Determining the convergence and correlation coefficient between two random variables which are sums of other random variables

Question

Random variables $X_1,\:...,\:X_n$ are independent and have the same variance $\sigma ^2$. Let $U=3X_1+X_2+...+X_n$ and $V=X_1+X_2+...+X_{n-1}+2X_n$. Determine the correlation coefficient between U and V

How should I do this? Note: We did not have the linearity of the covariance yet, we only know that we can only:

$Var\left(Xi\:+\:Xj\right)\:=\:Var\left(Xi\right)\:+\:Var\left(Xj\right)\:+\:2Cov\left(Xi,Xj\right)$

$Cov\left(Xi,Xj\right)\:=\:E\left(XiXj\right)\:−\:E\left(Xi\right)E\left(Xj\right)$

$Cov(aXi + b, cXj + d) = ac Cov(Xi,Xj)$

If Xi and Xj independent => Cov(Xi,Xj) = 0

$Var\left(X\right)\:=\:E\left[\left(X\:−\:EX\right)\left(X\:−\:EX\right)^T\right]$

That's it, can we still do this exercise with this knowledge? (we know how to calculate EX etc)

Answer 1

Hint: Let $m_i=EX_i$. We have $E(\sum\limits_{i=1}^{n} a_iX_i)(\sum\limits_{j=1}^{n} b_jX_j)=\sum_{i \neq j} a_ib_jm_im_j+\sum\limits_{k=1}^{n}a_ib_i(\sigma^{2}+m_i^{2})$.