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Let $X, Y$ be independent RV given the variance and no means what is correlation coefficient of $X$ and $Z=2X+Y$?

Given $var(X)=3, var(Y)=4$ and $\mathbf{E}[X]$ and $\mathbf{E}[Y]$ are not known, let $Z=2X+Y$. What is the correlation coefficient?

$Var(2X + Y) = Var(Z) = 2^2\times Var(X) + Var(Y) = 12 + 4 = 16$

I don't know where to go from here

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  • $\begingroup$ Calculate $cov(X,Z)$, then normalise $\endgroup$
    – gunes
    Jun 8 at 8:03
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By definition: \begin{equation} Corr(X,Z)=\frac{Cov(X,Z)}{\sqrt{Var(X)}\sqrt{Var(Z)}} \end{equation} You know that $Var(X)=3$ and $Var(Y)=4$. Since Z is a linear affin transformation of $X$ and $X$ and $Y$are independent, you get: \begin{align} Var(Z)&=Var(2X+Y)=4Var(X)+Var(Y)=12+4=16 \end{align} You can calculate the covariance between X and Z as: \begin{align} Cov(X,Z)=Cov(X,2X+Y)=2Cov(X,X)+Cov(X,Y)=2Var(X)=2\cdot3=6 \end{align} The result is: \begin{align} Corr(X,Z)=\frac{6}{\sqrt{3}\sqrt{16}} \end{align}

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