Conditional expectation of a random variable Y given a random variable X, denoted E(Y|X), is a function of X representing the best mean-squared-error prediction of Y using the information contained in X; its key properties include taking out what is known (factoring out functions of the conditioning variable), reduction to unconditional expectation under independence, and the tower property (iterated expectation, "Adam's Law"), which states that the expectation of the conditional expectation equals the unconditional expectation. A companion result, the law of total variance ("Eve's Law"), decomposes the total variance of Y into the expectation of the conditional variance (within-group variability) plus the variance of the conditional expectation (between-group variability). These are foundational tools in probability theory for reducing unconditional calculations to more tractable conditional ones, generalizing the law of total probability, and geometrically interpretable as orthogonal projection in an inner-product space of random variables.
Conditional expectation of a random variable Y given a random variable X, denoted E(Y|X), is a function of X representing the best mean-squared-error prediction of Y using the information contained i…