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Math Help - Correlation of samples

  1. #1
    Super Member Anonymous1's Avatar
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    Correlation of samples

    Let X_i, i= 1,..,n, be i.i.d. sampes from N(\mu, \sigma^2). Let \bar{X} = \frac{1}{n}\sum_{i=1}^{n} X_i.

    Prove that \bar{X} and  X_i - \bar{X} are uncorrelated for any i.
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  2. #2
    MHF Contributor matheagle's Avatar
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    for simplicity let i=1....

    Let's obtain the covariance between \bar X and X_1-\bar X

     Cov(\bar X,X_1-\bar X) ={1\over n} Cov\left(X_1+\cdots +X_n,\left(1-{1\over n}\right)X_1-{1\over n}[X_2+\cdots +X_n]\right)

    Now find the covariance between each pair, taking one term from each set.

    ALL we need is for the Cov(X_i,X_j)=0 for each i\ne j , we don't need NORMALITY...

     \left({1\over n}\right)\left[\left(1-{1\over n}\right)\sigma_1^2-{1\over n}\left[\sigma_2^2+\cdots +\sigma_n^2\right]\right)

    Next use the fact that all the variances are equal this becomes zero.
    Changing 1 to i is easy, just sum over all the terms that's not i in the second sum.
    Last edited by matheagle; March 6th 2010 at 07:51 AM.
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