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Say I have an observable z, which linearly correlated with two other variables x and y, like this:

z = ax + by

If you assume x and y are completely uncorrelated, I would think that the correlation coefficient Rx,z or Ry,z can not become one (or even close to). Is there a theoretical maximum for this correlation coefficient? And if yes, how can you prove it? And what is it? I'm guessing it would depend on a and b.

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    $\begingroup$ Consider $a = 0$. What's the correlation between $z$ and $y$? $\endgroup$
    – jbowman
    Commented Aug 6, 2015 at 17:19
  • $\begingroup$ Well, assuming b is not zero it would be one. But I'm more interested in the non-trivial cases. $\endgroup$
    – Lu Kas
    Commented Aug 6, 2015 at 17:45
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    $\begingroup$ $a=0$ isn't "trivial." If you want to exclude it, what would your answer be for $a=10^{-100}$? If you fix $a$ and $b$, then you're asking how to compute the correlation of $z$ in terms of the variances and covariances of $x$ and $y$--which is a standard formula you can readily find in many threads here. $\endgroup$
    – whuber
    Commented Aug 6, 2015 at 18:21

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