Question 19250: Choose a Strict All-n Chebyshev Tail Witness

Original question and givens

Use declared mean=24/25, positive finite variance=25/26, and iid_certificate=symbolic_all_finite_subfamilies. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive epsilon=1/26, delta=1/34. Give (epsilon, delta, N, chebyshev_bound_at_N, strict_delta_tail, certificate).

• G001 — Use declared mean=24/25, positive finite variance=25/26, and iid_certificate=symbolic_all_finite_subfamilies. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive epsilon=1/26, delta=1/34. Give (epsilon, delta, N, chebyshev_bound_at_N, strict_delta_tail, certificate).

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-weak-law-large-numbers

Step-by-step derivation

source operation=wlln-epsilon-delta-witness;original snapshot and mathematics retained

explicit ordered source output fields=[epsilon, delta, N, chebyshev_bound_at_N, strict_delta_tail, certificate]

original common finite mean=24/25;POSITIVE finite variance=25/26;whole-sequence iid_certificate=symbolic_all_finite_subfamilies

ASSUME for EVERY finite subfamily the declared independence and identical law/common moments;marker declares a premise,NOT independent evidence that a real sequence satisfies it

for ANY positive integer n:sample_mean_n=(X1+...+Xn)/n;every Xi has the SAME mean and variance;distinct Xi,Xj independent

linearity gives E(sample_mean_n)=mean;centered products have zero expectation for i!=j;Var(sample_mean_n)=(n*variance)/n^2=variance/n

WLLN strict-tail epsilon squared := mul(1/26, 1/26)

WLLN strict-tail epsilon squared = 1/676

WLLN strict-tail delta times epsilon squared := mul(1/34, 1/676)

WLLN strict-tail delta times epsilon squared = 1/22984

WLLN strict-tail rational threshold := div(25/26, 1/22984)

WLLN strict-tail rational threshold = 22100

WLLN first strict bound-based index := sum(22100, 1)

WLLN first strict bound-based index = 22101

WLLN cutoff evaluation epsilon squared := mul(1/26, 1/26)

WLLN cutoff evaluation epsilon squared = 1/676

WLLN cutoff evaluation positive n times epsilon squared := mul(22101, 1/676)

WLLN cutoff evaluation positive n times epsilon squared = 22101/676

WLLN cutoff evaluation raw Chebyshev upper bound := div(25/26, 22101/676)

WLLN cutoff evaluation raw Chebyshev upper bound = 650/22101

WLLN cutoff evaluation:same whole-sequence IID premise;positive n=22101;positive epsilon=1/26;common variance=25/26

P(abs(sample_mean_n-mean)>=epsilon)<=RAW bound=650/22101;NOT event probability,NOT equality,NOT clipped to1

WLLN previous positive index := sub(22101, 1)

WLLN previous positive index = 22100

WLLN previous bound evaluation epsilon squared := mul(1/26, 1/26)

WLLN previous bound evaluation epsilon squared = 1/676

WLLN previous bound evaluation positive n times epsilon squared := mul(22100, 1/676)

WLLN previous bound evaluation positive n times epsilon squared = 425/13

WLLN previous bound evaluation raw Chebyshev upper bound := div(25/26, 425/13)

WLLN previous bound evaluation raw Chebyshev upper bound = 1/34

WLLN previous bound evaluation:same whole-sequence IID premise;positive n=22100;positive epsilon=1/26;common variance=25/26

P(abs(sample_mean_n-mean)>=epsilon)<=RAW bound=1/34;NOT event probability,NOT equality,NOT clipped to1

positive epsilon=1/26;delta=1/34;EXACT threshold=22100;floor=22100;N=22101;bound_at_N=650/22101;previous bound=1/34

strict Chebyshev comparison holds at N and fails at preceding positive index;least BOUND-BASED N,not necessarily least true-probability cutoff

SAME declared whole IID premise;positive epsilon=1/26;positive delta=1/34;chosen positive N=22101;bound_at_N=650/22101<delta

for EVERY n>=N,Chebyshev bound variance/(n*epsilon^2)<=bound_at_N<delta;therefore convergence in PROBABILITY under the declared premise

Reverse checks

• S018 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S017 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S016 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S015 · premise_and_result_audit — Recompute WLLN previous bound evaluation raw Chebyshev upper bound with exact rational arithmetic, not rounded decimals.
• S014 · premise_and_result_audit — Recompute WLLN previous bound evaluation positive n times epsilon squared with exact rational arithmetic, not rounded decimals.
• S013 · premise_and_result_audit — Recompute WLLN previous bound evaluation epsilon squared with exact rational arithmetic, not rounded decimals.
• S012 · premise_and_result_audit — Recompute WLLN previous positive index with exact rational arithmetic, not rounded decimals.
• S011 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S010 · premise_and_result_audit — Recompute WLLN cutoff evaluation raw Chebyshev upper bound with exact rational arithmetic, not rounded decimals.
• S009 · premise_and_result_audit — Recompute WLLN cutoff evaluation positive n times epsilon squared with exact rational arithmetic, not rounded decimals.
• S008 · premise_and_result_audit — Recompute WLLN cutoff evaluation epsilon squared with exact rational arithmetic, not rounded decimals.
• S007 · premise_and_result_audit — Recompute WLLN first strict bound-based index with exact rational arithmetic, not rounded decimals.
• S006 · premise_and_result_audit — Recompute WLLN strict-tail rational threshold with exact rational arithmetic, not rounded decimals.
• S005 · premise_and_result_audit — Recompute WLLN strict-tail delta times epsilon squared with exact rational arithmetic, not rounded decimals.
• S004 · premise_and_result_audit — Recompute WLLN strict-tail epsilon squared with exact rational arithmetic, not rounded decimals.
• 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 — Check against the original data and this step's premises; a one-way implication is not an equivalence.

Result

(1/26, 1/34, 22101, 650/22101, Yes, forall_n_ge_N_monotone_Chebyshev_tail)

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.