Linearity of expectation versus "multiplicity of expectation" for dependent variables I'm having some difficulty understanding why we don't have (what I have chosen to call) muliplicity of expectation for dependent variables (but only for independent ones).
Can someone come up with a simple example that shows this fact?
More info:
We have $E[x+y] = E[x]+E[y]$ for both dependent and independent variables, but we only have $E[x \cdot y] = E[x] \cdot E[y]$ for independent variables.

$$
\mathbb{E}(XY) = \int xyf_{xy}(x,y)dxdy 
$$
where $f_{xy}$ is the joint distribution.
if we have an independent set of variables, then we have
$$
f_{xy}(x,y) = f_x(x)f_y(y)
$$
thus we can do this
$$
\mathbb{E}(XY) = \int xyf_{xy}(x,y)dxdy = \int xyf_x(x)f_y(y)dxdy = \left(\int xf_x(x)dx\right)\left(\int yf_y(y)dy\right) = \mathbb{E}(X) \mathbb{E}(Y) 
$$
otherwise you can not separate the integrals as shown above. so in general
$$
\mathbb{E}(XY) \neq \mathbb{E}(X) \mathbb{E}(Y) 
$$
as much as 
$$
f_{xy}(x,y) \neq f_x(x)f_y(y)
$$
