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Let be $(W,Y)$ a normal bivariate vector with correlation coefficient $\rho$ and variances $\sigma_W^2=\sigma_Y^2$. Proof the independence of variables $W, (Y-\rho W)$.

I´m trying to proof that $Cov(W,Y-\rho W)=0$, because that implies that the variables are independent (because both are normal distributions). I know that clearing the formula of correlation coefficient $Cov(W,Y)=\rho \sigma_W^2$, I think that maybe a variable change could find $Cov(W,Y-\rho W)$.

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    $\begingroup$ Be careful saying "because both are normal distributions". That's not quite strong enough: en.wikipedia.org/wiki/… $\endgroup$ – Barry Smith Feb 19 '16 at 0:17
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No need to change variables; use the definition of correlation coefficient: $$ \rho := {\mathop{Cov}(W,Y)\over\sigma_W\sigma_Y} $$

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For any random variables $P,Q,R$, recall that: $\mathsf {Cov}(P, Q+R) = \mathsf {Cov}(P, Q)+\mathsf {Cov}(P,R)$

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