# Calculation of the correlation coefficient of these quantities S=3X+3Y+2Z+U+V+W and T=9X+3Y+2Z+2U+V+W

Independent random variables $X,Y,X,U,V,W$ have variance equal to 1. Find $\rho \left(S,T\right)$ - the correlation coefficient of random variables $S=3X+3Y+2Z+U+V+W$ and $T=9X+3Y+2Z+2U+V+W$
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Step 1
Let $S=3X+3Y+2Z+U+V+W$ and $T=S+6X+U$ .
$\begin{array}{rl}С\mathrm{o}\mathrm{v}\left(S,T\right)& =С\mathrm{o}\mathrm{v}\left(S,S+6X+U\right)\\ & =С\mathrm{o}\mathrm{v}\left(S,S\right)+С\mathrm{o}\mathrm{v}\left(S,6X\right)+С\mathrm{o}\mathrm{v}\left(S,U\right)\\ & ={\sigma }_{S}^{2}+6С\mathrm{o}\mathrm{v}\left(S,X\right)+С\mathrm{o}\mathrm{v}\left(S,U\right)\\ & ={\sigma }_{S}^{2}+6\left(С\mathrm{o}\mathrm{v}\left(3X,X\right)+С\mathrm{o}\mathrm{v}\left(3Y,X\right)+С\mathrm{o}\mathrm{v}\left(2Z,X\right)+С\mathrm{o}\mathrm{v}\left(U,X\right)+С\mathrm{o}\mathrm{v}\left(V,X\right)+С\mathrm{o}\mathrm{v}\left(W,X\right)\right)\\ & +\left(С\mathrm{o}\mathrm{v}\left(3X,U\right)+С\mathrm{o}\mathrm{v}\left(3Y,U\right)+С\mathrm{o}\mathrm{v}\left(2Z,U\right)+С\mathrm{o}\mathrm{v}\left(U,U\right)+С\mathrm{o}\mathrm{v}\left(V,U\right)+С\mathrm{o}\mathrm{v}\left(W,U\right)\right)\end{array}$
$X,Y,Z,U,V,W$ are all independent. I guess it would be easy to carry this on.