Conditional expectation is a fundamental concept in probability theory and statistics that refers to the expected value of a random variable given that certain conditions or information are known. It captures the idea of updating our expectations based on additional information. Formally, if \( X \) is a random variable and \( Y \) is another random variable (or an event), the conditional expectation of \( X \) given \( Y \) is denoted as \( \mathbb{E}[X | Y] \).
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For an integrable random variable and a sub-sigma-algebra , the conditional expectation is the almost-everywhere unique -measurable integrable random variable satisfyingfor every .