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conditional expectation
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(Definition)
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Let $(\Omega,\mathcal{F},P)$ be a probability space and $X\colon \Omega \to \mathbb{R}$ a real random variable with $E[|X|]<\infty$ .
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 $$E[X | B]:=\frac{1}{P(B)}\int_B X dP.$$
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, $B_i=X^{-1}(\lbrace w_i\rbrace)$ and $w_i\in\mathbb{R}$ , then conditional expectation of $X$ given $B$ becomes \begin{eqnarray*} E[X|B]&=&\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) \\ &=& \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), \end{eqnarray*}where $P(B_i|B)$ is the conditional probability of $B_i$ given $B$ .
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$\colon$
$E[X|\mathcal{D}]$ is the function from $\Omega$ to $\mathbb{R}$ satisfying $\colon$
- $E[X|\mathcal{D}]$ is $\mathcal{D}$ -measurable
- $\displaystyle \int_{A}E[X|\mathcal{D}]dP=\int_{A}XdP$ ,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
- $E[\alpha X+\beta Y|\mathcal{D}]=\alpha E[X|\mathcal{D}]+\beta E[X|\mathcal{D}]$
- $E[E[X|\mathcal{D}]]=E[X]$
- $E[X|\mathcal{D}]=X$ if $X$ is $\mathcal{D}$ -measurable
- $E[X|\mathcal{D}]=E[X]$ if $X$ is independent of $\mathcal{D}$
- $E[YX|\mathcal{D}]=YE[X|\mathcal{D}]$ if $Y$ is $\mathcal{D}$ -measurable
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 $Y$ .
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"conditional expectation" is owned by georgiosl. [ full author list (2) ]
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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 11 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 4894 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) |
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Pending Errata and Addenda
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