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coefficient of correlation, regression

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This job explores coefficient of correlation, regression, and other terms.

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3.13 X1, X2, X3 are uncorrelated, i.e. cov(Xi,Xj)=0 for all I<>j
Let Var(Xi)=S
Then Var(X1+X2)=E([(X1+X2)-E(X1+X2)]^2)=E([X1+X2-E(X1)-E(X2)]^2)
= E(([X1-E(X1)]+[X2- E(X2)])^2)=E(([X1-E(X1)]^2+[X2- E(X2)]^2+[X1-E(X1)]* [X2- E(X2)]
= E([X1-E(X1)]^2)+E([X2- E(X2)]^2)+E([X1-E(X1)]* [X2- E(X2)])
= var(X1)+ var(X2)+cov(X1,X2)=S+S+0=2S
in the same way, we can prove that Var(X3+X2)=2S
Now, Cov[(X1+X2),( X3+X2)]=E([(X1+X2)-E(X1+X2)]*[(X3+X2)-E(X3+X2)])
=E{([X1-E(X1)]+[X2-E(X2)])* ([X3-E(X3)]+[X2-E(X2)])}
=E{[X1-E(X1)]* [X3-E(X3)]+ [X2-E(X2)]* [X3-E(X3)]+ [X2-E(X2)]* [X1-E(X1)]+[X2-E(X2)]* [X2-E(X2)]}= E{[X1-E(X1)]* [X3-E(X3)]}+ E{[X2-E(X2)]* [X3-E(X3)]}+ E{[X2-E(X2)]* ...

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