\begin{equation} cov(x,y) = E[(X-E[X])(Y-E[Y])] = E[XY]-E[X]E[Y] \end{equation}

(the derivation comes from FOIling the two terms and applying properties of expectation.

we want to consider: if a point goes way beyond its expectation, does the corresponding point change for another?

\begin{equation} (x-E[x])(y-E[y]) \end{equation}

if both points are varying .

Instead of using this unbounded value, we sometimes use a normalized value named correlation:

\begin{equation} \rho(X,Y) = \frac{Cov(X,Y)}{\sqrt{Var(X)Var(Y)}} \end{equation}