Web Sfog - Absorbin Everything interesting: Probability 2

Second post in the series.

Conditional expectation

Given a random variable $X$ over probability space of $(\Omega, \mathcal{F}, P)$ with $\mathbb{E}[X]<\infty$ we define conditional expectation $Y=\mathbb{E} [X|\mathcal{F}_1]$ where $\mathcal{F}_1 \subset \mathcal{F}$ as follows:

It is random variable measurable on $\mathcal{F}_1$
The following holds for any $A\in\mathcal{F}_1, \mathbb{E}[Y \mathbb{1}_{A}]=\mathbb{E}[X\mathbb{1}_{A}]$

What does those conditions imply?

Lets begin with the first condition. The measurability on $\mathcal{F}_1$ is the requirement that the level sets, $\{ \omega \mid \mathbb{E} [X|\mathcal{F}_1](\omega) \leq \alpha; \forall \alpha \in \mathbb{R}\}$ , of the random variable in a question, are in $\mathcal{F}_1$ .

To get a feeling of what’s going on, I think of $\mathbb{E} [X|\mathcal{F}_1]$ as a steps function over sets in $\mathcal{F}_1$ . The justification comes from the fact that any measurable function can be approximated as the limit of a sequence of steps functions. When I say “steps function” I refer to a function defined as $\sum_{A \in \mathcal{F}} \alpha_{A} \mathbb{1}_A$ usually called “simple function” (that notion should be familiar from Lebesgue integration theory).

To build the intuition for conditional expectation I will define the notion of minimal sets in $\mathcal{F}_1$ . Minimal set is one that can’t be further divided into smaller sets inside $\mathcal{F}_1$ . look at the very simple case below, where $\mathcal{F}_1$ consists only of $A$ and its completion $A^{C}$ . Since $A$ and $A^{C}$ are the only sets in $\mathcal{F}_1$ (except the whole $\Omega$ ), they are obviously the minimal sets there. The random variable $Y$ on $\mathcal{F}_1$ can change only on the boundaries of the minimal sets. It is easy to see why, suppose $Y$ did change inside, obviously, that would create level sets which are not in $\mathcal{F}_1$ in contradiction to $Y$ being measurable on $\mathcal{F}_1$ . So minimal sets dictate the best resolution of the random variables that live on $\mathcal{F}_1$ .

Now I will turn my attention to the second condition. It requires that the expectation of both the $X$ and the $Y$ , agree when calculated over sets in $\mathcal{F}_1$ .

This means that conditional expectation $Y$ is a coarse estimation of the original random variable $X$ . Fires note that the minimal sets in $\mathcal{F}_1$ are not in general minimal in $\mathcal{F}$ . It is possible to further divided them to get still smaller sets in $\mathcal{F}$ . Consequently the resolution of random variable on $\mathcal{F}$ is greater then on $\mathcal{F}_1$ . Thus the variable in $\mathcal{F}$ may appear smooth while variable in $\mathcal{F}_1$ may appear coarse in comparison.

Now think of $X$ as step function that is defined on so fine structure of $\mathcal{F}$ that it just appears as smooth function over $\Omega$ , while $Y$ is confined to the very course structure of $\mathcal{F}_1\subset \mathcal{F}$ :

In order to satisfy condition 2 the average of $X$ and $Y$ over sets in $\mathcal{F}_1$ should agree. $Y$ is constant on the minimal sets of $\mathcal{F}_1$ so it has to be equal to the average of $X$ over minimal set of $\mathcal{F}_1$ . If you still follow me, you should see by now why $Y$ is a sort of course version of $X$ by courser resolution of $\mathcal{F}_1$ .

What is the relation to the “undergraduate” notion of conditional expectation?

In introduction to the probability theory course we defined conditional expectation differently. Given random variable $X$ and event $A$ conditional expectation , $\mathbb{E}[X|A]$ , is $\int_{A} Xd\omega$ , so how this definition is related to the “graduate” conditional expectation?

First of all while $\mathbb{E}[X|A]$ is real number the “graduate” definition talks about a random variable.

All the rest is pretty straight forward, lets define $\mathcal{F}_1$ to be $(A,A^{C},\varnothing,\Omega)$ then $Y=\mathbb{E} [X|\mathcal{F}_1]$ equals to:

$\mathbb{E} [X|\mathcal{F}_1]=\begin{cases}\mathbb{E}[X|A],&\quad \forall \omega \in A\\ \mathbb{E}[X|A^{C}],&\quad \forall \omega \in A^{C}\end{cases}$