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Covariance/Independence Proof Help

  1. Jan 20, 2010 #1
    1. The problem statement, all variables and given/known data

    Suppose {X1,X2, ...,Xn} is an iid (independent, identically distributed) sample from N(μ, σ^2). Show that Cov(X_bar, Xi − X_bar ) = 0 for all i, and hence conclude that X_bar is independent of Xi − X_bar for every i.

    2. Relevant equations

    3. The attempt at a solution

    cov(X_bar ̅,Xi-X_bar)
    =cov(X_bar ̅,X_i ) - cov(X_bar ̅,X _bar )
    =cov((1/n)*∑Xj ,Xi ) - cov((1/n) ∑Xi, (1/n) ∑Xj)
    =∑(1/n) cov(Xj,Xi ) - ∑∑(1/n)^2 cov(Xi,Xj )
    =(n/n)*σ^2 - (n/n)^2*σ^2
    =σ^2 - σ^2=0

    not sure if what im doing is right, and i don't see how showing Cov( X_bar,Xi − X_bar ) = 0 for all i can help to arrive at the suggested conclusion.

    Thanks for ur help guys... :)
  2. jcsd
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