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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

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    $\begingroup$ What does "cor(xu,xv,yu,yv)" mean? $\endgroup$
    – Glen_b
    Jun 27, 2014 at 10:16
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    $\begingroup$ Since this is routine book-work, please add the self-study tag and read its tag-wiki info, altering your question if necessary. $\endgroup$
    – Glen_b
    Jun 27, 2014 at 10:17
  • $\begingroup$ thanks, didn't know that. it means that cor(x,u) cor(x,v) and so on is zero $\endgroup$
    – beginneR
    Jun 27, 2014 at 10:44
  • $\begingroup$ Just for a numerical check: the correlation between $A$ and $B$ is $0.92$. $\endgroup$ Jun 27, 2014 at 11:18

1 Answer 1

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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

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