Let $A$ denote an event and $X$ denote a random variable. Is it true that $$\mathbb{P}[A]\cdot\mathbb{E}[X|A] = \mathbb{E}\big[X\cdot\mathbf{1}[A]\big],$$ where $\mathbf{1}[A]$ is an indicator function that event $A$ happens? We assume $\mathbb{P}[A] > 0$.
2026-04-01 23:28:04.1775086084
$\mathbb{P}[A]\cdot\mathbb{E}[X|A] = \mathbb{E}\big[X\cdot\mathbf{1}[A]\big]$?
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