conditional expectation

Let (Ω,,P) be a probability spaceMathworldPlanetmath and X:Ω a real random variableMathworldPlanetmath with E[|X|]<.

Conditional Expectation Given an Event

Given an event B such that P(B)>0, then we define the conditional expectation of X given B, denoted by E[X|B] to be


When P(B)=0, E[X|B] is sometimes defaulted to 0.

If X is discrete, then we can write X=i=1wi1Bi, where 1Bi are the indicator functionsPlanetmathPlanetmath, Bi=X-1({wi}) and wi, then conditional expectation of X given B becomes

E[X|B] = 1P(B)B(i=1wi1Bi)𝑑P=1P(B)(i=1wiB1Bi𝑑P)
= 1P(B)(i=1wiP(BiB))=i=1wiP(Bi|B),

where P(Bi|B) is the conditional probabilityMathworldPlanetmath of Bi given B.

Conditional Expectation Given a Sigma Algebra

If 𝒟 is a sub σ-algebra, then the conditional expectation of X given D, denoted by E[X|𝒟] is defined as follows:


E[X|𝒟] is the function from Ω to satisfying :

  1. 1.

    E[X|𝒟] is 𝒟-measurable

  2. 2.

    AE[X|𝒟]dP=AXdP , for  all  A𝒟.

It can be shown, via Radon-Nikodym TheoremMathworldPlanetmath, that E[X|𝒟] always exists and is unique almost everywhere: any two 𝒟-measurable random variables Y,Z with


differ by a null event in 𝒟. We can in fact set up an equivalence relationMathworldPlanetmath on the set of all integrable 𝒟-measurable functionsMathworldPlanetmath satisfying condition 2 above. In this sense, E[X|𝒟] is an equivalence classMathworldPlanetmath of random variables, and any two members in E[X|𝒟] may qualify as conditional expectations of X given 𝒟 (they are often called versions of the conditional expectation). In practice, however, we often think of E[X|𝒟] as a function rather than a set of functions. As long as we realize that any two such functions are equal almost surely, we may blur such differences and abuse the languagePlanetmathPlanetmath.

Suppose Y:Ω is another random variable with E[|Y|]< and let α,β. Then

  1. 1.


  2. 2.


  3. 3.

    E[X|𝒟]=X if X is 𝒟-measurable

  4. 4.

    E[X|𝒟]=E[X] if X is independentPlanetmathPlanetmath ( of 𝒟

  5. 5.

    E[YX|𝒟]=YE[X|𝒟] if Y is 𝒟-measurable

Conditional Expectation Given a Random Variable

Given any real random variable Y:Ω, we define the conditional expectation of X given Y to be the conditional expectation of X given Y, the sigma algebra generated by Y (

Title conditional expectation
Canonical name ConditionalExpectation
Date of creation 2013-03-22 15:43:45
Last modified on 2013-03-22 15:43:45
Owner georgiosl (7242)
Last modified by georgiosl (7242)
Numerical id 13
Author georgiosl (7242)
Entry type Definition
Classification msc 60-00
Classification msc 60A10
Related topic ConditionalProbability
Related topic ConditionalExpectationUnderChangeOfMeasure
Related topic ConditionalExpectationsAreUniformlyIntegrable