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The first equality comes from v ar u i x i e u 2 i x

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The first equality comes from V ar ( u i | X i ) = E ( u 2 i | X i ) - E ( u i | X i ) 2 and the assumption E ( u i | X i ) = 0 while the second equality comes from the fact that given X i (i.e., if we know X i ), the variation of Y i only comes from the variation of u i . Therefore, we only need to find P ( Y i = 1 | X i ) and P ( Y i = 0 | X i ) to calculate E ( u 2 i | X i ). Let us take the conditional expectations on both sides of the model. From the model Y i = β 0 + β 1 X i + u i , we get E ( Y i | X i ) = E ( β 0 + β 1 X i + u i | X i ) = E ( β 0 | X i ) + E ( β 1 X i | X i ) + E ( u i | X i ) = β 0 + β 1 X i . Here we exploited the properties of conditional expectations (which should be familiar) and the assumption E ( u i | X i ) = 0. On the other hand, since Y i is a Bernoulli random variable given X i , E ( Y i | X i ) = 1 · P ( Y i = 1 | X i ) + 0 · P ( Y i = 0 | X i ) = P ( Y i = 1 | X i ). This combined with the above result yields, P ( Y i = 1 | X i ) = β 0 + β 1 X i P ( Y i = 0 | X i ) = 1 - P ( Y i = 1 | X i ) = 1 - β 0 - β 1 X i . The conditional variance of Y i given X i is then obtained from the product of P ( Y i = 1 | X i ) and P ( Y i = 0 | X i ): V ar ( Y i | X i ) = ( β 0 + β 1 X i )(1 - β 0 - β 1 X i ). (If Z is a random variable which is one with probability p and zero with probability q = 1 - p (i.e., Z is a Bernoulli random variable), then E ( Z ) = p and V ar ( Z ) = p · q . Likewise, the variance of Y i given X i is p · q where p is P ( Y i = 1 | X i ) and q is P ( Y i = 0 | X i )).
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