(adsbygoogle = window.adsbygoogle || []).push({}); 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... :)

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# Homework Help: Covariance/Independence Proof Help

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