Conditional variance

Dependencies:

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

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

Equivalently, $\Var(X \mid A) = \E((X-\E(X \mid A))^2 \mid A) = \E(X^2 \mid A) - \E(X \mid A)^2$.

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

Dependency for:

1. Var(Y) = Var(E(Y|X)) + E(Var(Y|X))

Info:

• Depth: 9
• Number of transitive dependencies: 22

Transitive dependencies:

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