Pearson Correlation Coefficient: How Did He Derive It?

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The discussion centers on the derivation of the Pearson correlation coefficient and its purpose in reflecting the linear relationship between two variables. Participants question whether the coefficient is merely a definitional construct or if there is a deeper mathematical foundation behind it. The formula involves discrete independent and dependent variables, along with their respective standard deviations. Clarifications about the mathematical origins and implications of the formula are sought. Understanding Pearson's derivation enhances comprehension of its application in statistical analysis.
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Now how exactly did Pearson derive his correlation coefficient?
 
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Is it simply a definition which reflects the linear relationship between two variables?
 
gimmytang said:
Is it simply a definition which reflects the linear relationship between two variables?

That's what it's supposed to do; I'm just wondering how did Pearson derived or came up with the formula for it, which I have in the attached GIF image file, (correlation.gif)

Where x=discrete independent variable, y=discrete dependent variable,
and Sx=standard deviation of x-set, and Sy=standard deviation of y-set
 

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Thread 'Formal derivation of statement from Peano Arithmetic system'

I was reading a Bachelor thesis on Peano Arithmetic (PA). PA has the following axioms (not including the induction schema): $$\begin{align} & (A1) ~~~~ \forall x \neg (x + 1 = 0) \nonumber \\ & (A2) ~~~~ \forall xy (x + 1 =y + 1 \to x = y) \nonumber \\ & (A3) ~~~~ \forall x (x + 0 = x) \nonumber \\ & (A4) ~~~~ \forall xy (x + (y +1) = (x + y ) + 1) \nonumber \\ & (A5) ~~~~ \forall x (x \cdot 0 = 0) \nonumber \\ & (A6) ~~~~ \forall xy (x \cdot (y + 1) = (x \cdot y) + x) \nonumber...
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