# Correlation between combinations of uncorrelated variables

If I have uncorrelated variables X,Y,Z how do I find the correlation between (X+Y) & (Y+Z)?

Ps. It's a theory question: ie, I can't compute the correlation from observed instances of X,Y & Z. What I can do is assume these variables have a standard deviation of a,b & c, and figure out the correlation of M & N (where M=X+Y & N=Y+Z) from there.

• Coul you specify in your question what you mean by "and so on"? + maybe you could first simply compute the covariance between $X+Y$ and $Y+Z$ and see where it goes... – Vincent Guillemot Jun 16 '15 at 15:38
• Edited the question. Essentially, I need to know how to compute covariance between (X+Y) & (Y+Z) given sd of X,Y & Z. – Retz Jun 16 '15 at 15:53
• Hint: Covariance is a bilinear function meaning that \begin{align}\operatorname{cov}(aW+bX,cY+dZ) &= (ac)\cdot\operatorname{cov}(W,Y)+(bd)\cdot\operatorname{cov}(X,Z)\\& ~~+(ad)\cdot\operatorname{cov}(W,Z)+(bc)\cdot\operatorname{cov}(X,Y)\end{align} unless, of course, those $\&$'s mean something else. – Dilip Sarwate Jun 16 '15 at 16:00
• Ok, I did not mean "compute" with a calculator, I meant: write down the definition of the correlation, or maybe first of the covariance, expand and simplify the terms that go away thanks to your hypotheses, and see what is left. – Vincent Guillemot Jun 16 '15 at 16:15
• In other words: where are you stuck? – Vincent Guillemot Jun 16 '15 at 16:16

## 1 Answer

Use bilinearity and calculate $$\DeclareMathOperator{\cov}{\mathbb{C}ov} \DeclareMathOperator{\var}{\mathbb{V}ar} \DeclareMathOperator{\cor}{corr} \cov(X+Y,Y+Z) =\cov(X,Y)+\cov(X,Z)+\cov(Y,Y)+\cov(Y,Z) =0+0+\var(Y)+0$$ so $$\cor(X+Y,Y+Z)=\frac{\cov(X+Y,Y+Z)}{\sqrt{\var(X+Y) \var(Y+Z)}} = \\ \frac{\var(Y)}{\sqrt{(\var(X)+\var(Y))(\var(Y)+\var(Z))}}$$ and you can take it from there.