Conditional expectation extends the definition of expected value by restricting the underlying probability distribution to the information given by a conditioning event or random variable, so E(Y|X) is a random variable (a function of X) representing the best prediction of Y given knowledge of X; it obeys the same axioms as ordinary expectation (linearity, etc.) because conditional probabilities are themselves probabilities. A key subtlety is that once a value is substituted for a conditioned variable, the conditioning information cannot be discarded unless independence is established, and this notion connects to the broader theory of dependence between random variables — illustrated by paradoxes in which naive symmetry or substitution arguments produce contradictory expected values. This material belongs to probability theory, specifically the theory of conditional distributions and expectation, and generalizes the law of total probability from probabilities to expectations (iterated expectation, a.k.a. Adam's Law).
Conditional expectation extends the definition of expected value by restricting the underlying probability distribution to the information given by a conditioning event or random variable, so E(Y|X) …