Let $Y = \beta_0 + \beta_1 X + \epsilon$ where $Y$ is a binary random variable. What is $Var[Y|X]$?
So since $Y$ takes on only 1 or 0, $E[Y|X] = \frac{1}{2}$
and $Var[Y|X] = Var[Y=1|X] + Var[Y=0|X]$, right? I'm trying to figure out how to go from here since I usually see $Var[Y|X=x]$ and I'm pretty sure $Var[Y=y|X]\neq Var[Y|X=x]$.