Can someone give me a formal rigorous proof of the following equation?
$$\frac{E\{X \cdot I(T=1) \}}{\Pr(T=1)}= E(X|T=1)$$
Many thanks!
2026-04-12 03:30:40.1775964640
Conditional probability and indicator function
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\begin{align} \mathsf{E}(X \cdot \mathbb{I}(T = 1)) &= \sum_t \mathsf{E}(X \cdot \mathbb{I}(T = 1) \mid T = t)\cdot\Pr(T = t) \\ &= \mathsf{E}(X \cdot \mathbb{I}(T = 1) \mid T = 1) \cdot \Pr(T = 1) \\ &= \mathsf{E}(X \mid T = 1) \cdot \Pr(T = 1) \end{align} where the first equality is by the law of total expectation.