Law of total probability: decomposing expectation over countable events

Dependencies:

1. Random variable
2. Conditional expectation
3. /measure-theory/linearity-of-lebesgue-integral

$\newcommand{\E}{\operatorname{E}}$ There are many results that are called 'Law of total probability'. The following is one of them.

Let $S$ be a countable set of pairwise-disjoint events such that $\bigcup_{A \in S} A = \Omega$ and $\Pr(\cdot \mid A)$ is defined $\forall A \in S$. Let $X$ be a random variable. Then \[ \E(X) = \sum_{A \in S} \E(X \mid A)\Pr(A). \]

Proof

\begin{align} \E(X) &= \int_{\omega \subseteq \Omega} X(\omega)\Pr(\omega) \\ &= \int_{\omega \subseteq \Omega} X(\omega)\left(\sum_{A \in S} \Pr(\omega \cap A)\right) \\ &= \int_{\omega \subseteq \Omega} X(\omega)\left(\sum_{A \in S} \Pr(\omega \mid A) \Pr(A)\right) \\ &= \sum_{A \in S} \Pr(A)\left(\int_{\omega \subseteq \Omega} X(\omega)\Pr(\omega \mid A)\right) \tag{linearity of Lebesgue integral} \\ &= \sum_{A \in S} \Pr(A)\E(X \mid A) \end{align}

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Info:

• Depth: 8
• Number of transitive dependencies: 21

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. Conditional expectation