# Theoretically maximum correlation coefficient between multiple correlated variables

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.

• Consider $a = 0$. What's the correlation between $z$ and $y$? Commented Aug 6, 2015 at 17:19
• Well, assuming b is not zero it would be one. But I'm more interested in the non-trivial cases. Commented Aug 6, 2015 at 17:45
• $a=0$ 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.
– whuber
Commented Aug 6, 2015 at 18:21