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

Info:

• Depth: 7
• Number of transitive dependencies: 19

Transitive dependencies:

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