PlanetMath: conditional expectation

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conditional expectation

(Definition)

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

Conditional Expectation Given an Event

Given an event $B\in \mathcal{F}$ such that

, then we define the conditional expectation of

given

, denoted by $E[X \vert B]$ to be

$\displaystyle E[X \vert B]:=\frac{1}{P(B)}\int_B X dP.$

When , $E[X\vert B]$ is sometimes defaulted to 0.

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

$\displaystyle E[X\vert B]$	$\displaystyle =$	$\displaystyle \frac{1}{P(B)}\int_B \Big( \sum_{i=1}^{\infty}w_i 1_{B_i} \Big) dP = \frac{1}{P(B)} \Big( \sum_{i=1}^{\infty}w_i \int_B 1_{B_i} dP\Big)$
	$\displaystyle =$	$\displaystyle \frac{1}{P(B)} \Big( \sum_{i=1}^{\infty}w_i P(B_i\cap B) \Big) = \sum_{i=1}^{\infty}w_i P(B_i\vert B),$

where $P(B_i\vert B)$ is the conditional probability of

given

Conditional Expectation Given a Sigma Algebra

If $\mathcal{D} \subset \mathcal{F}$ is a sub $\sigma$ -algebra, then the conditional expectation of

given $\mathcal{D}$ , denoted by $E[X\vert\mathcal{D}]$ is defined as follows $\colon$

Definition

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

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

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

with

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

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, $E[X\vert\mathcal{D}]$ is an equivalence class of random variables, and any two members in $E[X\vert\mathcal{D}]$ may qualify as conditional expectations of

given $\mathcal{D}$ (they are often called versions of the conditional expectation). In practice, however, we often think of $E[X\vert\mathcal{D}]$ 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 $E[\vert Y\vert]<\infty$ and let $\alpha,\beta \in \mathbb{R}$ . Then

$E[\alpha X+\beta Y\vert\mathcal{D}]=\alpha E[X\vert\mathcal{D}]+\beta E[X\vert\mathcal{D}]$
$E[E[X\vert\mathcal{D}]]=E[X]$
$E[X\vert\mathcal{D}]=X$ if is $\mathcal{D}$ -measurable
$E[X\vert\mathcal{D}]=E[X]$ if is independent of $\mathcal{D}$
$E[YX\vert\mathcal{D}]=YE[X\vert\mathcal{D}]$ if 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

given

to be the conditional expectation of

given $\mathcal{F}_Y$ , the sigma algebra generated by

"conditional expectation" is owned by georgiosl. [ full author list (2) ]

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Attachments:: conditional expectation under change of measure (Derivation) by stevecheng

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Cross-references: language, differences, equivalence class, equivalence relation, null, almost everywhere, Radon-Nikodym theorem, function, conditional probability, indicator functions, discrete, event, random variable, real, probability space
There are 6 references to this entry.

This is version 10 of conditional expectation, born on 2006-03-04, modified 2007-01-29.
Object id is 7679, canonical name is ConditionalExpectation.
Accessed 2864 times total.

Classification:

AMS MSC:	60-00 (Probability theory and stochastic processes :: General reference works )
	60A10 (Probability theory and stochastic processes :: Foundations of probability theory :: Probabilistic measure theory)

Pending Errata and Addenda

None.[ View all 2 ]

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