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

Dependencies:

1. Random variable
2. Expected value of a random variable
3. Variance of a random variable
4. Conditional expectation
5. Conditional variance
6. Linearity of expectation
7. Law of total probability: E(Y) = E(E(Y|X)) (incomplete)

$\newcommand{\E}{\operatorname{E}}$ $\newcommand{\Var}{\operatorname{Var}}$ Let $Y$ be a real-valued random variable. Let $X$ be a random variable. Then \[ \Var(Y) = \Var(\E(Y|X)) + \E(\Var(Y|X)). \]

Proof

Let $Z = \E(Y|X)$.

\begin{align} & \Var(Z) + \E(\Var(Y|X)) \\ &= (\E(Z^2) - \E(Z)^2) + \E(\E(Y^2|X) - \E(Y|X)^2) \tag{definition of $\Var$} \\ &= \E(Z^2) - \E(Z)^2 + \E(\E(Y^2|X)) - \E(Z^2) \tag{linearity of expectation} \\ &= \E(\E(Y^2|X)) - \E(Z)^2 \\ &= \E(Y^2) - \E(Y)^2 \tag{law of total probability} \\ &= \Var(Y) \end{align}

Dependency for: None

Info:

• Depth: 10
• Number of transitive dependencies: 25

Transitive dependencies:

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