202501141801

Tags : Measure Theoretic Probability

Conditional Expectation

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The goal is to incorporate partial information into your model for probabilities. This concept is ubiquitous in all of probability theory and is of fundamental importance.

Consider the case of $3$ consecutive coin tosses, this gives a sample space of $8$ options, namely $\Omega =\{\text{HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}\}$. And let $X$ be the number of heads. $E(X)=1.5$

If we now say have the information, that the first roll are $\text{H}$, then the possible set of events becomes $\{\text{HHH, HHT, HTH, HTT}\}$. With this restriction, we get that the $E(X|R_1=H)=2$. Similarly If the first toss was tails, then we get an expected value of $1$.

Here knowing the first toss, is the same as knowing which of the following sets occurred

$$\{\varnothing ,A_1,A_2,\Omega \}=\mathcal{F}$$

where

• $A_1=\{\text{HHH, HHT, HTH, HTT}\}$
• $A_2=\{\text{THH, THT, TTH, TTT}\}$

Expected value of $X$ given that we are observing the first throw could hence be thought of as a random variable over the $\sigma$ field $\mathcal{F}$.

The simple baysean definition of conditional expectation breaks down when to comes to continuous models. Where there can be events of probability $0$ that can add up to give a non-zero probability.

Definition

Let $⟨\Omega ,\mathcal{A},P⟩$ be a probability space. Let $\mathcal{G}$ be a sub-sigma field of $\mathcal{A}$. Let $X$ be an integrable r.v. Conditional expecation of $X$ given $G$; denoted as $E(X|G)$ is a random variable $X^{\ast }$ such that

• $X^{\ast }$ is $G$ measurable.
• $E(X{I}_A)=E(X^{\ast }{I}_A)$ for every $A$ in $G$.

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References