# Question 19245: Choose a Strict All-n Chebyshev Tail Witness [Return to catalogue page 97](https://ocme.evemisslab.com/en/questions/page/97/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261014-advance-20261105-r07-weak-law-large-numbers-foundations/#q-20261014-advance-20261105-r07-weak-law-large-numbers-0070) This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth. R0037-19245 · q-20261014-advance-20261105-r07-weak-law-large-numbers-0070 ## Original question and givens Use declared mean=19/20, positive finite variance=20/21, 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=1/21, delta=1/29. Give (epsilon, delta, N, chebyshev_bound_at_N, strict_delta_tail, certificate). - G001 — Use declared mean=19/20, positive finite variance=20/21, 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=1/21, delta=1/29. 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](https://ocme.evemisslab.com/en/mko/mko-weak-law-large-numbers/) ## Step-by-step derivation S001 · impliesBefore source operation=wlln-epsilon-delta-witness;original snapshot and mathematics retained After explicit ordered source output fields=[epsilon, delta, N, chebyshev_bound_at_N, strict_delta_tail, certificate] Conditions: none Retain the historical interface marker in the original snapshot while listing ordered source fields here,without JavaScript or external lookup.Field names are not mathematical evidence,and No retains the operation's specific claim scope. Step identifiers and dependenciesRule: source_ordered_interface_explicit_without_javascript · Uses: G001 S002 · impliesBefore original common finite mean=19/20;POSITIVE finite variance=20/21;whole-sequence iid_certificate=symbolic_all_finite_subfamilies After 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 Conditions: none The marker is a premise of this problem,not proof of IID by a string or finite data.What follows is conditional on a whole sequence satisfying that premise,not a Lean proof or certification of empirical data. Step identifiers and dependenciesRule: wlln_whole_IID_declared_premise_not_token_proof · Uses: G001, S001 S003 · impliesBefore for ANY positive integer n:sample_mean_n=(X1+...+Xn)/n;every Xi has the SAME mean and variance;distinct Xi,Xj independent After 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 Conditions: none Finite variances ensure integrability of the products.Independence factors every distinct centered cross term into0*0,leaving n diagonal variance terms.Scaling by1/n squared gives variance/n for arbitrary n,not a deduction from one observed average matching the mean. Step identifiers and dependenciesRule: wlln_universal_variance_sum_independence_not_one_sample · Uses: G001, S002 S004 · calculationBefore WLLN strict-tail epsilon squared := mul(1/21, 1/21) After WLLN strict-tail epsilon squared = 1/441 Conditions: none Substitute original roles and established intermediates exactly,retaining parameter,outcome,target and denominator roles without replacing the claim. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · calculationBefore WLLN strict-tail delta times epsilon squared := mul(1/29, 1/441) After WLLN strict-tail delta times epsilon squared = 1/12789 Conditions: none Substitute original roles and established intermediates exactly,retaining parameter,outcome,target and denominator roles without replacing the claim. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004 S006 · calculationBefore WLLN strict-tail rational threshold := div(20/21, 1/12789) After WLLN strict-tail rational threshold = 12180 Conditions: none Substitute original roles and established intermediates exactly,retaining parameter,outcome,target and denominator roles without replacing the claim. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005 S007 · calculationBefore WLLN first strict bound-based index := sum(12180, 1) After WLLN first strict bound-based index = 12181 Conditions: none Substitute original roles and established intermediates exactly,retaining parameter,outcome,target and denominator roles without replacing the claim. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S006 S008 · calculationBefore WLLN cutoff evaluation epsilon squared := mul(1/21, 1/21) After WLLN cutoff evaluation epsilon squared = 1/441 Conditions: none Substitute original roles and established intermediates exactly,retaining parameter,outcome,target and denominator roles without replacing the claim. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S007 S009 · calculationBefore WLLN cutoff evaluation positive n times epsilon squared := mul(12181, 1/441) After WLLN cutoff evaluation positive n times epsilon squared = 12181/441 Conditions: none Substitute original roles and established intermediates exactly,retaining parameter,outcome,target and denominator roles without replacing the claim. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S008 S010 · calculationBefore WLLN cutoff evaluation raw Chebyshev upper bound := div(20/21, 12181/441) After WLLN cutoff evaluation raw Chebyshev upper bound = 420/12181 Conditions: none Substitute original roles and established intermediates exactly,retaining parameter,outcome,target and denominator roles without replacing the claim. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S009 S011 · impliesBefore WLLN cutoff evaluation:same whole-sequence IID premise;positive n=12181;positive epsilon=1/21;common variance=20/21 After P(abs(sample_mean_n-mean)>=epsilon)<=RAW bound=420/12181;NOT event probability,NOT equality,NOT clipped to1 Conditions: none Chebyshev divides the variance of this same average by epsilon squared.An upper bound above1 is valid but loose,not an event probability.Do not clip the source value to1;the statement remains conditional on the declared whole-sequence premise. Step identifiers and dependenciesRule: wlln_Chebyshev_raw_upper_bound_not_probability_value · Uses: G001, S010 S012 · calculationBefore WLLN previous positive index := sub(12181, 1) After WLLN previous positive index = 12180 Conditions: none Substitute original roles and established intermediates exactly,retaining parameter,outcome,target and denominator roles without replacing the claim. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S011 S013 · calculationBefore WLLN previous bound evaluation epsilon squared := mul(1/21, 1/21) After WLLN previous bound evaluation epsilon squared = 1/441 Conditions: none Substitute original roles and established intermediates exactly,retaining parameter,outcome,target and denominator roles without replacing the claim. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S012 S014 · calculationBefore WLLN previous bound evaluation positive n times epsilon squared := mul(12180, 1/441) After WLLN previous bound evaluation positive n times epsilon squared = 580/21 Conditions: none Substitute original roles and established intermediates exactly,retaining parameter,outcome,target and denominator roles without replacing the claim. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S013 S015 · calculationBefore WLLN previous bound evaluation raw Chebyshev upper bound := div(20/21, 580/21) After WLLN previous bound evaluation raw Chebyshev upper bound = 1/29 Conditions: none Substitute original roles and established intermediates exactly,retaining parameter,outcome,target and denominator roles without replacing the claim. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S014 S016 · impliesBefore WLLN previous bound evaluation:same whole-sequence IID premise;positive n=12180;positive epsilon=1/21;common variance=20/21 After P(abs(sample_mean_n-mean)>=epsilon)<=RAW bound=1/29;NOT event probability,NOT equality,NOT clipped to1 Conditions: none Chebyshev divides the variance of this same average by epsilon squared.An upper bound above1 is valid but loose,not an event probability.Do not clip the source value to1;the statement remains conditional on the declared whole-sequence premise. Step identifiers and dependenciesRule: wlln_Chebyshev_raw_upper_bound_not_probability_value · Uses: G001, S015 S017 · impliesBefore positive epsilon=1/21;delta=1/29;EXACT threshold=12180;floor=12180;N=12181;bound_at_N=420/12181;previous bound=1/29 After strict Chebyshev comparison holds at N and fails at preceding positive index;least BOUND-BASED N,not necessarily least true-probability cutoff Conditions: none The bound is below delta iff n>variance/(delta epsilon squared),so floor+1 is required even at integral thresholds.Previous failure proves minimality for this Chebyshev bound,not for the actual unknown event probability. Step identifiers and dependenciesRule: wlln_exact_strict_bound_cutoff_not_actual_probability_minimum · Uses: G001, S016 S018 · impliesBefore SAME declared whole IID premise;positive epsilon=1/21;positive delta=1/29;chosen positive N=12181;bound_at_N=420/12181=N,Chebyshev bound variance/(n*epsilon^2)<=bound_at_N