Under which conditions does the linear least square regression equal conditional mathematical expectations?
In other words: under which conditions $$\mathbb{E}[z_1|z_2=a] = \mu_1 + \beta(a - \mu_2)$$ (with $\mu_1$ and $\mu_2$ known constants).
This is trivially true under the assumption of joint normality of the vector $(z_1, z_2)$, but holds also in more general cases (e.g. multivariate Laplace distribution). Which are those other cases?