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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 $Var[Y=y|X]\neq Var[Y|X=x]$.

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    To echo @DilipSarwate's puzzlement, let me mention that as soon as$X$and ϵ are independent and not degenerate and β1 is not zero, one can be sure that$Y$takes at least 3 values, hence Y is not Bernoulli. This question needs some heavy rewriting.2012-10-31

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