Equivalence of Pearson Correlation Coefficient Equations

I hope you all are doing well! I am currently trying to deepen my understanding of some equations connected to the Pearson correlation coefficient, but I am having a bit of trouble grasping how they are equivalent. Specifically, I'm looking at the following equations:

r xy = r yŷ (that is, the bivariate correlation between x and y equals the bivariate correlation between y and ŷ )
r ^2 xy =S^2ŷ /S^2y ((which says that the squared correlation between X and Y equals the ratio of the variance of the predicted scores over the variance of the observed scores on Y.)
I understand the basic idea of the Pearson correlation coefficient r and how it measures the linear relationship between two variables. However, I'm struggling to see how these equations transform and show that they are equivalent.

Could someone kindly break down these equations and show through transformations that they indeed represent the same elements respectively? Any insights, step-by-step explanations, or examples would be invaluable to my understanding.

Thank you so much in advance!

  • Savionf Savionf

    This is an interesting question, but providing a thorough, step-by-step explanation to demonstrate the equivalence of these equations can be quite time-consuming, and you should offer a fair bounty to incentivize users to spend their valuable time and expertise on this.

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