How to relate the formula of covariance to the interpretation that it measures how one variable changes with respect to another?

Question

I understand the covariance measures the tendency of how one variable changes with respect to another, i.e. if one variable increases, whether it is likely for another to increase. And the formula for covariance is $E[(X-\mu_x)(Y-\mu_y)]$. However, I cannot see why this formula relates to that interpretation. Does anyone has a good explanation or this?

Answer 1

How about this,

$(X-\mu_X)(Y-\mu_Y)>0$ means both $X$ and $Y$ are above their respective mean values or below theirs together, but when $(X-\mu_X)(Y-\mu_Y)<0$ it means one of them is below its mean value and another is above its own, and by taking expectation, it means we calculate the 'general' behavior of $X$ and $Y$,

if $E[(X-\mu_X)(Y-\mu_Y)]>0$ then we can interpret that in 'general' $X$ and $Y$ are above (or below) their mean values together more often than when $X$ and $Y$ are in the opposite position, and we can conclude their relation is positive,

but when $E[(X-\mu_X)(Y-\mu_Y)]<0$ we can conclude in general that $X$ and $Y$ are in the opposite position regarding their respective mean values (negative relation)

Similar for $E[(X-\mu_X)(Y-\mu_Y)]=0$

is it clear enough?