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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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There is something wrong here. Is $X$ a random variable? Are $\beta_0$ and $\beta_1$ constants? Is $\epsilon$ an error term that should be modeled as a random variable? –  Dilip Sarwate Oct 28 '12 at 20:43
    
Yes, X is a random variable and the beta coefficients are constants, and $\epsilon$ is indeed the error term –  Emir Oct 28 '12 at 22:59
    
looks like I found the model en.wikipedia.org/wiki/Linear_probability_model –  Emir Oct 28 '12 at 23:06
    
So, $Y$ is a Bernoulli random variable taking on values $0$ and $1$ only. What, if anything is known about $X$? –  Dilip Sarwate Oct 29 '12 at 2:42
    
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. –  Did Oct 31 '12 at 7:54

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