Independent identically distributed sequence

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

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 — Every finite subfamily must be mutually independent.
• Discrete-time stochastic process — The variables are indexed by discrete time.
• Sequence — The infinite variable family is represented as a sequence.
• Finite joint distribution — Each finite cylinder has an exact joint law.
• Expectation in finite models — 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.