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Math Help - double summation question

  1. #1
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    double summation question

    double summation question-double-sum.jpg



    In the above pic xbar is a constant and so can come outside the summation and the second expression in the first line is equal to zero so the second line is just a rearrangement of  \sum u_i x_i \sum u_i

    I can't figure out how we get from  \sum u_i x_i \sum u_i to the \sum ui^2x_i plus the double summation of i and i not equal to j of u_iu_jx_j
    Last edited by kingsolomonsgrave; October 10th 2013 at 04:31 PM.
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  2. #2
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    Re: double summation question

    Hey kingsolomonsgrave.

    You should try doing a quick expansion of the product. You will find that you can group the terms where both summands have the same index (i = j) and you will have a different expression where i != j.

    As an example consider n = 2. You have

    sigma (UiXi) * sigma(Ui) [1 to 2] = (U1X1 + U2X2)(U1 + U2)
    = [U1^2X1 + U2^2X2] + [U1U2X1 + U1U2X2]
    = sigma [1 to 2] (Ui^2*Xi) + sigma [i != j] [1 to 2] [1 to n] Ui*Uj*Xj

    Use that pattern to show it for the general case of n > 1.
    Thanks from kingsolomonsgrave
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