# Law of total expectation

This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.

On the same finite probability space, weighting every partition-cell conditional mean by its cell probability gives E[X].

mko-law-of-total-expectation · theorem · v0.10.0



## Mathematical statement

Let G be the sigma-algebra generated by a complete partition. The tower identity is E(E(X|G))=E(X). Cells are pairwise disjoint and cover the sample space. Positive-mass cells contribute P(B_i)E[X|B_i]. Conditional expectation is determined only almost surely: a null-cell quotient is undefined, but any integrable version there has zero weighted contribution. Do not divide by zero. No independence is required, and this is not Bayes formula.





## Representative formula (exact TeX)

E(E(X\mid G))=E(X)





## Assumptions


- The normalized finite law, integrable variable X and complete partition are explicit; G is generated by that partition.
- Do not compute a conditional quotient on zero-mass cells.





## Symbol scope


- E — expectation and conditional-expectation operator
- X — integrable random variable
- G — partition-generated sigma-algebra





## Intuition, definition, and misconceptions

Check partition completeness and disjointness first.

Compute exact conditional means on positive-mass cells.

Null cells contribute zero and the tower expectation is preserved.

Do not divide on a zero-mass cell or demand independence of partition events.





## Adopted derivation guide

[Total expectation with a nonempty null block](https://ocme.evemisslab.com/en/derivations/derivation-mko-law-of-total-expectation-01/)The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.





## Prerequisites


- [Finite conditional expectation](https://ocme.evemisslab.com/en/mko/mko-conditional-expectation/) — Cell conditional means are required.
- [Expectation in finite models](https://ocme.evemisslab.com/en/mko/mko-expectation/) — Use the global mean under the same law.
- [Event](https://ocme.evemisslab.com/en/mko/mko-event/) — Cells are events of the partition.





## Verification boundary

This slice uses complete finite models; the general sigma-algebra tower identity needs a separate measure-theoretic proof.

not_formalized

This does not mean the content is false; it means no replayable computational or formal evidence is registered yet.





## Next obligation

Add the law of total variance.





## Same-source structured data

This page and the interactive view use the same object and existing language edition; this adds no proof or human review.

[Canonical MKO JSON](https://ocme.evemisslab.com/data/mko/law-of-total-expectation.json) · [Plain Markdown](https://ocme.evemisslab.com/en/mko/mko-law-of-total-expectation/index.md)