Conditional Expectation

Last updated Nov 1, 2022

TODO I am going to rework this page into

1. Conditional Expectation with respect to Sigma Field
2. Conditional Expectation with respect to Random Element
3. Conditional Expectation with respect to Event

Let $(\Omega, \mathcal{B}, \mathbb{P})$ be a Probability Space and let $X$ be a Random Variable in $\bar{L^{1}}(\mathcal{B})$. Let $\mathcal{G} \subset \mathcal{B}$ be a sub-Sigma Algebra. Then the Conditional Expectation of $X$ (denoted $\mathbb{E}[X | \mathcal{G}]$) with respect to $\mathcal{G}$ is a Random Variable in $\bar{L^{1}}(\mathcal{G})$ such that $\forall A \in \mathcal{G}$

$$\int\limits_{A} \mathbb{E}[X | \mathcal{G}] d \mathbb{P}(\omega) = \int\limits_{A} X d \mathbb{P}(\omega)$$

When we talk about Conditional Expectation with respect to another Random Variable $Y$, use the shorthand $$\mathbb{E}[X | Y] = \mathbb{E}[X | \sigma(Y)]$$

When we talk about Conditional Expectation with respect to a Random Variable $Y$ equal to $y \in \mathbb{R}$, we use the shorthand $$\mathbb{E}[X | Y=y] = \mathbb{E} X | Y$$ for (any) $\omega \in Y^{-1}({y})$. Note that this value may not be Well-Defined, but given a particular Conditional Expectation, we can get a reasonable answer Almost Surely.

When we talk about Conditional Expectation with respect to a Set $B \in \mathcal{B}$, we use the shorthand: $$\mathbb{E}[X|B] := \mathbb{E} X|1_{B}$$ for (any) $\omega \in B$. Note that this value is only Well-Defined if $\mathbb{P}(B) > 0$. For details on its exact form, see Conditional Expectation with respect to a Set is Integral divided by Probability.

1. We require $X$ to be in $\bar{L^{1}}$ so the above integral is defined.
2. With Conditional Expectation with respect to a Set, it does not matter what $\omega \in B$ we choose, the value is the same. To see this, recall that $\sigma(1_{B}) = {\emptyset, B, B^{C}, \Omega}$. Since $\mathbb{E}[X|1_{B}]$ is $\sigma(1_{B})$-measureable, $\sigma(\mathbb{E}[X|1_{B}]) \subset \sigma(1_{B})$. Let $\omega \in B$. Since Singletons are Closed in the Real Numbers, we know ${\mathbb{E} X|1_{B}} \in \mathcal{B}(\mathbb{R})$. Thus we have that $\mathbb{E}[X|1_{B}]^{-1}({\mathbb{E} X|1_{B}}) \in \sigma(1_{B})$ and it is not disjoint with $B$. The only two sets that satisfy this criterion in $\sigma(1_{B})$ are $\Omega$ and $B$ itself. In either case, $\mathbb{E} X|1_{B} = {\mathbb{E} X|1_{B}}$ and $\mathbb{E}[X|1_{B}]$ is constant on $B$. $\blacksquare$

1. Conditional Expectation Exists and is Almost Surely Unique
2. Conditioning on known information is Idempotent
3. Conditional Expectation over the Trivial Sigma Algebra is Expectation
4. Smoothing
5. Conditional Expectation is Linear
6. Conditional Expectation is Non-Decreasing
7. Conditional Expectation satisfies Jensen’s Inequality
8. Conditional Expectation with respect to a Set is Integral divided by Probability
9. Monotone Convergence Theorem for Conditional Expectation
10. Fatou’s Lemma for Conditional Expectation
11. Dominated Convergence Theorem for Conditional Expectation
12. Conditioning on Independent Information is just Expectation

I want to connect to the ways I think of Conditional Expectation. For example

1. I think of conditional expectation (especially in the sense discrete random variables) as expectation over the conditional probability. Maybe it would be good to define conditional probability then connect it to conditional expectation through there.

Let $(\Omega, \mathcal{B}, \mathbb{P})$ be a Probability Space and let $X$ be a Discrete Random Variable on $\Omega$ with Support of a Random Element ${k_{n}}_{n} \subset \mathbb{R}$. Let $Y$

• Extended L1 Functions
• Sigma Algebra induced by Random Element