Conditional Expectation

NAVIGATION

Home

Research

Bookshelf

Garden

FIND ME ON

GitHub

Home

Research

Bookshelf

Garden

Conditional Expectation

Definition (Conditional Expectation)

Let $(\Omega,\mathcal{F},P)$ be a probability space. Let $X$ be integrable RV. Let $\mathcal{G}\subset \mathcal{F}$ be a sub-σ-algebra. $E[X|\mathcal{G}]$ is called the conditional expectation given $\mathcal{G}$ if

$E[X|\mathcal{G}]$ is $\mathcal{G}$ -measurable (i.e. a measurable function on $\mathcal{G}$ )
$\forall A\in\mathcal{G}$ $\int\limits _{A}E[X|\mathcal{G}] \, dP=\int\limits _{A}X \, dP$

Proposition

$E[X|\mathcal{G}]=X\text{ a.s.}$ $\forall \mathcal{G}\subset \mathcal{F}$ .

Theorem (Law of Total Expectation)

Let $X$ be an integrable RV and $\mathcal{G}\subset \mathcal{F}$ sub-σ-algebra then $E[E[X|\mathcal{G}]]=E[X]$

Theorem (Iterated Expectation)

Let $X$ be an integrable RV, let $\mathcal{G}_{1}\subset \mathcal{G}_{2}\subset \mathcal{F}$ sub-σ-algebras. Then $E[E[X|\mathcal{G_{2}}]|\mathcal{G}_{1}]=E[X|\mathcal{G_{1}}]\text{ a.s.}$

Proposition

Let $X\in\mathscr{L}^{1}(\Omega,\mathcal{F},P)$ be an integrable RV and $Y$ an RV on $(\Omega,\mathcal{F},P)$ such that $XY\in\mathscr{L}^{1}(\Omega,\mathcal{F},P)$ . Let $\mathcal{G}\subset \mathcal{F}$ be a sub-σ-algebra. Then, $Y \ \mathcal{G}\text{-measurable}\implies E[XY|\mathcal{G}]=YE[X|\mathcal{G}]\text{ a.s.}$

Theorem (Conditional Expectation is Uniformly Integrable)

Let $X$ be an integrable RV. Then the family $(E[X|\mathcal{G}])_{\mathcal{G}\subset\mathcal{F}}$ is uniformly integrable i.e. $X\in\mathscr{L}^{1}(\Omega,\mathcal{F},P)\implies(E[X|\mathcal{G}])_{\mathcal{G}\subset\mathcal{F}}\text{ u.i.}$

Theorem (Independence of Conditional Expectation)

Let $X\in\mathscr{L}^{1}(\Omega,\mathcal{F},P)$ , $\mathcal{G}\subset \mathcal{F}$ sub σ-algebra. Assume $\sigma(X)$ is independent of $\mathcal{G}$ then $E[X|\mathcal{G}]=E[X]\text{ a.s.}$

Proposition (Positivity of Conditional Expectation)

Let $X\in\mathscr{L}^{1}(\Omega,\mathcal{F},P)$ , $\mathcal{G}\subset \mathcal{F}$ sub-σ-algebra. Assume $X\ge0$ , then $E[X|\mathcal{G}]\ge 0\text{ a.s. }$

Proposition (Linearity of Conditional Expectation)

Let $X,Y\in\mathscr{L}^{1}(\Omega,\mathcal{F},P)$ , let $\mathcal{G}\subset \mathcal{F}$ be a sub-σ-algebra, then $\forall\alpha,\beta\in\mathbb{R}$ $E[\alpha X+\beta Y|\mathcal{G}]=\alpha E[X|\mathcal{G}]+\beta E[Y|\mathcal{G}]$

2 Discrete RVs

Let $X$ and $Y$ be two discrete RVs. Given $Y=y$ , the conditional expectation of $X$ is $E[X|Y=y]:=\sum_{x\in\mathscr{X}}x \ p_{X|Y}(x|y)$ if $P(Y=y)>0$ and == $E[|X|]<\infty$ ==.

2 Discrete RVs and Function

Let $X$ and $Y$ be two discrete RVs. Given $Y=y$ , the conditional expectation of $f(X)$ is $E[f(X)|Y=y]:=\sum_{x\in\mathscr{X}}f(x) \ p_{X|Y}(x|y)$ if $P(Y=y)>0$ and $E[|g(X)|]<\infty$ .

2 Continuous RVs

Let $X$ and $Y$ be two continuous RVs. Given $Y=y$ , the conditional expectation of $X$ is $E[X|Y=y]=\int_\mathbb{R}xp_{X|Y}(x|y)dx$ if $p_Y(y)>0$ and $E[|X|]<\infty$ .

Linked from

Kalman Filter