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

Question

Independent random variables $X, Y, X, U, V, W$ have variance equal to $1$. Find $ρ(S, T)$ - the correlation coefficient of random variables $S = 3X + 3Y + 2Z + U + V + W$ and $T = 9X + 3Y + 2Z + 2U + V + W$

Solution

First, I started calculating the standard deviation for $S$ and $T$

$\sigma(S) = \sqrt{3^2+3^2+2^2+1^2+1^2} = 2\sqrt{6}$

$\sigma(T) = \sqrt{9^2+3^2+2^2+2^2+1^2} = \sqrt {99}$

Then I need to calculate the covariance $\operatorname{Сov}(S,T)$.But if I start multiplying, I will get a very large expression, where nothing can be reduced. I thought to represent it in the form of a matrix, but I also did not think of how to calculate it later. Tell me please!

Answer 1

Let $S=3X+3Y+2Z+U+V+W$ and $T=S+6X+U$.

\begin{align} \operatorname{Сov}(S,T) &=\operatorname{Сov}(S,S+6X+U)\\ &= \operatorname{Сov}(S,S)+\operatorname{Сov}(S,6X)+\operatorname{Сov}(S,U)\\ &= \sigma_S^2+6\operatorname{Сov}(S,X)+\operatorname{Сov}(S,U)\\ &=\sigma_S^2+6\left(\operatorname{Сov}(3X,X)+\operatorname{Сov}(3Y,X)+\operatorname{Сov}(2Z,X)+\operatorname{Сov}(U,X)+\operatorname{Сov}(V,X)+\operatorname{Сov}(W,X)\right)\\ &+\left(\operatorname{Сov}(3X,U)+\operatorname{Сov}(3Y,U)+\operatorname{Сov}(2Z,U)+\operatorname{Сov}(U,U)+\operatorname{Сov}(V,U)+\operatorname{Сov}(W,U)\right)\\ \end{align}

$X,Y,Z,U,V,W$ are all independent. I guess it would be easy to carry this on.

I hope this helps.