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.!

We know that
$E[XY]=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}xyf_{XY}(x,y)dxdy$
If the RVs are independent then
$E[XY]=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}xyf_{X}(x)f_{Y}(y)dxdy \\
\qquad \quad=\int_{-\infty}^{\infty}xf_{X}(x)dx\int_{-\infty}^{\infty}yf_{Y}(y)dx \\
\qquad \quad = E[X]E[Y]$
For the two jointly distributed Gaussian $X,Y$ RVs it means uncorrelatedness $(\rho_{xy}=0,\Sigma = \mathrm{diag}\{\sigma_{x}^2,\sigma_{y}^2\})$ $(f_{XY}(x,y)=f_{X}(x)f_{Y}(y))$ implying independence (Not true for other densities). But it tells us that one variable does not contain information about the other random variable.
