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I understand that correlation coefficient can only capture the linear dependence.
I have just read a book saying that:

“As a normalized covariance, the correlation coefficient captures only one particular aspect of dependence: The strength of linear dependence between the underlying random variables.”

I really do not understand: Why does normalized covariance lead the correlation coefficient to only captures the linear dependency structures? So, is that mean, without normalization, we can measure non-linear dependency?!

Peter Flom
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Mary
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  • @ttnphns Thank you for comment. My question is why (because the normalized covariance) the correlation coefficient captures only one linear dependence. – Mary Jun 26 '19 at 09:45
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    Mary, my answer in the linked thread perhaps answers it (see e.g. the very last point): normalizarion of variances to unit makes two variances equal, and under this condition cov (now it is corr) will be maximal when (for centered variables) $X_i=Y_i$ (linearity), so we might say that corr measures how much close the values of the two variables to this relation. – ttnphns Jun 26 '19 at 10:04

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In fact, the correlation coefficient is a normalized covariance. Because $Corr(X,Y) = \frac{Cov(X,Y)}{\sqrt{V(X)V(Y)}}$.

As covariance mesure the strength of linear dependence between the underlying random variables, it is the same for the correlation coefficient which is just a normalization of covariance.

Abdoul Haki
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  • It is not true that `covariance measure the strength of [specifically] linear dependence`. Correlation - is true, and this is _because_ it is normalized. – ttnphns Jun 26 '19 at 09:29
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    Correlation and covariance measure exactly the same thing. The only difference between them is that correlation is normalized then vary between -1 and 1. After that, Covariance is influenced by the unit and the magnitude of the variables X and Y and it is not the cas of Correlation. – Abdoul Haki Jun 26 '19 at 09:41
  • This is your stance, ok. But you may want to read my answer in the link provided to the question – ttnphns Jun 26 '19 at 09:44
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    Covariance measure _covariation_ which depends _both_ on linearity strength and on variances. Therefore I wouldn't rush to say covariance and correlstion coefficients "measure exactly the same thing". Which "exactly the same thing" then - one should clarify/define. – ttnphns Jun 26 '19 at 09:48