# Intuition Between Product Moment of 2 Random Variables

I am struggling to grasp the intuition behind the E[XY] of 2 Random variables that enters into the Covariance Formula:

Cov[X,Y]=E[XY]-E[X]E[Y]

I’m having a tough time connecting this equation with the information it gives you on how the 2 RV’s move around their means with respect to one another.

Even in its expanded (discrete) form this is making little sense to me as to what the math is saying:

iΣj xiyj pdf(xi,yj)] - [Σixi pdf(xi)] [Σjyj pdf(yj)]

If anyone could provide an intuitive explanation of what E[XY] means by itself, it would be greatly appreciated. Thanks to all. - SDH

\begin{align}\text{Cov}(X,Y) &= E[(X-E[X])(Y-E[Y])]\\[2ex]&= E[XY-X~E(Y)-Y~E(X)+E(X)~E(Y)]\\[2ex]&=E(XY)-E(X)~E(Y)\end{align}