Consider two real valued random variables $X,Z$ and a dummy variable $D$. Is it true that $$ E(XD|Z)=E(X|Z, D=1)Pr(D=1|Z) $$ ? Why?
2026-04-07 21:20:59.1775596859
Show $E(XD|Z)=E(X|Z, D=1)Pr(D=1|Z)$
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A dummy variable is also known as an indicator variable. It takes on boolean values ($\{0, 1\}$) in response to some specified criteria.
Then by the Law of Iterated Expectation:
$\begin{align}\mathsf E(XD\mid Z) ~=~& \mathsf E(\mathsf E(XD\mid Z, D)\mid Z) \\[1ex] ~=~& \mathsf E(X\cdot 0\mid Z, D=0)~\mathsf P(D=0\mid Z)+\mathsf E(X\cdot 1\mid Z, D=1)~\mathsf P(D=1\mid Z) \\[1ex] ~=~& \mathsf E(X \mid Z, D=1)~\mathsf P(D=1\mid Z) \end{align}$