Get it from someone else but don't quite know how to answer.

  1. If $\rho_{X,Z}=0.4$, $\rho_{Y,Z}=0.3$, what is the range of $\rho_{X,Y}$? Here $\rho$ is the Pearson correlation coefficient.

  2. We run a simple linear regression (with or without an intercept, so 2 questions actually) on a dataset from $X$ to $Z$ and $Y$ to $Z$ respectively. The R-squared value for the first regression is 0.16, and 0.09 for the second one. What is the range of the R-squared value if we regress from $X$ to $Y$?

My intuition is that we can think of $Z=X+\epsilon_X$ and $Z=Y+\epsilon_Y$, with $\epsilon$ having 0 mean and some kind of variances. By considering $\epsilon_X,\epsilon_Y$ positively or negatively related we can somehow get a range. I don't have a rigorous solution though - are there any possible ways to do this?



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