4.4.1. Recuritment to a Program

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$$ ETT=E[Y_1-Y_0|X=1] $$

$$ P(Y_x=y|x\prime)=\sum_zP(Y_x=y|z,x\prime)P(z|x\prime) $$

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$$ P(Y_x=y|X=x\prime)=\sum_zP(Y=y|X=x,Z=z)P(Z=z|X=x\prime) $$

$$ P(Y=y|do(X=x))=\sum P(Y=y|X=x, Z=z)P(Z=z) $$

$$ ETT=E[Y_1-Y_0|X=1] $$

$$ =E[Y_1|X=1]-E[Y_0|X=1] $$