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Math Help - A simple identity to verify

  1. #1
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    A simple identity to verify

    Hello,

    I am having a hard time with this problem (picture attached). I always have difficulty with proof-like problems such as this one. I guess I just don't really have the mathematical intuition needed for these types of problems.

    I am not seeing how this identity works, because I thought my professor said that mew is equivalent to x bar, so I don't even understand how this identity makes sense. Can somebody help me with this problem?? I am supposed to verify the identity shown in the picture.

    Attached Thumbnails Attached Thumbnails A simple identity to verify-photo.jpg  
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  2. #2
    MHF Contributor
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    Hi

    \sum_{i=1}^n \left(X_i - \bar{X}\right)^2 + n \left(\bar{X}-\mu \right)^2 = \sum_{i=1}^n \left(X_i \right)^2 - 2 \bar{X} \sum_{i=1}^n X_i  + n \bar{X}^2 + n \bar{X}^2 - 2 n \mu \bar{X}  + n \mu^2

    \bar{X} = \frac{1}{n} \sum_{i=1}^n X_i

    therefore - 2 \bar{X} \sum_{i=1}^n X_i = - 2 n \bar{X}^2 cancels out with n \bar{X}^2 + n \bar{X}^2

    \sum_{i=1}^n \left(X_i - \bar{X}\right)^2 + n \left(\bar{X}-\mu \right)^2 = \sum_{i=1}^n \left(X_i \right)^2 - 2 n \mu \bar{X} + n \mu^2

    \sum_{i=1}^n \left(X_i - \bar{X}\right)^2 + n \left(\bar{X}-\mu \right)^2 = \sum_{i=1}^n \left(X_i \right)^2 - 2 \mu  \sum_{i=1}^n X_i + n \mu^2

    \sum_{i=1}^n \left(X_i - \bar{X}\right)^2 + n \left(\bar{X}-\mu \right)^2 = \sum_{i=1}^n \left(X_i - \mu \right)^2
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  3. #3
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    Thanks for the help!
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