conditional expectation

Let $(\Omega,\mathcal{F},P)$ be a probability space and $X\colon\Omega\to\mathbb{R}$ a real random variable with $E[|X|]<\infty$ .

Conditional Expectation Given an Event

Given an event $B\in\mathcal{F}$ 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]:=\frac{1}{P(B)}\int_{B}XdP.

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

If $X$ is discrete, then we can write $X=\sum_{i=1}^{\infty}w_{i}1_{B_{i}}$ , where $1_{B_{i}}$ are the indicator functions, $B_{i}=X^{-1}(\{w_{i}\})$ and $w_{i}\in\mathbb{R}$ , then conditional expectation of $X$ given $B$ becomes

	$\displaystyle E[X\|B]$	$\displaystyle=$	$\displaystyle\frac{1}{P(B)}\int_{B}\Big{(}\sum_{i=1}^{\infty}w_{i}1_{B_{i}}% \Big{)}dP=\frac{1}{P(B)}\Big{(}\sum_{i=1}^{\infty}w_{i}\int_{B}1_{B_{i}}dP\Big% {)}$
		$\displaystyle=$	$\displaystyle\frac{1}{P(B)}\Big{(}\sum_{i=1}^{\infty}w_{i}P(B_{i}\cap B)\Big{)% }=\sum_{i=1}^{\infty}w_{i}P(B_{i}\|B),$

where $P(B_{i}|B)$ is the conditional probability of $B_{i}$ given $B$ .

Conditional Expectation Given a Sigma Algebra

If $\mathcal{D}\subset\mathcal{F}$ is a sub $\sigma$ -algebra, then the conditional expectation of $X$ given $\mathcal{D}$ , denoted by $E[X|\mathcal{D}]$ is defined as follows $\colon$

Definition

$E[X|\mathcal{D}]$ is the function from $\Omega$ to $\mathbb{R}$ satisfying $\colon$

1.

$E[X|\mathcal{D}]$ is $\mathcal{D}$ -measurable
2.

$\displaystyle\int_{A}E[X|\mathcal{D}]dP=\int_{A}XdP$ , for all $A\in\mathcal{D}$ .

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

\displaystyle\int_{A}YdP=\int_{A}ZdP=\int_{A}XdP

differ by a null event in $\mathcal{D}$ . We can in fact set up an equivalence relation on the set of all integrable $\mathcal{D}$ -measurable functions satisfying condition 2 above. In this sense, $E[X|\mathcal{D}]$ is an equivalence class of random variables, and any two members in $E[X|\mathcal{D}]$ may qualify as conditional expectations of $X$ given $\mathcal{D}$ (they are often called versions of the conditional expectation). In practice, however, we often think of $E[X|\mathcal{D}]$ 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 $Y\colon\Omega\to\mathbb{R}$ is another random variable with $E[|Y|]<\infty$ and let $\alpha,\beta\in\mathbb{R}$ . Then

1.

$E[\alpha X+\beta Y|\mathcal{D}]=\alpha E[X|\mathcal{D}]+\beta E[X|\mathcal{D}]$
2.

$E[E[X|\mathcal{D}]]=E[X]$
3.

$E[X|\mathcal{D}]=X$ if $X$ is $\mathcal{D}$ -measurable
4.

$E[X|\mathcal{D}]=E[X]$ if $X$ is independent (http://planetmath.org/IndependentSigmaAlgebras) of $\mathcal{D}$
5.

$E[YX|\mathcal{D}]=YE[X|\mathcal{D}]$ if $Y$ is $\mathcal{D}$ -measurable

Conditional Expectation Given a Random Variable

Given any real random variable $Y:\Omega\to\mathbb{R}$ , we define the conditional expectation of $X$ given $Y$ to be the conditional expectation of $X$ given $\mathcal{F}_{Y}$ , 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