Given a very simple linear projection model $Y=X\beta+e$ with $E(e|X)=0$ and $X$ a scalar. Notice that this is a simple linear model with no intercept.
Then $\beta = E(XY)/E(X^2)$ from the least squares formula.
On the other hand, if I take expectation on both sides of the original model, I get $E(Y)=\beta E(X)+E(e)=\beta E(X)$. Therefore, $\beta=E(Y)/E(X)$. (Of course, suppose $E(X)\neq 0$.)
I just don't see how the two quantities could be identical. Did I do something wrong?