Use declared mean=18/19, positive finite variance=19/20, 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=23. Give (mean, variance, n, average_variance, certificate).
G001 — Use declared mean=18/19, positive finite variance=19/20, 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=23. 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.
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.
original common finite mean=18/19;POSITIVE finite variance=19/20;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.
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.
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.