Intuition Between Product Moment of 2 Random Variables

Question

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:

[Σ_iΣ_j x_iy_j pdf(x_i,y_j)] - [Σ_ix_i pdf(x_i)] [Σ_jy_j pdf(y_j)]

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

Answer 1

If you want intuition about the covariance representing "how the two random variables move around their means with respect to one another," it is better to use the following different (but equivalent) formula.

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