I am trying to figure out how one can deduce from the formula for Covariance:
$ Cov(X,Y) \:=E[(X-\mu_X)(Y-\mu_Y)]= E(XY)-\mu_X\mu_Y $
Correlation measures the degree of linear dependence. Does Covariance too measure the level of linear dependence between $X$ and $Y$? Does it follow from the formula/definition, given $Corr(X,Y)$ is just a standardized version of $Corr(X,Y)$?
Moreover, I am surprised that , given a set of points $(x,y)$ (uniformly-distributed) on the circle, so that we have $y=\pm \sqrt{1-x^2} $ that the correlation between $(Y,X)$ is exactly $0$. Is there someway of seeing this non-analytically? In this sense, a circle or "circular relation" is the opposite ofa linear relation in that If $Y=aX+b $ then $(X,Y)=1$ , while if $Y= \pm \sqrt{ 1- X^2}$ then $Corr(X,Y)=0$?
EDIT: I am trying to understand better just what covariance/correlation measure. So I would appreciate some input on: 1)How can we gather from the formulas for Covariance, Correlation that they measure linear dependence?
2)Is there a way of seeing non-analytically that e.g., points in a circle are uncorrelated? Thanks.