Intuition behind $E(XY) = E(X) E(Y) $ for independent random variables $X,Y$ I have been wondering what's the intuition behind a well known result:  $E(XY) = E(X) E(Y) $ for independent random variables $X,Y$
I found this post: here which kinda solves the problem.
But, the explanation given there seems to be not clear enough for me.
What I think:
Without loss of generality, we know that besides independence we can assume that both random variables, $X$ and $Y$ are simple random variables, and so, it is possible to represent them as, i.e. taking X first:
$X = \sum^n_{i=1} a_i 1_{A_i}$, then compute the product $XY$ and take expectation. 
But could somebody please explain the intuition behind it to me? 
I really want to get the notion of how to understand the result of that post (which i believe is correct)
Thank you all  guys.!

It is hard to give precise answer since you are asking for intuition.
Suppose for a certain number b you will compute bX. What’s the expected value of this computation? Well, if the realization of the variable X was done independently of the choice of the number b, then your computation will produce on average b.EX. Now make b random...
