The proposition in probability theory known as the law of total expectation, the law of iterated expectations (LIE), Adam's law, the tower rule, and the smoothing theorem, among other names, states that if X is a random variable whose expected value is defined, and Y is any random variable on the same probability space, then i.e., the expected value of the conditional expected value of X given Y is the same as the expected value of X. The conditional expected value, with Y a random variable, is not a simple number; it is a random variable whose value depends on the value of Y. That is, the conditional expected value of X given the event Y = y is a number and it is a function of y. If we write g(y) for the value of then the random variable is g( Y ). One special case states that if is a finite or countable partition of the sample space, then

Example

Suppose that only two factories supply light bulbs to the market. Factory X ' s bulbs work for an average of 5000 hours, whereas factory Y ' s bulbs work for an average of 4000 hours. It is known that factory X supplies 60% of the total bulbs available. What is the expected length of time that a purchased bulb will work for? Applying the law of total expectation, we have: where Thus each purchased light bulb has an expected lifetime of 4600 hours.

Informal proof

When a joint probability density function is well defined and the expectations are integrable, we write for the general case A similar derivation works for discrete distributions using summation instead of integration. For the specific case of a partition, give each cell of the partition a unique label and let the random variable Y be the function of the sample space that assigns a cell's label to each point in that cell.

Proof in the general case

Let be a probability space on which two sub σ-algebras are defined. For a random variable X on such a space, the smoothing law states that if is defined, i.e., then Proof. Since a conditional expectation is a Radon–Nikodym derivative, verifying the following two properties establishes the smoothing law: The first of these properties holds by definition of the conditional expectation. To prove the second one, so the integral is defined (not equal ). The second property thus holds since implies Corollary. In the special case when and, the smoothing law reduces to **Alternative proof for ** This is a simple consequence of the measure-theoretic definition of conditional expectation. By definition, is a \sigma(Y)-measurable random variable that satisfies for every measurable set. Taking A = \Omega proves the claim.