Law of total expectation

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The proposition in probability theory known as the law of total expectation,[1] the law of iterated expectations,[2] the tower rule,[3], Adam's law, and the smoothing theorem,[4] 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

Example[edit]

Suppose that 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.

Proof in the finite and countable cases[edit]

Let the random variables and , defined on the same probability space, assume a finite or countably infinite set of finite values. Assume that is defined, i.e. . If is a partition of the probability space , then

Proof.

If the series is finite, then we can switch the summations around, and the previous expression will become

If, on the other hand, the series is infinite, then its convergence cannot be conditional, due to the assumption that The series converges absolutely if both and are finite, and diverges to an infinity when either or is infinite. In both scenarios, the above summations may be exchanged without affecting the sum.

Proof in the general case[edit]

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, note that

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

Proof of partition formula[edit]

where is the indicator function of the set .

If the partition is finite, then, by linearity, the previous expression becomes

and we are done.

If, however, the partition is infinite, then we use the dominated convergence theorem to show that

Indeed, for every ,

Since every element of the set falls into a specific partition , it is straightforward to verify that the sequence pointwise-converges to . By initial assumption, . Applying the dominated convergence theorem yields the desired.

See also[edit]

References[edit]

  1. ^ Weiss, Neil A. (2005). A Course in Probability. Boston: Addison–Wesley. pp. 380–383. ISBN 0-321-18954-X.
  2. ^ "Law of Iterated Expectation | Brilliant Math & Science Wiki". brilliant.org. Retrieved 2018-03-28.
  3. ^ Rhee, Chang-han (Sep 20, 2011). "Probability and Statistics" (PDF).
  4. ^ Wolpert, Robert (November 18, 2010). "Conditional Expectation" (PDF).