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Math Help - Maximising a sum of squared differences.

  1. #1
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    Maximising a sum of squared differences.

    for what value of c is the quantity sigma(Xi-c)^2 minimized?
    [Hint: Take the derivative with respect to c , set equal to 0 and solve]


    I don't know where to start...
    Last edited by mr fantastic; March 11th 2010 at 06:57 PM. Reason: Changed post title
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  2. #2
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    Quote Originally Posted by questionboy View Post
    for what value of c is the quantity sigma(Xi-c)^2 minimized?
    [Hint: Take the derivative with respect to c , set equal to 0 and solve]


    I don't know where to start...
    Start by noting that

    \sum_{i = 1}^n (X_i - c)^2 = \sum_{i = 1}^n (X_i^2 - 2 c X_i - c^2) = \sum_{i = 1}^n X_i^2 - 2 c \sum_{i = 1}^nX_i - \sum_{i = 1}^n c^2

     = \sum_{i = 1}^n X_i^2 - 2 c \sum_{i = 1}^nX_i - n c^2

    and then use the hint.
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  3. #3
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    Thank you Mr fantastic

    Do I set derivative of C equal to zero?
    Last edited by mr fantastic; March 12th 2010 at 02:30 AM. Reason: restored original post so the reply from matheagle makes sense.
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  4. #4
    MHF Contributor matheagle's Avatar
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    why don't you use the hint.........
    Take the derivative with respect to c , set equal to 0 and solve

    {d\over dc} \sum_{i=1}^n(X_i-c)^2=-2 \sum_{i=1}^n(X_i-c)=-2(n\bar X-nc)

    set this equal to zero, shows that c=\bar X

    And it is a min, that's why you should obtain the second derivative.
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