it eventually. C u, MD
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