I have two functions $w_0 = \mu_0 + \epsilon_0$ and $w_1 = \mu_1 + \epsilon_1$ Where $\epsilon_0 \sim N(0, \sigma_0^2)$ and $\epsilon_1 \sim N(0, \sigma_1^2)$
Assuming that $\epsilon_1 - \epsilon_0 = v$
What is $E(\epsilon_0|v)$ ?
I have seen it solved as $\frac{\sigma_{0v}}{\sigma_v^2}v$
But it looks like it should be $\frac{\sigma_{0}^2}{\sigma_0^2 + \sigma_1^2}v$
Or could you go from one solution to the other?