Question 14485: Identical Distribution Means the Complete Marginal, Not Just Its Mean

Original question and givens

Use the explicitly declared Bernoulli product-law model with exact p=10/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. left_index=9, right_index=26. Give (left_index,right_index,[0,zero_mass],[1,one_mass],identical?,full_Bernoulli_marginal_not_mean_only).

• G001 — Use the explicitly declared Bernoulli product-law model with exact p=10/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. left_index=9, right_index=26. Give (left_index,right_index,[0,zero_mass],[1,one_mass],identical?,full_Bernoulli_marginal_not_mean_only).

Definition and scope

Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate.

• mko-iid-sequence

Step-by-step derivation

Bernoulli zero-outcome mass := sub(1, 10/29)

Bernoulli zero-outcome mass = 19/29

raw Bernoulli parameter=[10, 29]

normalized p=10/29;mass at0=19/29;mass at1=10/29;symbolic product-law model explicitly declared

indices=[9, 26];same declared Bernoulli product law

both complete marginals:mass0=19/29;mass1=10/29;identical=Yes

Reverse checks

• S003 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S001 · premise_and_result_audit — Recompute Bernoulli zero-outcome mass with exact rational arithmetic, not rounded decimals.

Result

(9, 26, [0, 19/29], [1, 10/29], Yes, full_Bernoulli_marginal_not_mean_only)

The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.

This question review and source binding

Review date: 2026-10-26

Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.