↑ arvau

Wikipedia / 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 property of conditional expectation, is a random variable is any random variable on the same probability space, then

i.e., the expected value of the conditional expected value.

The conditional expected value given the event.

is a finite or countable partition of the sample space, then

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;

  • ;

  • ;

  • ;

  • .

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 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

be a probability space on which two sub σ-algebras, 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,

).

Corollary.

This is a simple consequence of the measure-theoretic definition of conditional expectation proves the claim.

See also

References

Further reading