correlation of two sums of random variables

Question

Imagine two random variables $X$ and $Y$ which are correlated with $\rho = 1$.

Both have a mean of $100$ and a standard deviation of $40$. Two other random variables $U$ and $V$ are correlated at $\rho=0.8$. Both have a mean of $0$ and standard deviation of $20$.

Now, I wonder if there is a formula to compute the correlation of $A = X+U$ and $B=Y+V$?

$\text{cor}(X,U)=\text{cor}(X,V)=\text{cor}(Y,U)=\text{cor}(Y,V)=0$

Any ideas on how to compute $\text{cor}(A,B)$?

Thanks in advance

Answer 1

1) write the correlation as a ratio (covariance is on the numerator, the denominator is a product of standard deviations)

2) Write covariance as an expectation and use elementary properties of expectation to compute $\text{cov}(X+U,Y+V)$.

3) Use elementary properties of variances to deal with the terms in the denominator

http://en.wikipedia.org/wiki/Expected_value#Properties

http://en.wikipedia.org/wiki/Variance#Basic_properties