Law of total expectation

Law of total expectation
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 is a random variable whose expected value is defined, and is any random variable on the same probability space, then
i.e., the expected value of the conditional expected value of given is the same as the expected value of .
One special case states that if is a finite or countable partition of the sample space, then
Note: The conditional expected value E(X | Y), 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 E(X | Y = y) then the random variable E(X | Y) is g(Y).
Example
Suppose that only two factories supply light bulbs to the market. Factory 's bulbs work for an average of 5000 hours, whereas factory 's bulbs work for an average of 4000 hours. It is known that factory 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
is the expected life of the bulb;
is the probability that the purchased bulb was manufactured by factory ;
is the probability that the purchased bulb was manufactured by factory ;
is the expected lifetime of a bulb manufactured by ;
is the expected lifetime of a bulb manufactured by .
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 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:
-measurable
for all
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 -measurable random variable that satisfies
for every measurable set . Taking proves the claim.
See also
The fundamental theorem of poker for one practical application.
Law of total probability
Law of total variance
Law of total covariance
Law of total cumulance
Product distribution#expectation (application of the Law for proving that the product expectation is the product of expectations)
References
(Theorem 34.4)
Christopher Sims, "Notes on Random Variables, Expectations, Probability Densities, and Martingales", especially equations (16) through (18)
Algebra of random variables
Theory of probability distributions
Statistical laws