The Population

We will use the fact that the outcome, $Y$, and the potential outcome $\tilde{Y}i(1)$ are equal almost surely with respect to the probability measure $\mathbb{P}{D=1}$.

$$ Y \overset{\mathbb{P}_{D=1} \ a.s}{=} \tilde{Y}(1) $$

The Sample

In the sample

$$ \begin{align*}= &\mathbb{E}\Big[\frac{1}{n_1}\sum Y_iD_i - \frac{1}{n_o}\sum Y_i(1-D_i)\Big] \\ =& \frac{1}{n_1}\sum \mathbb{E}[Y_iD_i] - \frac{1}{n_o}\sum \mathbb{E}[Y_i(1-D_i)] \\\end{align*} $$