# Independent identically distributed sequence

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

An IID sequence requires every variable to have the same distribution and every finite subfamily to be mutually independent.

mko-iid-sequence · definition · v0.10.0



## Mathematical statement

This slice uses a symbolic Bernoulli(p) infinite product-law cylinder certificate: each finite binary word has probability p raised to its number of ones times (1−p) raised to its number of zeros. A finite prefix can only certify a finite IID family; by itself it does not certify the infinite sequence.





## Representative formula (exact TeX)

I(X)





## Assumptions


- p is an exact rational number in [0,1], and a cylinder word is a finite binary sequence.
- An infinite IID claim explicitly requires the symbolic product-law certificate; equal expectations or one checked prefix are insufficient.





## Symbol scope


- I — IID-sequence record
- X — sequence of random variables
- p — common Bernoulli parameter





## Intuition, definition, and misconceptions

Identical distribution checks the full common marginal, not just its mean.

Independence quantifies over every finite subfamily.

The Bernoulli product law supplies a uniform cylinder formula for every finite binary word.

Equal expectations, pairwise independence or a fixed checked prefix do not establish an infinite IID sequence.





## Adopted derivation guide

[IID sequence: proving IID from the Bernoulli(1/3) cylinder product formula](https://ocme.evemisslab.com/en/derivations/derivation-mko-iid-sequence-01/)The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.





## Prerequisites


- [Independence of random variables](https://ocme.evemisslab.com/en/mko/mko-random-variable-independence/) — Every finite subfamily must be mutually independent.
- [Discrete-time stochastic process](https://ocme.evemisslab.com/en/mko/mko-discrete-stochastic-process/) — The variables are indexed by discrete time.
- [Sequence](https://ocme.evemisslab.com/en/mko/mko-sequence/) — The infinite variable family is represented as a sequence.
- [Finite joint distribution](https://ocme.evemisslab.com/en/mko/mko-finite-joint-distribution/) — Each finite cylinder has an exact joint law.
- [Expectation in finite models](https://ocme.evemisslab.com/en/mko/mko-expectation/) — A common mean is a consequence, not a certificate of identical distribution.





## Verification boundary

The infinite claim relies only on the explicitly declared Bernoulli symbolic product-law certificate; no LLN, CLT or additional Markov conclusion is inferred.

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 weak law of large numbers and central limit theorem.





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