conditional expectation
Let (Ω,ℱ,P) be a probability space and X:Ω→ℝ a real random variable
with E[|X|]<∞.
Conditional Expectation Given an Event
Given an event B∈ℱ such that P(B)>0, then we define the conditional expectation of X given B, denoted by E[X|B] to be
E[X|B]:=1P(B)∫BXdP. |
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 functions, Bi=X-1({wi}) and wi∈ℝ, then conditional expectation of X given B becomes
E[X|B] | = | 1P(B)∫B(∞∑i=1wi1Bi)𝑑P=1P(B)(∞∑i=1wi∫B1Bi𝑑P) | ||
= | 1P(B)(∞∑i=1wiP(Bi∩B))=∞∑i=1wiP(Bi|B), |
where P(Bi|B) is the conditional probability of Bi given B.
Conditional Expectation Given a Sigma Algebra
If 𝒟⊂ℱ is a sub σ-algebra, then the conditional expectation of X given D, denoted by E[X|𝒟] is defined as follows:
Definition
E[X|𝒟] is the function from Ω to ℝ satisfying :
-
1.
E[X|𝒟] is 𝒟-measurable
-
2.
∫AE[X|𝒟]dP=∫AXdP , for all A∈𝒟.
It can be shown, via Radon-Nikodym Theorem, that E[X|𝒟] always exists and is unique almost everywhere: any two 𝒟-measurable random variables Y,Z with
∫AY𝑑P=∫AZ𝑑P=∫AX𝑑P |
differ by a null event in 𝒟. We can in fact set up an equivalence relation on the set of all integrable 𝒟-measurable functions
satisfying condition 2 above. In this sense, E[X|𝒟] is an equivalence class
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 language
.
Suppose Y:Ω→ℝ is another random variable with E[|Y|]<∞ and let α,β∈ℝ. Then
-
1.
E[αX+βY|𝒟]=αE[X|𝒟]+βE[X|𝒟]
-
2.
E[E[X|𝒟]]=E[X]
-
3.
E[X|𝒟]=X if X is 𝒟-measurable
-
4.
E[X|𝒟]=E[X] if X is independent
(http://planetmath.org/IndependentSigmaAlgebras) of 𝒟
-
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→ℝ, we define the conditional expectation of X given Y to be the conditional expectation of X given ℱY, the sigma algebra generated by Y (http://planetmath.org/MathcalFMeasurableFunction).
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 |