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

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

• Calculate $cov(X,Z)$, then normalise Jun 8 at 8:03

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}