I do realize that there were several questions regarding the expected value of the product of two discrete variables. However, I have not found any natural derivation of the covariance terms that appear in the expectation of the product. Hence,
Out of mathematical curiosity...
We can define the expected value of the product of discrete random variables $X,Y:\Omega\rightarrow \mathbb{R}$ as the following sum (where summation is over the elements in the image of the variable in question):
$$\mathbb{E}(XY)=\sum\limits_{x\in\text{Im}\left(X\right)}\sum\limits_{y\in\text{Im}\left(Y\right)}\mathbb{P}(X=x\wedge Y=y)\cdot xy$$
Then if the events are independent, the situation is simplified because the probability of the intersection of two sets would be just the probability of the product of the individual sets. My question is, is there any way to go from this to the form where the covariance terms pop up naturally in the process? Is there a way to turn this sum into something that resembles "expectation of the product as if the variables were independent plus the corrective term" i.e. $\mathbb{E}(X)\mathbb{E}(Y)+\text{cov}(X,Y)$?