# Expectation in finite models

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

Finite expectation is the weighted sum of random-variable values under a valid probability law; linearity does not require independence.

mko-expectation · definition · v0.10.0



## Mathematical statement

For nonempty finite S, normalized nonnegative weights w_s, and a real random variable X, define E[X]=Σ_(s∈S)w_s X(s). On the same probability model, E[aX+b]=aE[X]+b and E[X+Y]=E[X]+E[Y] without assuming independence of X and Y; a constant c has expectation c.





## Representative formula (exact TeX)

E(X)





## Assumptions


- Weights are nonnegative and sum exactly to one, and the random variables are total on the same finite S.
- Exercises use integer values and transformation coefficients with explicit rational weights and exact rational arithmetic, not sampling estimates.





## Symbol scope


- E — finite expectation operation
- X — real random variable on the same probability model





## Intuition, definition, and misconceptions

Validate the weights, then multiply each outcome value by its probability and sum. Averaging distinct values once each is generally wrong. Expectation may be negative and need not itself be an attained value.

Affine transformations and sums of two variables on the same model can be formed pointwise before weighting. Linearity does not require independence and does not establish that a product expectation equals the product of expectations.

This is mathematical expectation in a known finite probability model, rather than an observed sample mean, a long-run convergence result, or a statement of estimation accuracy. General integral existence and infinite expectations have separate obligations.

Common errors include averaging all values without declared equiprobability, or inferring independence or a product formula from linearity.





## Adopted derivation guide

[Expectation in finite models: proving linearity, with an instance of dependent variables](https://ocme.evemisslab.com/en/derivations/derivation-mko-expectation-01/)The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.





## Prerequisites


- [Real random variables on finite sample spaces](https://ocme.evemisslab.com/en/mko/mko-random-variable/) — Expectation weights the values of a specified random variable.
- [Probability on finite sample spaces](https://ocme.evemisslab.com/en/mko/mko-probability/) — First fix nonnegative weights with total one.





## Verification boundary

Finite weighted sums can be checked exactly; formal Evidence for general expectation and integrability has not been established.

not_formalized

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





## Next obligation

Bind finite sums to integral expectation through an integrability bridge, retaining linearity and the common-model premise.





## 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/expectation.json) · [Plain Markdown](https://ocme.evemisslab.com/en/mko/mko-expectation/index.md)