Conditional Mean and Linearity The conditional mean of $Y$
  given $X$
  is:$$E[Y|X]=\int yf(y|x)dy$$
  What is the relation to the linear model? I have read somewhere that when X
  and $Y$
  are normal (their marginal distributions), then the conditional mean of $Y$
  becomes a linear function of $X$.
  How can I see this?
If X and Y are normal, then the distribution of X conditional on Y is:
$$ X | (Y = y) = N(\mu_x + \rho \frac{\sigma_x}{\sigma_y}(y - \mu_y), (1-\rho)^2 \sigma^2_x )$$ 

Therefore, $E[X|Y = y] = \mu_x + \rho \frac{\sigma_x}{\sigma_y}(y - \mu_y) = (\mu_x - \mu_y (\rho \frac{\sigma_x}{\sigma_y})) + (\rho \frac{\sigma_x}{\sigma_y})y $ which is linear in $y$.
Symmetrically:
$$E[Y|X = x] = (\mu_y - \mu_x (\rho \frac{\sigma_y}{\sigma_x})) + (\rho \frac{\sigma_y}{\sigma_x})x   $$
which is also linear in $x$