Conditional expectation

Dependencies:

1. Probability
2. Random variable
3. Expected value of a random variable
4. Conditional probability (incomplete)

$\newcommand{\E}{\operatorname{E}}$ Let $X$ be a random variable on $(\Omega, \mathcal{F}, \Pr)$ and $A$ be an event. Then $\E(X \mid A)$ is the same as $\E(X)$, except that the probability measure for the Lebesgue integral is $\Pr_{|A}$ instead of $\Pr$.

Let $X$ be a real-valued random variable, and $Y$ be a random variable. Define $g(y) = \E(X \mid Y=y)$. Then $\E(X \mid Y)$ is defined as the random variable $g(Y)$.

Dependency for:

1. Law of total probability: decomposing expectation over countable events
2. Law of total probability: E(Y) = E(E(Y|X)) (incomplete)
3. Var(Y) = Var(E(Y|X)) + E(Var(Y|X))

Info:

• Depth: 7
• Number of transitive dependencies: 19

Transitive dependencies:

1. /analysis/topological-space
2. /sets-and-relations/countable-set
3. /sets-and-relations/de-morgan-laws
4. /measure-theory/lebesgue-integral
5. σ-algebra
6. Generated σ-algebra
7. Borel algebra
8. Measurable function
9. Generators of the real Borel algebra (incomplete)
10. Measure
11. σ-algebra is closed under countable intersections
12. Group
13. Ring
14. Field
15. Vector Space
16. Probability
17. Conditional probability (incomplete)
18. Random variable
19. Expected value of a random variable