# Question 19280: Independence Gives Variance of the Average, Not a Finite-Sample Limit

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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.

R0037-19280 · q-20261014-advance-20261105-r07-weak-law-large-numbers-0005



## Original question and givens

Use declared mean=4/5, positive finite variance=5/6, 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 sample size n=9. Give (mean, variance, n, average_variance, certificate).


- G001 — Use declared mean=4/5, positive finite variance=5/6, 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 sample size n=9. Give (mean, variance, n, average_variance, 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-variance-average;original snapshot and mathematics retained

After

explicit ordered source output fields=[mean, variance, n, average_variance, 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=4/5;POSITIVE finite variance=5/6;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 selected sum of diagonal variances := mul(9, 5/6)

After

WLLN selected sum of diagonal variances = 15/2

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 selected average denominator squared := mul(9, 9)

After

WLLN selected average denominator squared = 81

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 selected average variance := div(15/2, 81)

After

WLLN selected average variance = 5/54

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









## Reverse checks


- S006 · premise_and_result_audit — Recompute WLLN selected average variance with exact rational arithmetic, not rounded decimals.
- S005 · premise_and_result_audit — Recompute WLLN selected average denominator squared with exact rational arithmetic, not rounded decimals.
- S004 · premise_and_result_audit — Recompute WLLN selected sum of diagonal variances 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

(4/5, 5/6, 9, 5/54, independence_gives_variance_sigma_squared_over_n)

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.

Source hashes and review receipts
- input commit: 0f7be61e2c22317a4075561dc822c87d881a50db
- original question SHA-256: b764bd23e7484d61eb553041cc175b4324267df75092a535a636ecb221b2c47b
- candidate record SHA-256: d8dd1b7bdab7f0a09ff7abec14b734daf1cde73867dccb234cb2d429db8d4c93
- semantic review receipt SHA-256: 065507b602ceee60cf47a700d387a51940e0c6724c35a3336a7a7948a199ed80
- question-first witness SHA-256: 7203f33b030792559673b0fee520a8c0e70903c4dd970b35293aa40852f333d6
- Lean: NOT_RUN · human review: NOT_PERFORMED

[Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261014-advance-20261105-r07-weak-law-large-numbers-0005.json)