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The covariance between X and Y measures how "strongly related" X and Y are and is given by
Cov(X,Y) = E[(X-m_X)(Y-m_Y)]. Thus

Cov(X,Y) =

A simpler formula for covariance is
Cov(X,Y) = E[X× Y] - m_X×m_Y

If Cov(X,Y)>0, then X tends to be large if Y is large, & small if Y is small.

Conversely, if Cov(X,Y)<0, then X tends to be small if Y is large, & large if Y is small.

Covariance depends on the units used to measure X and Y. A better indicator of the relationship between X and Y is the correlation coefficient of X and Y which is given by r_X,Y (or more commonly, r)

Corr(X,Y) =

The value of r is always between -1 and +1.

Note: If X and Y are independent, then E[XY]=E[X]× E[Y] = m_Xm_Y so that Cov(X,Y) and Corr(X,Y) =r_X,Y = 0
(i.e., X and Y are uncorrelated).

HOWEVER, if r_X,Y=0 it DOES NOT mean that X and Y are independent !
Thus, two independent variables are uncorrelated, but two uncorrelated variables are not necessarily independent.

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