# 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. 1.

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

2. 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. 1.

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

2. 2.

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

3. 3.

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

4. 4.

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

5. 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 ConditionalExpectation 2013-03-22 15:43:45 2013-03-22 15:43:45 georgiosl (7242) georgiosl (7242) 13 georgiosl (7242) Definition msc 60-00 msc 60A10 ConditionalProbability ConditionalExpectationUnderChangeOfMeasure ConditionalExpectationsAreUniformlyIntegrable