Conditional Expectation

Abstract

Let X and Y be two random variables with Y taking values in R with X taking on only countably many values. It often arises that we know already the value of X and want to calculate the expected value of Y taking into account the knowledge of X. That is, suppose we know that the event {X = j} for some value j has occurred. The expectation of Y may change given this knowledge. Indeed, if Q(Λ) = P(Λ|X = j), it makes more sense to calculate Eq{Y} than it does to calculate E _P {Y} (E _R {·} denotes expectation with respect to the Probability measure R.)