Probability Theory 22 | Conditional Expectation (given random variables) [dark version]

Conditional expectation given a random variable extends the familiar idea of “averaging with information” from conditioning on an event to conditioning on another random variable. For discrete random variables, it becomes a weighted average where the weights are conditional probabilities of X’s values given that Y takes a specific value; the result is not just a number but a new random variable, written as E[X | Y]. This matters because it’s the bridge from basic conditional probability to the machinery used in stochastic processes, where the “known information” often arrives through random variables rather than fixed events.

The setup starts with the earlier definition of E[X | B] for a discrete random variable X and an event B with positive probability: it’s computed by summing over all possible values x of X, multiplying x by P(X = x | B). The new step replaces the event B with the event “Y equals y.” That yields E[X | Y = y], and by varying y, the conditional expectation becomes a function of y. Since Y itself is random (it maps outcomes ω to real numbers), composing that function with Y produces a new random variable E[X | Y] defined on the same probability space.

A concrete die example makes the construction tangible. Let X indicate whether the die result is even (X = 1 for {2,4,6}, otherwise 0). Let Y indicate whether the die result is 6 (Y = 1 only when the outcome is 6, otherwise 0). When Y = 0, the conditional expectation E[X | Y = 0] averages X over the remaining outcomes {1,2,3,4,5}. Only {2,4} make X = 1, so the conditional expectation is 2/5. When Y = 1, the only possible outcome is 6, which forces X = 1, so E[X | Y = 1] = 1. Putting both cases together shows how E[X | Y] “tracks” the uncertainty about X while accounting for what Y reveals.

For continuous random variables, the same idea generalizes by replacing sums with integrals and probability mass functions with joint densities. When X and Y are absolutely continuous and have a joint density f(x,y), the conditional expectation E[X | Y] is computed using an integral over x: essentially x multiplied by the conditional density (joint density divided by the marginal density of Y). The result again is a function of Y, hence a new random variable obtained by composing that function with Y.

Several key properties are highlighted. If X and Y are independent, conditioning on Y removes any dependence: conditional probabilities or densities factorize, and E[X | Y] reduces to the ordinary expectation E[X] (a constant random variable). A related simplification holds for products like E[X·Y | Y], where the Y term cancels, leaving E[X]. Conditioning on X itself gives E[X | X] = X. Finally, averaging the conditional expectations recovers the unconditional expectation: E(E[X | Y]) = E[X], tied to the law of total probability. These properties set up the later use of conditional expectation in stochastic processes, where conditioning on random information is routine.