Declared IID CLT Premises, Exact Sum Moments and Finite-Lattice Boundaries
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This question bank has corresponding bilingual step-by-step derivations. Source candidate and English ai_candidate states are retained. AI review, finite computation and Lean formal proof are labelled separately.
batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations · 2026-10-15 · 100 questions
This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.
Field order uses a source-bound presentation supplement; canonical JSONL and mathematical answers are unchanged.
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1. Finite Marginal Moments Qualify the Declared IID CLT Premises
Use the aligned finite law values=[-2, -1], probabilities=[1/2, 1/2], with declared process certificate=infinite_iid. The token is an explicit infinite IID premise, not a process construction or proof by this finite calculation. Give (law_rows, mean, variance, positive_finite_variance, declared_process_certificate, qualification_statement).
infinite_iid is a declared premise token, not a constructed or proved infinite IID process. Finite marginal moments are qualification checks only.
View answer and explanation
([[-2, 1/2], [-1, 1/2]], -3/2, 1/4, Yes, infinite_iid, CLT_hypotheses_qualified)
Compute the exact common mean and centered second moment, requiring strictly positive finite variance. This qualifies a declared marginal law; it does not verify independence or an infinite process from samples, nor prove the CLT. The exact result, retaining the requested order, is ([[-2, 1/2], [-1, 1/2]], -3/2, 1/4, Yes, infinite_iid, CLT_hypotheses_qualified).
Reviewed step-by-step derivation
Central limit theorem for finite-variance IID sequences · qf-mod49-clt-finite-law-moments · candidate
2. Finite Marginal Moments Qualify the Declared IID CLT Premises
Use the aligned finite law values=[-1, 0], probabilities=[1/2, 1/2], with declared process certificate=infinite_iid. The token is an explicit infinite IID premise, not a process construction or proof by this finite calculation. Give (law_rows, mean, variance, positive_finite_variance, declared_process_certificate, qualification_statement).
infinite_iid is a declared premise token, not a constructed or proved infinite IID process. Finite marginal moments are qualification checks only.
View answer and explanation
([[-1, 1/2], [0, 1/2]], -1/2, 1/4, Yes, infinite_iid, CLT_hypotheses_qualified)
Compute the exact common mean and centered second moment, requiring strictly positive finite variance. This qualifies a declared marginal law; it does not verify independence or an infinite process from samples, nor prove the CLT. The exact result, retaining the requested order, is ([[-1, 1/2], [0, 1/2]], -1/2, 1/4, Yes, infinite_iid, CLT_hypotheses_qualified).
Reviewed step-by-step derivation
Central limit theorem for finite-variance IID sequences · qf-mod49-clt-finite-law-moments · candidate
3. Finite Marginal Moments Qualify the Declared IID CLT Premises
Use the aligned finite law values=[0, 1], probabilities=[1/2, 1/2], with declared process certificate=infinite_iid. The token is an explicit infinite IID premise, not a process construction or proof by this finite calculation. Give (law_rows, mean, variance, positive_finite_variance, declared_process_certificate, qualification_statement).
infinite_iid is a declared premise token, not a constructed or proved infinite IID process. Finite marginal moments are qualification checks only.
View answer and explanation
([[0, 1/2], [1, 1/2]], 1/2, 1/4, Yes, infinite_iid, CLT_hypotheses_qualified)
Compute the exact common mean and centered second moment, requiring strictly positive finite variance. This qualifies a declared marginal law; it does not verify independence or an infinite process from samples, nor prove the CLT. The exact result, retaining the requested order, is ([[0, 1/2], [1, 1/2]], 1/2, 1/4, Yes, infinite_iid, CLT_hypotheses_qualified).
Reviewed step-by-step derivation
Central limit theorem for finite-variance IID sequences · qf-mod49-clt-finite-law-moments · candidate
4. Finite Marginal Moments Qualify the Declared IID CLT Premises
Use the aligned finite law values=[1, 2], probabilities=[1/2, 1/2], with declared process certificate=infinite_iid. The token is an explicit infinite IID premise, not a process construction or proof by this finite calculation. Give (law_rows, mean, variance, positive_finite_variance, declared_process_certificate, qualification_statement).
infinite_iid is a declared premise token, not a constructed or proved infinite IID process. Finite marginal moments are qualification checks only.
View answer and explanation
([[1, 1/2], [2, 1/2]], 3/2, 1/4, Yes, infinite_iid, CLT_hypotheses_qualified)
Compute the exact common mean and centered second moment, requiring strictly positive finite variance. This qualifies a declared marginal law; it does not verify independence or an infinite process from samples, nor prove the CLT. The exact result, retaining the requested order, is ([[1, 1/2], [2, 1/2]], 3/2, 1/4, Yes, infinite_iid, CLT_hypotheses_qualified).
Reviewed step-by-step derivation
Central limit theorem for finite-variance IID sequences · qf-mod49-clt-finite-law-moments · candidate
5. Finite Marginal Moments Qualify the Declared IID CLT Premises
Use the aligned finite law values=[2, 3], probabilities=[1/2, 1/2], with declared process certificate=infinite_iid. The token is an explicit infinite IID premise, not a process construction or proof by this finite calculation. Give (law_rows, mean, variance, positive_finite_variance, declared_process_certificate, qualification_statement).
infinite_iid is a declared premise token, not a constructed or proved infinite IID process. Finite marginal moments are qualification checks only.
View answer and explanation
([[2, 1/2], [3, 1/2]], 5/2, 1/4, Yes, infinite_iid, CLT_hypotheses_qualified)
Compute the exact common mean and centered second moment, requiring strictly positive finite variance. This qualifies a declared marginal law; it does not verify independence or an infinite process from samples, nor prove the CLT. The exact result, retaining the requested order, is ([[2, 1/2], [3, 1/2]], 5/2, 1/4, Yes, infinite_iid, CLT_hypotheses_qualified).
Reviewed step-by-step derivation
Central limit theorem for finite-variance IID sequences · qf-mod49-clt-finite-law-moments · candidate
6. Finite Marginal Moments Qualify the Declared IID CLT Premises
Use the aligned finite law values=[-2, -1], probabilities=[1/3, 2/3], with declared process certificate=infinite_iid. The token is an explicit infinite IID premise, not a process construction or proof by this finite calculation. Give (law_rows, mean, variance, positive_finite_variance, declared_process_certificate, qualification_statement).
infinite_iid is a declared premise token, not a constructed or proved infinite IID process. Finite marginal moments are qualification checks only.
View answer and explanation
([[-2, 1/3], [-1, 2/3]], -4/3, 2/9, Yes, infinite_iid, CLT_hypotheses_qualified)
Compute the exact common mean and centered second moment, requiring strictly positive finite variance. This qualifies a declared marginal law; it does not verify independence or an infinite process from samples, nor prove the CLT. The exact result, retaining the requested order, is ([[-2, 1/3], [-1, 2/3]], -4/3, 2/9, Yes, infinite_iid, CLT_hypotheses_qualified).
Reviewed step-by-step derivation
Central limit theorem for finite-variance IID sequences · qf-mod49-clt-finite-law-moments · candidate
7. Finite Marginal Moments Qualify the Declared IID CLT Premises
Use the aligned finite law values=[-1, 0], probabilities=[1/3, 2/3], with declared process certificate=infinite_iid. The token is an explicit infinite IID premise, not a process construction or proof by this finite calculation. Give (law_rows, mean, variance, positive_finite_variance, declared_process_certificate, qualification_statement).
infinite_iid is a declared premise token, not a constructed or proved infinite IID process. Finite marginal moments are qualification checks only.
View answer and explanation
([[-1, 1/3], [0, 2/3]], -1/3, 2/9, Yes, infinite_iid, CLT_hypotheses_qualified)
Compute the exact common mean and centered second moment, requiring strictly positive finite variance. This qualifies a declared marginal law; it does not verify independence or an infinite process from samples, nor prove the CLT. The exact result, retaining the requested order, is ([[-1, 1/3], [0, 2/3]], -1/3, 2/9, Yes, infinite_iid, CLT_hypotheses_qualified).
Reviewed step-by-step derivation
Central limit theorem for finite-variance IID sequences · qf-mod49-clt-finite-law-moments · candidate
8. Finite Marginal Moments Qualify the Declared IID CLT Premises
Use the aligned finite law values=[0, 1], probabilities=[1/3, 2/3], with declared process certificate=infinite_iid. The token is an explicit infinite IID premise, not a process construction or proof by this finite calculation. Give (law_rows, mean, variance, positive_finite_variance, declared_process_certificate, qualification_statement).
infinite_iid is a declared premise token, not a constructed or proved infinite IID process. Finite marginal moments are qualification checks only.
View answer and explanation
([[0, 1/3], [1, 2/3]], 2/3, 2/9, Yes, infinite_iid, CLT_hypotheses_qualified)
Compute the exact common mean and centered second moment, requiring strictly positive finite variance. This qualifies a declared marginal law; it does not verify independence or an infinite process from samples, nor prove the CLT. The exact result, retaining the requested order, is ([[0, 1/3], [1, 2/3]], 2/3, 2/9, Yes, infinite_iid, CLT_hypotheses_qualified).
Reviewed step-by-step derivation
Central limit theorem for finite-variance IID sequences · qf-mod49-clt-finite-law-moments · candidate
9. Finite Marginal Moments Qualify the Declared IID CLT Premises
Use the aligned finite law values=[1, 2], probabilities=[1/3, 2/3], with declared process certificate=infinite_iid. The token is an explicit infinite IID premise, not a process construction or proof by this finite calculation. Give (law_rows, mean, variance, positive_finite_variance, declared_process_certificate, qualification_statement).
infinite_iid is a declared premise token, not a constructed or proved infinite IID process. Finite marginal moments are qualification checks only.
View answer and explanation
([[1, 1/3], [2, 2/3]], 5/3, 2/9, Yes, infinite_iid, CLT_hypotheses_qualified)
Compute the exact common mean and centered second moment, requiring strictly positive finite variance. This qualifies a declared marginal law; it does not verify independence or an infinite process from samples, nor prove the CLT. The exact result, retaining the requested order, is ([[1, 1/3], [2, 2/3]], 5/3, 2/9, Yes, infinite_iid, CLT_hypotheses_qualified).
Reviewed step-by-step derivation
Central limit theorem for finite-variance IID sequences · qf-mod49-clt-finite-law-moments · candidate
10. Finite Marginal Moments Qualify the Declared IID CLT Premises
Use the aligned finite law values=[2, 3], probabilities=[1/3, 2/3], with declared process certificate=infinite_iid. The token is an explicit infinite IID premise, not a process construction or proof by this finite calculation. Give (law_rows, mean, variance, positive_finite_variance, declared_process_certificate, qualification_statement).
infinite_iid is a declared premise token, not a constructed or proved infinite IID process. Finite marginal moments are qualification checks only.
View answer and explanation
([[2, 1/3], [3, 2/3]], 8/3, 2/9, Yes, infinite_iid, CLT_hypotheses_qualified)
Compute the exact common mean and centered second moment, requiring strictly positive finite variance. This qualifies a declared marginal law; it does not verify independence or an infinite process from samples, nor prove the CLT. The exact result, retaining the requested order, is ([[2, 1/3], [3, 2/3]], 8/3, 2/9, Yes, infinite_iid, CLT_hypotheses_qualified).
Reviewed step-by-step derivation
Central limit theorem for finite-variance IID sequences · qf-mod49-clt-finite-law-moments · candidate
11. Finite Marginal Moments Qualify the Declared IID CLT Premises
Use the aligned finite law values=[-2, -1], probabilities=[1/4, 3/4], with declared process certificate=infinite_iid. The token is an explicit infinite IID premise, not a process construction or proof by this finite calculation. Give (law_rows, mean, variance, positive_finite_variance, declared_process_certificate, qualification_statement).
infinite_iid is a declared premise token, not a constructed or proved infinite IID process. Finite marginal moments are qualification checks only.
View answer and explanation
([[-2, 1/4], [-1, 3/4]], -5/4, 3/16, Yes, infinite_iid, CLT_hypotheses_qualified)
Compute the exact common mean and centered second moment, requiring strictly positive finite variance. This qualifies a declared marginal law; it does not verify independence or an infinite process from samples, nor prove the CLT. The exact result, retaining the requested order, is ([[-2, 1/4], [-1, 3/4]], -5/4, 3/16, Yes, infinite_iid, CLT_hypotheses_qualified).
Reviewed step-by-step derivation
Central limit theorem for finite-variance IID sequences · qf-mod49-clt-finite-law-moments · candidate
12. Finite Marginal Moments Qualify the Declared IID CLT Premises
Use the aligned finite law values=[-1, 0], probabilities=[1/4, 3/4], with declared process certificate=infinite_iid. The token is an explicit infinite IID premise, not a process construction or proof by this finite calculation. Give (law_rows, mean, variance, positive_finite_variance, declared_process_certificate, qualification_statement).
infinite_iid is a declared premise token, not a constructed or proved infinite IID process. Finite marginal moments are qualification checks only.
View answer and explanation
([[-1, 1/4], [0, 3/4]], -1/4, 3/16, Yes, infinite_iid, CLT_hypotheses_qualified)
Compute the exact common mean and centered second moment, requiring strictly positive finite variance. This qualifies a declared marginal law; it does not verify independence or an infinite process from samples, nor prove the CLT. The exact result, retaining the requested order, is ([[-1, 1/4], [0, 3/4]], -1/4, 3/16, Yes, infinite_iid, CLT_hypotheses_qualified).
Reviewed step-by-step derivation
Central limit theorem for finite-variance IID sequences · qf-mod49-clt-finite-law-moments · candidate
13. Finite Marginal Moments Qualify the Declared IID CLT Premises
Use the aligned finite law values=[0, 1], probabilities=[1/4, 3/4], with declared process certificate=infinite_iid. The token is an explicit infinite IID premise, not a process construction or proof by this finite calculation. Give (law_rows, mean, variance, positive_finite_variance, declared_process_certificate, qualification_statement).
infinite_iid is a declared premise token, not a constructed or proved infinite IID process. Finite marginal moments are qualification checks only.
View answer and explanation
([[0, 1/4], [1, 3/4]], 3/4, 3/16, Yes, infinite_iid, CLT_hypotheses_qualified)
Compute the exact common mean and centered second moment, requiring strictly positive finite variance. This qualifies a declared marginal law; it does not verify independence or an infinite process from samples, nor prove the CLT. The exact result, retaining the requested order, is ([[0, 1/4], [1, 3/4]], 3/4, 3/16, Yes, infinite_iid, CLT_hypotheses_qualified).
Reviewed step-by-step derivation
Central limit theorem for finite-variance IID sequences · qf-mod49-clt-finite-law-moments · candidate
14. Finite Marginal Moments Qualify the Declared IID CLT Premises
Use the aligned finite law values=[1, 2], probabilities=[1/4, 3/4], with declared process certificate=infinite_iid. The token is an explicit infinite IID premise, not a process construction or proof by this finite calculation. Give (law_rows, mean, variance, positive_finite_variance, declared_process_certificate, qualification_statement).
infinite_iid is a declared premise token, not a constructed or proved infinite IID process. Finite marginal moments are qualification checks only.
View answer and explanation
([[1, 1/4], [2, 3/4]], 7/4, 3/16, Yes, infinite_iid, CLT_hypotheses_qualified)
Compute the exact common mean and centered second moment, requiring strictly positive finite variance. This qualifies a declared marginal law; it does not verify independence or an infinite process from samples, nor prove the CLT. The exact result, retaining the requested order, is ([[1, 1/4], [2, 3/4]], 7/4, 3/16, Yes, infinite_iid, CLT_hypotheses_qualified).
Reviewed step-by-step derivation
Central limit theorem for finite-variance IID sequences · qf-mod49-clt-finite-law-moments · candidate
15. Finite Marginal Moments Qualify the Declared IID CLT Premises
Use the aligned finite law values=[2, 3], probabilities=[1/4, 3/4], with declared process certificate=infinite_iid. The token is an explicit infinite IID premise, not a process construction or proof by this finite calculation. Give (law_rows, mean, variance, positive_finite_variance, declared_process_certificate, qualification_statement).
infinite_iid is a declared premise token, not a constructed or proved infinite IID process. Finite marginal moments are qualification checks only.
View answer and explanation
([[2, 1/4], [3, 3/4]], 11/4, 3/16, Yes, infinite_iid, CLT_hypotheses_qualified)
Compute the exact common mean and centered second moment, requiring strictly positive finite variance. This qualifies a declared marginal law; it does not verify independence or an infinite process from samples, nor prove the CLT. The exact result, retaining the requested order, is ([[2, 1/4], [3, 3/4]], 11/4, 3/16, Yes, infinite_iid, CLT_hypotheses_qualified).
Reviewed step-by-step derivation
Central limit theorem for finite-variance IID sequences · qf-mod49-clt-finite-law-moments · candidate
16. Finite Marginal Moments Qualify the Declared IID CLT Premises
Use the aligned finite law values=[-2, -1], probabilities=[2/5, 3/5], with declared process certificate=infinite_iid. The token is an explicit infinite IID premise, not a process construction or proof by this finite calculation. Give (law_rows, mean, variance, positive_finite_variance, declared_process_certificate, qualification_statement).
infinite_iid is a declared premise token, not a constructed or proved infinite IID process. Finite marginal moments are qualification checks only.
View answer and explanation
([[-2, 2/5], [-1, 3/5]], -7/5, 6/25, Yes, infinite_iid, CLT_hypotheses_qualified)
Compute the exact common mean and centered second moment, requiring strictly positive finite variance. This qualifies a declared marginal law; it does not verify independence or an infinite process from samples, nor prove the CLT. The exact result, retaining the requested order, is ([[-2, 2/5], [-1, 3/5]], -7/5, 6/25, Yes, infinite_iid, CLT_hypotheses_qualified).
Reviewed step-by-step derivation
Central limit theorem for finite-variance IID sequences · qf-mod49-clt-finite-law-moments · candidate
17. Finite Marginal Moments Qualify the Declared IID CLT Premises
Use the aligned finite law values=[-1, 0], probabilities=[2/5, 3/5], with declared process certificate=infinite_iid. The token is an explicit infinite IID premise, not a process construction or proof by this finite calculation. Give (law_rows, mean, variance, positive_finite_variance, declared_process_certificate, qualification_statement).
infinite_iid is a declared premise token, not a constructed or proved infinite IID process. Finite marginal moments are qualification checks only.
View answer and explanation
([[-1, 2/5], [0, 3/5]], -2/5, 6/25, Yes, infinite_iid, CLT_hypotheses_qualified)
Compute the exact common mean and centered second moment, requiring strictly positive finite variance. This qualifies a declared marginal law; it does not verify independence or an infinite process from samples, nor prove the CLT. The exact result, retaining the requested order, is ([[-1, 2/5], [0, 3/5]], -2/5, 6/25, Yes, infinite_iid, CLT_hypotheses_qualified).
Reviewed step-by-step derivation
Central limit theorem for finite-variance IID sequences · qf-mod49-clt-finite-law-moments · candidate
18. Finite Marginal Moments Qualify the Declared IID CLT Premises
Use the aligned finite law values=[0, 1], probabilities=[2/5, 3/5], with declared process certificate=infinite_iid. The token is an explicit infinite IID premise, not a process construction or proof by this finite calculation. Give (law_rows, mean, variance, positive_finite_variance, declared_process_certificate, qualification_statement).
infinite_iid is a declared premise token, not a constructed or proved infinite IID process. Finite marginal moments are qualification checks only.
View answer and explanation
([[0, 2/5], [1, 3/5]], 3/5, 6/25, Yes, infinite_iid, CLT_hypotheses_qualified)
Compute the exact common mean and centered second moment, requiring strictly positive finite variance. This qualifies a declared marginal law; it does not verify independence or an infinite process from samples, nor prove the CLT. The exact result, retaining the requested order, is ([[0, 2/5], [1, 3/5]], 3/5, 6/25, Yes, infinite_iid, CLT_hypotheses_qualified).
Reviewed step-by-step derivation
Central limit theorem for finite-variance IID sequences · qf-mod49-clt-finite-law-moments · candidate
19. Finite Marginal Moments Qualify the Declared IID CLT Premises
Use the aligned finite law values=[1, 2], probabilities=[2/5, 3/5], with declared process certificate=infinite_iid. The token is an explicit infinite IID premise, not a process construction or proof by this finite calculation. Give (law_rows, mean, variance, positive_finite_variance, declared_process_certificate, qualification_statement).
infinite_iid is a declared premise token, not a constructed or proved infinite IID process. Finite marginal moments are qualification checks only.
View answer and explanation
([[1, 2/5], [2, 3/5]], 8/5, 6/25, Yes, infinite_iid, CLT_hypotheses_qualified)
Compute the exact common mean and centered second moment, requiring strictly positive finite variance. This qualifies a declared marginal law; it does not verify independence or an infinite process from samples, nor prove the CLT. The exact result, retaining the requested order, is ([[1, 2/5], [2, 3/5]], 8/5, 6/25, Yes, infinite_iid, CLT_hypotheses_qualified).
Reviewed step-by-step derivation
Central limit theorem for finite-variance IID sequences · qf-mod49-clt-finite-law-moments · candidate
20. Finite Marginal Moments Qualify the Declared IID CLT Premises
Use the aligned finite law values=[2, 3], probabilities=[2/5, 3/5], with declared process certificate=infinite_iid. The token is an explicit infinite IID premise, not a process construction or proof by this finite calculation. Give (law_rows, mean, variance, positive_finite_variance, declared_process_certificate, qualification_statement).
infinite_iid is a declared premise token, not a constructed or proved infinite IID process. Finite marginal moments are qualification checks only.
View answer and explanation
([[2, 2/5], [3, 3/5]], 13/5, 6/25, Yes, infinite_iid, CLT_hypotheses_qualified)
Compute the exact common mean and centered second moment, requiring strictly positive finite variance. This qualifies a declared marginal law; it does not verify independence or an infinite process from samples, nor prove the CLT. The exact result, retaining the requested order, is ([[2, 2/5], [3, 3/5]], 13/5, 6/25, Yes, infinite_iid, CLT_hypotheses_qualified).
Reviewed step-by-step derivation
Central limit theorem for finite-variance IID sequences · qf-mod49-clt-finite-law-moments · candidate
21. Finite Marginal Moments Qualify the Declared IID CLT Premises
Use the aligned finite law values=[-2, -1], probabilities=[1/5, 4/5], with declared process certificate=infinite_iid. The token is an explicit infinite IID premise, not a process construction or proof by this finite calculation. Give (law_rows, mean, variance, positive_finite_variance, declared_process_certificate, qualification_statement).
infinite_iid is a declared premise token, not a constructed or proved infinite IID process. Finite marginal moments are qualification checks only.
View answer and explanation
([[-2, 1/5], [-1, 4/5]], -6/5, 4/25, Yes, infinite_iid, CLT_hypotheses_qualified)
Compute the exact common mean and centered second moment, requiring strictly positive finite variance. This qualifies a declared marginal law; it does not verify independence or an infinite process from samples, nor prove the CLT. The exact result, retaining the requested order, is ([[-2, 1/5], [-1, 4/5]], -6/5, 4/25, Yes, infinite_iid, CLT_hypotheses_qualified).
Reviewed step-by-step derivation
Central limit theorem for finite-variance IID sequences · qf-mod49-clt-finite-law-moments · candidate
22. Finite Marginal Moments Qualify the Declared IID CLT Premises
Use the aligned finite law values=[-1, 0], probabilities=[1/5, 4/5], with declared process certificate=infinite_iid. The token is an explicit infinite IID premise, not a process construction or proof by this finite calculation. Give (law_rows, mean, variance, positive_finite_variance, declared_process_certificate, qualification_statement).
infinite_iid is a declared premise token, not a constructed or proved infinite IID process. Finite marginal moments are qualification checks only.
View answer and explanation
([[-1, 1/5], [0, 4/5]], -1/5, 4/25, Yes, infinite_iid, CLT_hypotheses_qualified)
Compute the exact common mean and centered second moment, requiring strictly positive finite variance. This qualifies a declared marginal law; it does not verify independence or an infinite process from samples, nor prove the CLT. The exact result, retaining the requested order, is ([[-1, 1/5], [0, 4/5]], -1/5, 4/25, Yes, infinite_iid, CLT_hypotheses_qualified).
Reviewed step-by-step derivation
Central limit theorem for finite-variance IID sequences · qf-mod49-clt-finite-law-moments · candidate
23. Finite Marginal Moments Qualify the Declared IID CLT Premises
Use the aligned finite law values=[0, 1], probabilities=[1/5, 4/5], with declared process certificate=infinite_iid. The token is an explicit infinite IID premise, not a process construction or proof by this finite calculation. Give (law_rows, mean, variance, positive_finite_variance, declared_process_certificate, qualification_statement).
infinite_iid is a declared premise token, not a constructed or proved infinite IID process. Finite marginal moments are qualification checks only.
View answer and explanation
([[0, 1/5], [1, 4/5]], 4/5, 4/25, Yes, infinite_iid, CLT_hypotheses_qualified)
Compute the exact common mean and centered second moment, requiring strictly positive finite variance. This qualifies a declared marginal law; it does not verify independence or an infinite process from samples, nor prove the CLT. The exact result, retaining the requested order, is ([[0, 1/5], [1, 4/5]], 4/5, 4/25, Yes, infinite_iid, CLT_hypotheses_qualified).
Reviewed step-by-step derivation
Central limit theorem for finite-variance IID sequences · qf-mod49-clt-finite-law-moments · candidate
24. Finite Marginal Moments Qualify the Declared IID CLT Premises
Use the aligned finite law values=[1, 2], probabilities=[1/5, 4/5], with declared process certificate=infinite_iid. The token is an explicit infinite IID premise, not a process construction or proof by this finite calculation. Give (law_rows, mean, variance, positive_finite_variance, declared_process_certificate, qualification_statement).
infinite_iid is a declared premise token, not a constructed or proved infinite IID process. Finite marginal moments are qualification checks only.
View answer and explanation
([[1, 1/5], [2, 4/5]], 9/5, 4/25, Yes, infinite_iid, CLT_hypotheses_qualified)
Compute the exact common mean and centered second moment, requiring strictly positive finite variance. This qualifies a declared marginal law; it does not verify independence or an infinite process from samples, nor prove the CLT. The exact result, retaining the requested order, is ([[1, 1/5], [2, 4/5]], 9/5, 4/25, Yes, infinite_iid, CLT_hypotheses_qualified).
Reviewed step-by-step derivation
Central limit theorem for finite-variance IID sequences · qf-mod49-clt-finite-law-moments · candidate
25. Finite Marginal Moments Qualify the Declared IID CLT Premises
Use the aligned finite law values=[2, 3], probabilities=[1/5, 4/5], with declared process certificate=infinite_iid. The token is an explicit infinite IID premise, not a process construction or proof by this finite calculation. Give (law_rows, mean, variance, positive_finite_variance, declared_process_certificate, qualification_statement).
infinite_iid is a declared premise token, not a constructed or proved infinite IID process. Finite marginal moments are qualification checks only.
View answer and explanation
([[2, 1/5], [3, 4/5]], 14/5, 4/25, Yes, infinite_iid, CLT_hypotheses_qualified)
Compute the exact common mean and centered second moment, requiring strictly positive finite variance. This qualifies a declared marginal law; it does not verify independence or an infinite process from samples, nor prove the CLT. The exact result, retaining the requested order, is ([[2, 1/5], [3, 4/5]], 14/5, 4/25, Yes, infinite_iid, CLT_hypotheses_qualified).
Reviewed step-by-step derivation
Central limit theorem for finite-variance IID sequences · qf-mod49-clt-finite-law-moments · candidate
26. Complete Fixed-n Sum Moments Do Not Prove a Distributional Limit
Use the aligned finite law values=[-2, -1], probabilities=[1/2, 1/2], with declared process certificate=infinite_iid. The token is an explicit infinite IID premise, not a process construction or proof by this finite calculation. Use sample size n=1 and S_n as the sum of all n coordinates. Give (n, sum_distribution, total_probability, sum_mean, sum_variance, n_times_mean, n_times_variance, standardized_mean, standardized_variance, standardization_description, finite_scope).
The complete finite joint law gives standardized moments0 and1; it neither makes Z_n normal at finite n nor proves a distributional limit.
View answer and explanation
(1, [[-2, 1/2], [-1, 1/2]], 1, -3/2, 1/4, -3/2, 1/4, 0, 1, Z_n=(S_n-n*mu)/(sigma*sqrt(n)), complete_joint_law_moments_do_not_prove_distributional_convergence)
Aggregate the entire finite IID product law by sum. Retain total mass, E[S_n]=n*mu and Var(S_n)=n*sigma^2. The centered normalization Z_n=(S_n−n*mu)/(sigma*sqrt(n)) has mean0 and variance1. Those moments do not make a finite-n law normal or prove convergence in distribution. The exact result, retaining the requested order, is (1, [[-2, 1/2], [-1, 1/2]], 1, -3/2, 1/4, -3/2, 1/4, 0, 1, Z_n=(S_n-n*mu)/(sigma*sqrt(n)), complete_joint_law_moments_do_not_prove_distributional_convergence).
Reviewed step-by-step derivation
Central limit theorem for finite-variance IID sequences · qf-mod49-clt-standardized-sum-moments · candidate
27. Complete Fixed-n Sum Moments Do Not Prove a Distributional Limit
Use the aligned finite law values=[-1, 0], probabilities=[1/2, 1/2], with declared process certificate=infinite_iid. The token is an explicit infinite IID premise, not a process construction or proof by this finite calculation. Use sample size n=1 and S_n as the sum of all n coordinates. Give (n, sum_distribution, total_probability, sum_mean, sum_variance, n_times_mean, n_times_variance, standardized_mean, standardized_variance, standardization_description, finite_scope).
The complete finite joint law gives standardized moments0 and1; it neither makes Z_n normal at finite n nor proves a distributional limit.
View answer and explanation
(1, [[-1, 1/2], [0, 1/2]], 1, -1/2, 1/4, -1/2, 1/4, 0, 1, Z_n=(S_n-n*mu)/(sigma*sqrt(n)), complete_joint_law_moments_do_not_prove_distributional_convergence)
Aggregate the entire finite IID product law by sum. Retain total mass, E[S_n]=n*mu and Var(S_n)=n*sigma^2. The centered normalization Z_n=(S_n−n*mu)/(sigma*sqrt(n)) has mean0 and variance1. Those moments do not make a finite-n law normal or prove convergence in distribution. The exact result, retaining the requested order, is (1, [[-1, 1/2], [0, 1/2]], 1, -1/2, 1/4, -1/2, 1/4, 0, 1, Z_n=(S_n-n*mu)/(sigma*sqrt(n)), complete_joint_law_moments_do_not_prove_distributional_convergence).
Reviewed step-by-step derivation
Central limit theorem for finite-variance IID sequences · qf-mod49-clt-standardized-sum-moments · candidate
28. Complete Fixed-n Sum Moments Do Not Prove a Distributional Limit
Use the aligned finite law values=[0, 1], probabilities=[1/2, 1/2], with declared process certificate=infinite_iid. The token is an explicit infinite IID premise, not a process construction or proof by this finite calculation. Use sample size n=1 and S_n as the sum of all n coordinates. Give (n, sum_distribution, total_probability, sum_mean, sum_variance, n_times_mean, n_times_variance, standardized_mean, standardized_variance, standardization_description, finite_scope).
The complete finite joint law gives standardized moments0 and1; it neither makes Z_n normal at finite n nor proves a distributional limit.
View answer and explanation
(1, [[0, 1/2], [1, 1/2]], 1, 1/2, 1/4, 1/2, 1/4, 0, 1, Z_n=(S_n-n*mu)/(sigma*sqrt(n)), complete_joint_law_moments_do_not_prove_distributional_convergence)
Aggregate the entire finite IID product law by sum. Retain total mass, E[S_n]=n*mu and Var(S_n)=n*sigma^2. The centered normalization Z_n=(S_n−n*mu)/(sigma*sqrt(n)) has mean0 and variance1. Those moments do not make a finite-n law normal or prove convergence in distribution. The exact result, retaining the requested order, is (1, [[0, 1/2], [1, 1/2]], 1, 1/2, 1/4, 1/2, 1/4, 0, 1, Z_n=(S_n-n*mu)/(sigma*sqrt(n)), complete_joint_law_moments_do_not_prove_distributional_convergence).
Reviewed step-by-step derivation
Central limit theorem for finite-variance IID sequences · qf-mod49-clt-standardized-sum-moments · candidate
29. Complete Fixed-n Sum Moments Do Not Prove a Distributional Limit
Use the aligned finite law values=[1, 2], probabilities=[1/2, 1/2], with declared process certificate=infinite_iid. The token is an explicit infinite IID premise, not a process construction or proof by this finite calculation. Use sample size n=1 and S_n as the sum of all n coordinates. Give (n, sum_distribution, total_probability, sum_mean, sum_variance, n_times_mean, n_times_variance, standardized_mean, standardized_variance, standardization_description, finite_scope).
The complete finite joint law gives standardized moments0 and1; it neither makes Z_n normal at finite n nor proves a distributional limit.
View answer and explanation
(1, [[1, 1/2], [2, 1/2]], 1, 3/2, 1/4, 3/2, 1/4, 0, 1, Z_n=(S_n-n*mu)/(sigma*sqrt(n)), complete_joint_law_moments_do_not_prove_distributional_convergence)
Aggregate the entire finite IID product law by sum. Retain total mass, E[S_n]=n*mu and Var(S_n)=n*sigma^2. The centered normalization Z_n=(S_n−n*mu)/(sigma*sqrt(n)) has mean0 and variance1. Those moments do not make a finite-n law normal or prove convergence in distribution. The exact result, retaining the requested order, is (1, [[1, 1/2], [2, 1/2]], 1, 3/2, 1/4, 3/2, 1/4, 0, 1, Z_n=(S_n-n*mu)/(sigma*sqrt(n)), complete_joint_law_moments_do_not_prove_distributional_convergence).
Reviewed step-by-step derivation
Central limit theorem for finite-variance IID sequences · qf-mod49-clt-standardized-sum-moments · candidate
30. Complete Fixed-n Sum Moments Do Not Prove a Distributional Limit
Use the aligned finite law values=[2, 3], probabilities=[1/2, 1/2], with declared process certificate=infinite_iid. The token is an explicit infinite IID premise, not a process construction or proof by this finite calculation. Use sample size n=1 and S_n as the sum of all n coordinates. Give (n, sum_distribution, total_probability, sum_mean, sum_variance, n_times_mean, n_times_variance, standardized_mean, standardized_variance, standardization_description, finite_scope).
The complete finite joint law gives standardized moments0 and1; it neither makes Z_n normal at finite n nor proves a distributional limit.
View answer and explanation
(1, [[2, 1/2], [3, 1/2]], 1, 5/2, 1/4, 5/2, 1/4, 0, 1, Z_n=(S_n-n*mu)/(sigma*sqrt(n)), complete_joint_law_moments_do_not_prove_distributional_convergence)
Aggregate the entire finite IID product law by sum. Retain total mass, E[S_n]=n*mu and Var(S_n)=n*sigma^2. The centered normalization Z_n=(S_n−n*mu)/(sigma*sqrt(n)) has mean0 and variance1. Those moments do not make a finite-n law normal or prove convergence in distribution. The exact result, retaining the requested order, is (1, [[2, 1/2], [3, 1/2]], 1, 5/2, 1/4, 5/2, 1/4, 0, 1, Z_n=(S_n-n*mu)/(sigma*sqrt(n)), complete_joint_law_moments_do_not_prove_distributional_convergence).
Reviewed step-by-step derivation
Central limit theorem for finite-variance IID sequences · qf-mod49-clt-standardized-sum-moments · candidate
31. Complete Fixed-n Sum Moments Do Not Prove a Distributional Limit
Use the aligned finite law values=[-2, -1], probabilities=[1/2, 1/2], with declared process certificate=infinite_iid. The token is an explicit infinite IID premise, not a process construction or proof by this finite calculation. Use sample size n=2 and S_n as the sum of all n coordinates. Give (n, sum_distribution, total_probability, sum_mean, sum_variance, n_times_mean, n_times_variance, standardized_mean, standardized_variance, standardization_description, finite_scope).
The complete finite joint law gives standardized moments0 and1; it neither makes Z_n normal at finite n nor proves a distributional limit.
View answer and explanation
(2, [[-4, 1/4], [-3, 1/2], [-2, 1/4]], 1, -3, 1/2, -3, 1/2, 0, 1, Z_n=(S_n-n*mu)/(sigma*sqrt(n)), complete_joint_law_moments_do_not_prove_distributional_convergence)
Aggregate the entire finite IID product law by sum. Retain total mass, E[S_n]=n*mu and Var(S_n)=n*sigma^2. The centered normalization Z_n=(S_n−n*mu)/(sigma*sqrt(n)) has mean0 and variance1. Those moments do not make a finite-n law normal or prove convergence in distribution. The exact result, retaining the requested order, is (2, [[-4, 1/4], [-3, 1/2], [-2, 1/4]], 1, -3, 1/2, -3, 1/2, 0, 1, Z_n=(S_n-n*mu)/(sigma*sqrt(n)), complete_joint_law_moments_do_not_prove_distributional_convergence).
Reviewed step-by-step derivation
Central limit theorem for finite-variance IID sequences · qf-mod49-clt-standardized-sum-moments · candidate
32. Complete Fixed-n Sum Moments Do Not Prove a Distributional Limit
Use the aligned finite law values=[-1, 0], probabilities=[1/2, 1/2], with declared process certificate=infinite_iid. The token is an explicit infinite IID premise, not a process construction or proof by this finite calculation. Use sample size n=2 and S_n as the sum of all n coordinates. Give (n, sum_distribution, total_probability, sum_mean, sum_variance, n_times_mean, n_times_variance, standardized_mean, standardized_variance, standardization_description, finite_scope).
The complete finite joint law gives standardized moments0 and1; it neither makes Z_n normal at finite n nor proves a distributional limit.
View answer and explanation
(2, [[-2, 1/4], [-1, 1/2], [0, 1/4]], 1, -1, 1/2, -1, 1/2, 0, 1, Z_n=(S_n-n*mu)/(sigma*sqrt(n)), complete_joint_law_moments_do_not_prove_distributional_convergence)
Aggregate the entire finite IID product law by sum. Retain total mass, E[S_n]=n*mu and Var(S_n)=n*sigma^2. The centered normalization Z_n=(S_n−n*mu)/(sigma*sqrt(n)) has mean0 and variance1. Those moments do not make a finite-n law normal or prove convergence in distribution. The exact result, retaining the requested order, is (2, [[-2, 1/4], [-1, 1/2], [0, 1/4]], 1, -1, 1/2, -1, 1/2, 0, 1, Z_n=(S_n-n*mu)/(sigma*sqrt(n)), complete_joint_law_moments_do_not_prove_distributional_convergence).
Reviewed step-by-step derivation
Central limit theorem for finite-variance IID sequences · qf-mod49-clt-standardized-sum-moments · candidate
33. Complete Fixed-n Sum Moments Do Not Prove a Distributional Limit
Use the aligned finite law values=[0, 1], probabilities=[1/2, 1/2], with declared process certificate=infinite_iid. The token is an explicit infinite IID premise, not a process construction or proof by this finite calculation. Use sample size n=2 and S_n as the sum of all n coordinates. Give (n, sum_distribution, total_probability, sum_mean, sum_variance, n_times_mean, n_times_variance, standardized_mean, standardized_variance, standardization_description, finite_scope).
The complete finite joint law gives standardized moments0 and1; it neither makes Z_n normal at finite n nor proves a distributional limit.
View answer and explanation
(2, [[0, 1/4], [1, 1/2], [2, 1/4]], 1, 1, 1/2, 1, 1/2, 0, 1, Z_n=(S_n-n*mu)/(sigma*sqrt(n)), complete_joint_law_moments_do_not_prove_distributional_convergence)
Aggregate the entire finite IID product law by sum. Retain total mass, E[S_n]=n*mu and Var(S_n)=n*sigma^2. The centered normalization Z_n=(S_n−n*mu)/(sigma*sqrt(n)) has mean0 and variance1. Those moments do not make a finite-n law normal or prove convergence in distribution. The exact result, retaining the requested order, is (2, [[0, 1/4], [1, 1/2], [2, 1/4]], 1, 1, 1/2, 1, 1/2, 0, 1, Z_n=(S_n-n*mu)/(sigma*sqrt(n)), complete_joint_law_moments_do_not_prove_distributional_convergence).
Reviewed step-by-step derivation
Central limit theorem for finite-variance IID sequences · qf-mod49-clt-standardized-sum-moments · candidate
34. Complete Fixed-n Sum Moments Do Not Prove a Distributional Limit
Use the aligned finite law values=[1, 2], probabilities=[1/2, 1/2], with declared process certificate=infinite_iid. The token is an explicit infinite IID premise, not a process construction or proof by this finite calculation. Use sample size n=2 and S_n as the sum of all n coordinates. Give (n, sum_distribution, total_probability, sum_mean, sum_variance, n_times_mean, n_times_variance, standardized_mean, standardized_variance, standardization_description, finite_scope).
The complete finite joint law gives standardized moments0 and1; it neither makes Z_n normal at finite n nor proves a distributional limit.
View answer and explanation
(2, [[2, 1/4], [3, 1/2], [4, 1/4]], 1, 3, 1/2, 3, 1/2, 0, 1, Z_n=(S_n-n*mu)/(sigma*sqrt(n)), complete_joint_law_moments_do_not_prove_distributional_convergence)
Aggregate the entire finite IID product law by sum. Retain total mass, E[S_n]=n*mu and Var(S_n)=n*sigma^2. The centered normalization Z_n=(S_n−n*mu)/(sigma*sqrt(n)) has mean0 and variance1. Those moments do not make a finite-n law normal or prove convergence in distribution. The exact result, retaining the requested order, is (2, [[2, 1/4], [3, 1/2], [4, 1/4]], 1, 3, 1/2, 3, 1/2, 0, 1, Z_n=(S_n-n*mu)/(sigma*sqrt(n)), complete_joint_law_moments_do_not_prove_distributional_convergence).
Reviewed step-by-step derivation
Central limit theorem for finite-variance IID sequences · qf-mod49-clt-standardized-sum-moments · candidate
35. Complete Fixed-n Sum Moments Do Not Prove a Distributional Limit
Use the aligned finite law values=[2, 3], probabilities=[1/2, 1/2], with declared process certificate=infinite_iid. The token is an explicit infinite IID premise, not a process construction or proof by this finite calculation. Use sample size n=2 and S_n as the sum of all n coordinates. Give (n, sum_distribution, total_probability, sum_mean, sum_variance, n_times_mean, n_times_variance, standardized_mean, standardized_variance, standardization_description, finite_scope).
The complete finite joint law gives standardized moments0 and1; it neither makes Z_n normal at finite n nor proves a distributional limit.
View answer and explanation
(2, [[4, 1/4], [5, 1/2], [6, 1/4]], 1, 5, 1/2, 5, 1/2, 0, 1, Z_n=(S_n-n*mu)/(sigma*sqrt(n)), complete_joint_law_moments_do_not_prove_distributional_convergence)
Aggregate the entire finite IID product law by sum. Retain total mass, E[S_n]=n*mu and Var(S_n)=n*sigma^2. The centered normalization Z_n=(S_n−n*mu)/(sigma*sqrt(n)) has mean0 and variance1. Those moments do not make a finite-n law normal or prove convergence in distribution. The exact result, retaining the requested order, is (2, [[4, 1/4], [5, 1/2], [6, 1/4]], 1, 5, 1/2, 5, 1/2, 0, 1, Z_n=(S_n-n*mu)/(sigma*sqrt(n)), complete_joint_law_moments_do_not_prove_distributional_convergence).
Reviewed step-by-step derivation
Central limit theorem for finite-variance IID sequences · qf-mod49-clt-standardized-sum-moments · candidate
36. Complete Fixed-n Sum Moments Do Not Prove a Distributional Limit
Use the aligned finite law values=[-2, -1], probabilities=[1/2, 1/2], with declared process certificate=infinite_iid. The token is an explicit infinite IID premise, not a process construction or proof by this finite calculation. Use sample size n=3 and S_n as the sum of all n coordinates. Give (n, sum_distribution, total_probability, sum_mean, sum_variance, n_times_mean, n_times_variance, standardized_mean, standardized_variance, standardization_description, finite_scope).
The complete finite joint law gives standardized moments0 and1; it neither makes Z_n normal at finite n nor proves a distributional limit.
View answer and explanation
(3, [[-6, 1/8], [-5, 3/8], [-4, 3/8], [-3, 1/8]], 1, -9/2, 3/4, -9/2, 3/4, 0, 1, Z_n=(S_n-n*mu)/(sigma*sqrt(n)), complete_joint_law_moments_do_not_prove_distributional_convergence)
Aggregate the entire finite IID product law by sum. Retain total mass, E[S_n]=n*mu and Var(S_n)=n*sigma^2. The centered normalization Z_n=(S_n−n*mu)/(sigma*sqrt(n)) has mean0 and variance1. Those moments do not make a finite-n law normal or prove convergence in distribution. The exact result, retaining the requested order, is (3, [[-6, 1/8], [-5, 3/8], [-4, 3/8], [-3, 1/8]], 1, -9/2, 3/4, -9/2, 3/4, 0, 1, Z_n=(S_n-n*mu)/(sigma*sqrt(n)), complete_joint_law_moments_do_not_prove_distributional_convergence).
Reviewed step-by-step derivation
Central limit theorem for finite-variance IID sequences · qf-mod49-clt-standardized-sum-moments · candidate
37. Complete Fixed-n Sum Moments Do Not Prove a Distributional Limit
Use the aligned finite law values=[-1, 0], probabilities=[1/2, 1/2], with declared process certificate=infinite_iid. The token is an explicit infinite IID premise, not a process construction or proof by this finite calculation. Use sample size n=3 and S_n as the sum of all n coordinates. Give (n, sum_distribution, total_probability, sum_mean, sum_variance, n_times_mean, n_times_variance, standardized_mean, standardized_variance, standardization_description, finite_scope).
The complete finite joint law gives standardized moments0 and1; it neither makes Z_n normal at finite n nor proves a distributional limit.
View answer and explanation
(3, [[-3, 1/8], [-2, 3/8], [-1, 3/8], [0, 1/8]], 1, -3/2, 3/4, -3/2, 3/4, 0, 1, Z_n=(S_n-n*mu)/(sigma*sqrt(n)), complete_joint_law_moments_do_not_prove_distributional_convergence)
Aggregate the entire finite IID product law by sum. Retain total mass, E[S_n]=n*mu and Var(S_n)=n*sigma^2. The centered normalization Z_n=(S_n−n*mu)/(sigma*sqrt(n)) has mean0 and variance1. Those moments do not make a finite-n law normal or prove convergence in distribution. The exact result, retaining the requested order, is (3, [[-3, 1/8], [-2, 3/8], [-1, 3/8], [0, 1/8]], 1, -3/2, 3/4, -3/2, 3/4, 0, 1, Z_n=(S_n-n*mu)/(sigma*sqrt(n)), complete_joint_law_moments_do_not_prove_distributional_convergence).
Reviewed step-by-step derivation
Central limit theorem for finite-variance IID sequences · qf-mod49-clt-standardized-sum-moments · candidate
38. Complete Fixed-n Sum Moments Do Not Prove a Distributional Limit
Use the aligned finite law values=[0, 1], probabilities=[1/2, 1/2], with declared process certificate=infinite_iid. The token is an explicit infinite IID premise, not a process construction or proof by this finite calculation. Use sample size n=3 and S_n as the sum of all n coordinates. Give (n, sum_distribution, total_probability, sum_mean, sum_variance, n_times_mean, n_times_variance, standardized_mean, standardized_variance, standardization_description, finite_scope).
The complete finite joint law gives standardized moments0 and1; it neither makes Z_n normal at finite n nor proves a distributional limit.
View answer and explanation
(3, [[0, 1/8], [1, 3/8], [2, 3/8], [3, 1/8]], 1, 3/2, 3/4, 3/2, 3/4, 0, 1, Z_n=(S_n-n*mu)/(sigma*sqrt(n)), complete_joint_law_moments_do_not_prove_distributional_convergence)
Aggregate the entire finite IID product law by sum. Retain total mass, E[S_n]=n*mu and Var(S_n)=n*sigma^2. The centered normalization Z_n=(S_n−n*mu)/(sigma*sqrt(n)) has mean0 and variance1. Those moments do not make a finite-n law normal or prove convergence in distribution. The exact result, retaining the requested order, is (3, [[0, 1/8], [1, 3/8], [2, 3/8], [3, 1/8]], 1, 3/2, 3/4, 3/2, 3/4, 0, 1, Z_n=(S_n-n*mu)/(sigma*sqrt(n)), complete_joint_law_moments_do_not_prove_distributional_convergence).
Reviewed step-by-step derivation
Central limit theorem for finite-variance IID sequences · qf-mod49-clt-standardized-sum-moments · candidate
39. Complete Fixed-n Sum Moments Do Not Prove a Distributional Limit
Use the aligned finite law values=[1, 2], probabilities=[1/2, 1/2], with declared process certificate=infinite_iid. The token is an explicit infinite IID premise, not a process construction or proof by this finite calculation. Use sample size n=3 and S_n as the sum of all n coordinates. Give (n, sum_distribution, total_probability, sum_mean, sum_variance, n_times_mean, n_times_variance, standardized_mean, standardized_variance, standardization_description, finite_scope).
The complete finite joint law gives standardized moments0 and1; it neither makes Z_n normal at finite n nor proves a distributional limit.
View answer and explanation
(3, [[3, 1/8], [4, 3/8], [5, 3/8], [6, 1/8]], 1, 9/2, 3/4, 9/2, 3/4, 0, 1, Z_n=(S_n-n*mu)/(sigma*sqrt(n)), complete_joint_law_moments_do_not_prove_distributional_convergence)
Aggregate the entire finite IID product law by sum. Retain total mass, E[S_n]=n*mu and Var(S_n)=n*sigma^2. The centered normalization Z_n=(S_n−n*mu)/(sigma*sqrt(n)) has mean0 and variance1. Those moments do not make a finite-n law normal or prove convergence in distribution. The exact result, retaining the requested order, is (3, [[3, 1/8], [4, 3/8], [5, 3/8], [6, 1/8]], 1, 9/2, 3/4, 9/2, 3/4, 0, 1, Z_n=(S_n-n*mu)/(sigma*sqrt(n)), complete_joint_law_moments_do_not_prove_distributional_convergence).
Reviewed step-by-step derivation
Central limit theorem for finite-variance IID sequences · qf-mod49-clt-standardized-sum-moments · candidate
40. Complete Fixed-n Sum Moments Do Not Prove a Distributional Limit
Use the aligned finite law values=[2, 3], probabilities=[1/2, 1/2], with declared process certificate=infinite_iid. The token is an explicit infinite IID premise, not a process construction or proof by this finite calculation. Use sample size n=3 and S_n as the sum of all n coordinates. Give (n, sum_distribution, total_probability, sum_mean, sum_variance, n_times_mean, n_times_variance, standardized_mean, standardized_variance, standardization_description, finite_scope).
The complete finite joint law gives standardized moments0 and1; it neither makes Z_n normal at finite n nor proves a distributional limit.
View answer and explanation
(3, [[6, 1/8], [7, 3/8], [8, 3/8], [9, 1/8]], 1, 15/2, 3/4, 15/2, 3/4, 0, 1, Z_n=(S_n-n*mu)/(sigma*sqrt(n)), complete_joint_law_moments_do_not_prove_distributional_convergence)
Aggregate the entire finite IID product law by sum. Retain total mass, E[S_n]=n*mu and Var(S_n)=n*sigma^2. The centered normalization Z_n=(S_n−n*mu)/(sigma*sqrt(n)) has mean0 and variance1. Those moments do not make a finite-n law normal or prove convergence in distribution. The exact result, retaining the requested order, is (3, [[6, 1/8], [7, 3/8], [8, 3/8], [9, 1/8]], 1, 15/2, 3/4, 15/2, 3/4, 0, 1, Z_n=(S_n-n*mu)/(sigma*sqrt(n)), complete_joint_law_moments_do_not_prove_distributional_convergence).
Reviewed step-by-step derivation
Central limit theorem for finite-variance IID sequences · qf-mod49-clt-standardized-sum-moments · candidate
41. Complete Fixed-n Sum Moments Do Not Prove a Distributional Limit
Use the aligned finite law values=[-2, -1], probabilities=[1/2, 1/2], with declared process certificate=infinite_iid. The token is an explicit infinite IID premise, not a process construction or proof by this finite calculation. Use sample size n=4 and S_n as the sum of all n coordinates. Give (n, sum_distribution, total_probability, sum_mean, sum_variance, n_times_mean, n_times_variance, standardized_mean, standardized_variance, standardization_description, finite_scope).
The complete finite joint law gives standardized moments0 and1; it neither makes Z_n normal at finite n nor proves a distributional limit.
View answer and explanation
(4, [[-8, 1/16], [-7, 1/4], [-6, 3/8], [-5, 1/4], [-4, 1/16]], 1, -6, 1, -6, 1, 0, 1, Z_n=(S_n-n*mu)/(sigma*sqrt(n)), complete_joint_law_moments_do_not_prove_distributional_convergence)
Aggregate the entire finite IID product law by sum. Retain total mass, E[S_n]=n*mu and Var(S_n)=n*sigma^2. The centered normalization Z_n=(S_n−n*mu)/(sigma*sqrt(n)) has mean0 and variance1. Those moments do not make a finite-n law normal or prove convergence in distribution. The exact result, retaining the requested order, is (4, [[-8, 1/16], [-7, 1/4], [-6, 3/8], [-5, 1/4], [-4, 1/16]], 1, -6, 1, -6, 1, 0, 1, Z_n=(S_n-n*mu)/(sigma*sqrt(n)), complete_joint_law_moments_do_not_prove_distributional_convergence).
Reviewed step-by-step derivation
Central limit theorem for finite-variance IID sequences · qf-mod49-clt-standardized-sum-moments · candidate
42. Complete Fixed-n Sum Moments Do Not Prove a Distributional Limit
Use the aligned finite law values=[-1, 0], probabilities=[1/2, 1/2], with declared process certificate=infinite_iid. The token is an explicit infinite IID premise, not a process construction or proof by this finite calculation. Use sample size n=4 and S_n as the sum of all n coordinates. Give (n, sum_distribution, total_probability, sum_mean, sum_variance, n_times_mean, n_times_variance, standardized_mean, standardized_variance, standardization_description, finite_scope).
The complete finite joint law gives standardized moments0 and1; it neither makes Z_n normal at finite n nor proves a distributional limit.
View answer and explanation
(4, [[-4, 1/16], [-3, 1/4], [-2, 3/8], [-1, 1/4], [0, 1/16]], 1, -2, 1, -2, 1, 0, 1, Z_n=(S_n-n*mu)/(sigma*sqrt(n)), complete_joint_law_moments_do_not_prove_distributional_convergence)
Aggregate the entire finite IID product law by sum. Retain total mass, E[S_n]=n*mu and Var(S_n)=n*sigma^2. The centered normalization Z_n=(S_n−n*mu)/(sigma*sqrt(n)) has mean0 and variance1. Those moments do not make a finite-n law normal or prove convergence in distribution. The exact result, retaining the requested order, is (4, [[-4, 1/16], [-3, 1/4], [-2, 3/8], [-1, 1/4], [0, 1/16]], 1, -2, 1, -2, 1, 0, 1, Z_n=(S_n-n*mu)/(sigma*sqrt(n)), complete_joint_law_moments_do_not_prove_distributional_convergence).
Reviewed step-by-step derivation
Central limit theorem for finite-variance IID sequences · qf-mod49-clt-standardized-sum-moments · candidate
43. Complete Fixed-n Sum Moments Do Not Prove a Distributional Limit
Use the aligned finite law values=[0, 1], probabilities=[1/2, 1/2], with declared process certificate=infinite_iid. The token is an explicit infinite IID premise, not a process construction or proof by this finite calculation. Use sample size n=4 and S_n as the sum of all n coordinates. Give (n, sum_distribution, total_probability, sum_mean, sum_variance, n_times_mean, n_times_variance, standardized_mean, standardized_variance, standardization_description, finite_scope).
The complete finite joint law gives standardized moments0 and1; it neither makes Z_n normal at finite n nor proves a distributional limit.
View answer and explanation
(4, [[0, 1/16], [1, 1/4], [2, 3/8], [3, 1/4], [4, 1/16]], 1, 2, 1, 2, 1, 0, 1, Z_n=(S_n-n*mu)/(sigma*sqrt(n)), complete_joint_law_moments_do_not_prove_distributional_convergence)
Aggregate the entire finite IID product law by sum. Retain total mass, E[S_n]=n*mu and Var(S_n)=n*sigma^2. The centered normalization Z_n=(S_n−n*mu)/(sigma*sqrt(n)) has mean0 and variance1. Those moments do not make a finite-n law normal or prove convergence in distribution. The exact result, retaining the requested order, is (4, [[0, 1/16], [1, 1/4], [2, 3/8], [3, 1/4], [4, 1/16]], 1, 2, 1, 2, 1, 0, 1, Z_n=(S_n-n*mu)/(sigma*sqrt(n)), complete_joint_law_moments_do_not_prove_distributional_convergence).
Reviewed step-by-step derivation
Central limit theorem for finite-variance IID sequences · qf-mod49-clt-standardized-sum-moments · candidate
44. Complete Fixed-n Sum Moments Do Not Prove a Distributional Limit
Use the aligned finite law values=[1, 2], probabilities=[1/2, 1/2], with declared process certificate=infinite_iid. The token is an explicit infinite IID premise, not a process construction or proof by this finite calculation. Use sample size n=4 and S_n as the sum of all n coordinates. Give (n, sum_distribution, total_probability, sum_mean, sum_variance, n_times_mean, n_times_variance, standardized_mean, standardized_variance, standardization_description, finite_scope).
The complete finite joint law gives standardized moments0 and1; it neither makes Z_n normal at finite n nor proves a distributional limit.
View answer and explanation
(4, [[4, 1/16], [5, 1/4], [6, 3/8], [7, 1/4], [8, 1/16]], 1, 6, 1, 6, 1, 0, 1, Z_n=(S_n-n*mu)/(sigma*sqrt(n)), complete_joint_law_moments_do_not_prove_distributional_convergence)
Aggregate the entire finite IID product law by sum. Retain total mass, E[S_n]=n*mu and Var(S_n)=n*sigma^2. The centered normalization Z_n=(S_n−n*mu)/(sigma*sqrt(n)) has mean0 and variance1. Those moments do not make a finite-n law normal or prove convergence in distribution. The exact result, retaining the requested order, is (4, [[4, 1/16], [5, 1/4], [6, 3/8], [7, 1/4], [8, 1/16]], 1, 6, 1, 6, 1, 0, 1, Z_n=(S_n-n*mu)/(sigma*sqrt(n)), complete_joint_law_moments_do_not_prove_distributional_convergence).
Reviewed step-by-step derivation
Central limit theorem for finite-variance IID sequences · qf-mod49-clt-standardized-sum-moments · candidate
45. Complete Fixed-n Sum Moments Do Not Prove a Distributional Limit
Use the aligned finite law values=[2, 3], probabilities=[1/2, 1/2], with declared process certificate=infinite_iid. The token is an explicit infinite IID premise, not a process construction or proof by this finite calculation. Use sample size n=4 and S_n as the sum of all n coordinates. Give (n, sum_distribution, total_probability, sum_mean, sum_variance, n_times_mean, n_times_variance, standardized_mean, standardized_variance, standardization_description, finite_scope).
The complete finite joint law gives standardized moments0 and1; it neither makes Z_n normal at finite n nor proves a distributional limit.
View answer and explanation
(4, [[8, 1/16], [9, 1/4], [10, 3/8], [11, 1/4], [12, 1/16]], 1, 10, 1, 10, 1, 0, 1, Z_n=(S_n-n*mu)/(sigma*sqrt(n)), complete_joint_law_moments_do_not_prove_distributional_convergence)
Aggregate the entire finite IID product law by sum. Retain total mass, E[S_n]=n*mu and Var(S_n)=n*sigma^2. The centered normalization Z_n=(S_n−n*mu)/(sigma*sqrt(n)) has mean0 and variance1. Those moments do not make a finite-n law normal or prove convergence in distribution. The exact result, retaining the requested order, is (4, [[8, 1/16], [9, 1/4], [10, 3/8], [11, 1/4], [12, 1/16]], 1, 10, 1, 10, 1, 0, 1, Z_n=(S_n-n*mu)/(sigma*sqrt(n)), complete_joint_law_moments_do_not_prove_distributional_convergence).
Reviewed step-by-step derivation
Central limit theorem for finite-variance IID sequences · qf-mod49-clt-standardized-sum-moments · candidate
46. Complete Fixed-n Sum Moments Do Not Prove a Distributional Limit
Use the aligned finite law values=[-2, -1], probabilities=[1/2, 1/2], with declared process certificate=infinite_iid. The token is an explicit infinite IID premise, not a process construction or proof by this finite calculation. Use sample size n=5 and S_n as the sum of all n coordinates. Give (n, sum_distribution, total_probability, sum_mean, sum_variance, n_times_mean, n_times_variance, standardized_mean, standardized_variance, standardization_description, finite_scope).
The complete finite joint law gives standardized moments0 and1; it neither makes Z_n normal at finite n nor proves a distributional limit.
View answer and explanation
(5, [[-10, 1/32], [-9, 5/32], [-8, 5/16], [-7, 5/16], [-6, 5/32], [-5, 1/32]], 1, -15/2, 5/4, -15/2, 5/4, 0, 1, Z_n=(S_n-n*mu)/(sigma*sqrt(n)), complete_joint_law_moments_do_not_prove_distributional_convergence)
Aggregate the entire finite IID product law by sum. Retain total mass, E[S_n]=n*mu and Var(S_n)=n*sigma^2. The centered normalization Z_n=(S_n−n*mu)/(sigma*sqrt(n)) has mean0 and variance1. Those moments do not make a finite-n law normal or prove convergence in distribution. The exact result, retaining the requested order, is (5, [[-10, 1/32], [-9, 5/32], [-8, 5/16], [-7, 5/16], [-6, 5/32], [-5, 1/32]], 1, -15/2, 5/4, -15/2, 5/4, 0, 1, Z_n=(S_n-n*mu)/(sigma*sqrt(n)), complete_joint_law_moments_do_not_prove_distributional_convergence).
Reviewed step-by-step derivation
Central limit theorem for finite-variance IID sequences · qf-mod49-clt-standardized-sum-moments · candidate
47. Complete Fixed-n Sum Moments Do Not Prove a Distributional Limit
Use the aligned finite law values=[-1, 0], probabilities=[1/2, 1/2], with declared process certificate=infinite_iid. The token is an explicit infinite IID premise, not a process construction or proof by this finite calculation. Use sample size n=5 and S_n as the sum of all n coordinates. Give (n, sum_distribution, total_probability, sum_mean, sum_variance, n_times_mean, n_times_variance, standardized_mean, standardized_variance, standardization_description, finite_scope).
The complete finite joint law gives standardized moments0 and1; it neither makes Z_n normal at finite n nor proves a distributional limit.
View answer and explanation
(5, [[-5, 1/32], [-4, 5/32], [-3, 5/16], [-2, 5/16], [-1, 5/32], [0, 1/32]], 1, -5/2, 5/4, -5/2, 5/4, 0, 1, Z_n=(S_n-n*mu)/(sigma*sqrt(n)), complete_joint_law_moments_do_not_prove_distributional_convergence)
Aggregate the entire finite IID product law by sum. Retain total mass, E[S_n]=n*mu and Var(S_n)=n*sigma^2. The centered normalization Z_n=(S_n−n*mu)/(sigma*sqrt(n)) has mean0 and variance1. Those moments do not make a finite-n law normal or prove convergence in distribution. The exact result, retaining the requested order, is (5, [[-5, 1/32], [-4, 5/32], [-3, 5/16], [-2, 5/16], [-1, 5/32], [0, 1/32]], 1, -5/2, 5/4, -5/2, 5/4, 0, 1, Z_n=(S_n-n*mu)/(sigma*sqrt(n)), complete_joint_law_moments_do_not_prove_distributional_convergence).
Reviewed step-by-step derivation
Central limit theorem for finite-variance IID sequences · qf-mod49-clt-standardized-sum-moments · candidate
48. Complete Fixed-n Sum Moments Do Not Prove a Distributional Limit
Use the aligned finite law values=[0, 1], probabilities=[1/2, 1/2], with declared process certificate=infinite_iid. The token is an explicit infinite IID premise, not a process construction or proof by this finite calculation. Use sample size n=5 and S_n as the sum of all n coordinates. Give (n, sum_distribution, total_probability, sum_mean, sum_variance, n_times_mean, n_times_variance, standardized_mean, standardized_variance, standardization_description, finite_scope).
The complete finite joint law gives standardized moments0 and1; it neither makes Z_n normal at finite n nor proves a distributional limit.
View answer and explanation
(5, [[0, 1/32], [1, 5/32], [2, 5/16], [3, 5/16], [4, 5/32], [5, 1/32]], 1, 5/2, 5/4, 5/2, 5/4, 0, 1, Z_n=(S_n-n*mu)/(sigma*sqrt(n)), complete_joint_law_moments_do_not_prove_distributional_convergence)
Aggregate the entire finite IID product law by sum. Retain total mass, E[S_n]=n*mu and Var(S_n)=n*sigma^2. The centered normalization Z_n=(S_n−n*mu)/(sigma*sqrt(n)) has mean0 and variance1. Those moments do not make a finite-n law normal or prove convergence in distribution. The exact result, retaining the requested order, is (5, [[0, 1/32], [1, 5/32], [2, 5/16], [3, 5/16], [4, 5/32], [5, 1/32]], 1, 5/2, 5/4, 5/2, 5/4, 0, 1, Z_n=(S_n-n*mu)/(sigma*sqrt(n)), complete_joint_law_moments_do_not_prove_distributional_convergence).
Reviewed step-by-step derivation
Central limit theorem for finite-variance IID sequences · qf-mod49-clt-standardized-sum-moments · candidate
49. Complete Fixed-n Sum Moments Do Not Prove a Distributional Limit
Use the aligned finite law values=[1, 2], probabilities=[1/2, 1/2], with declared process certificate=infinite_iid. The token is an explicit infinite IID premise, not a process construction or proof by this finite calculation. Use sample size n=5 and S_n as the sum of all n coordinates. Give (n, sum_distribution, total_probability, sum_mean, sum_variance, n_times_mean, n_times_variance, standardized_mean, standardized_variance, standardization_description, finite_scope).
The complete finite joint law gives standardized moments0 and1; it neither makes Z_n normal at finite n nor proves a distributional limit.
View answer and explanation
(5, [[5, 1/32], [6, 5/32], [7, 5/16], [8, 5/16], [9, 5/32], [10, 1/32]], 1, 15/2, 5/4, 15/2, 5/4, 0, 1, Z_n=(S_n-n*mu)/(sigma*sqrt(n)), complete_joint_law_moments_do_not_prove_distributional_convergence)
Aggregate the entire finite IID product law by sum. Retain total mass, E[S_n]=n*mu and Var(S_n)=n*sigma^2. The centered normalization Z_n=(S_n−n*mu)/(sigma*sqrt(n)) has mean0 and variance1. Those moments do not make a finite-n law normal or prove convergence in distribution. The exact result, retaining the requested order, is (5, [[5, 1/32], [6, 5/32], [7, 5/16], [8, 5/16], [9, 5/32], [10, 1/32]], 1, 15/2, 5/4, 15/2, 5/4, 0, 1, Z_n=(S_n-n*mu)/(sigma*sqrt(n)), complete_joint_law_moments_do_not_prove_distributional_convergence).
Reviewed step-by-step derivation
Central limit theorem for finite-variance IID sequences · qf-mod49-clt-standardized-sum-moments · candidate
50. Complete Fixed-n Sum Moments Do Not Prove a Distributional Limit
Use the aligned finite law values=[2, 3], probabilities=[1/2, 1/2], with declared process certificate=infinite_iid. The token is an explicit infinite IID premise, not a process construction or proof by this finite calculation. Use sample size n=5 and S_n as the sum of all n coordinates. Give (n, sum_distribution, total_probability, sum_mean, sum_variance, n_times_mean, n_times_variance, standardized_mean, standardized_variance, standardization_description, finite_scope).
The complete finite joint law gives standardized moments0 and1; it neither makes Z_n normal at finite n nor proves a distributional limit.
View answer and explanation
(5, [[10, 1/32], [11, 5/32], [12, 5/16], [13, 5/16], [14, 5/32], [15, 1/32]], 1, 25/2, 5/4, 25/2, 5/4, 0, 1, Z_n=(S_n-n*mu)/(sigma*sqrt(n)), complete_joint_law_moments_do_not_prove_distributional_convergence)
Aggregate the entire finite IID product law by sum. Retain total mass, E[S_n]=n*mu and Var(S_n)=n*sigma^2. The centered normalization Z_n=(S_n−n*mu)/(sigma*sqrt(n)) has mean0 and variance1. Those moments do not make a finite-n law normal or prove convergence in distribution. The exact result, retaining the requested order, is (5, [[10, 1/32], [11, 5/32], [12, 5/16], [13, 5/16], [14, 5/32], [15, 1/32]], 1, 25/2, 5/4, 25/2, 5/4, 0, 1, Z_n=(S_n-n*mu)/(sigma*sqrt(n)), complete_joint_law_moments_do_not_prove_distributional_convergence).
Reviewed step-by-step derivation
Central limit theorem for finite-variance IID sequences · qf-mod49-clt-standardized-sum-moments · candidate
51. An Exact Binomial Lattice Is Not a Normal Approximation
Use Bernoulli success probability p=1/2, sample size n=2, integer threshold=0, and independent coordinates for this finite sum. Give (n, success_probability, binomial_distribution, total_probability, threshold, cdf_at_threshold, normalized, standardized_threshold_description, finite_scope).
View answer and explanation
(2, 1/2, [[0, 1/4], [1, 1/2], [2, 1/4]], 1, 0, 1/4, Yes, z_threshold=(0-n*p)/sqrt(n*p*(1-p)), exact_lattice_not_normal_approximation_or_convergence_evidence)
Compute the exact probabilities for every integer count0 through n and the inclusive CDF at the threshold. Retain the symbolic standardized threshold; neither that expression nor a finite lattice table establishes an asymptotic limit or an approximation error. The exact result, retaining the requested order, is (2, 1/2, [[0, 1/4], [1, 1/2], [2, 1/4]], 1, 0, 1/4, Yes, z_threshold=(0-n*p)/sqrt(n*p*(1-p)), exact_lattice_not_normal_approximation_or_convergence_evidence).
Reviewed step-by-step derivation
Central limit theorem for finite-variance IID sequences · qf-mod49-clt-binomial-lattice · candidate
52. An Exact Binomial Lattice Is Not a Normal Approximation
Use Bernoulli success probability p=1/2, sample size n=2, integer threshold=1, and independent coordinates for this finite sum. Give (n, success_probability, binomial_distribution, total_probability, threshold, cdf_at_threshold, normalized, standardized_threshold_description, finite_scope).
View answer and explanation
(2, 1/2, [[0, 1/4], [1, 1/2], [2, 1/4]], 1, 1, 3/4, Yes, z_threshold=(1-n*p)/sqrt(n*p*(1-p)), exact_lattice_not_normal_approximation_or_convergence_evidence)
Compute the exact probabilities for every integer count0 through n and the inclusive CDF at the threshold. Retain the symbolic standardized threshold; neither that expression nor a finite lattice table establishes an asymptotic limit or an approximation error. The exact result, retaining the requested order, is (2, 1/2, [[0, 1/4], [1, 1/2], [2, 1/4]], 1, 1, 3/4, Yes, z_threshold=(1-n*p)/sqrt(n*p*(1-p)), exact_lattice_not_normal_approximation_or_convergence_evidence).
Reviewed step-by-step derivation
Central limit theorem for finite-variance IID sequences · qf-mod49-clt-binomial-lattice · candidate
53. An Exact Binomial Lattice Is Not a Normal Approximation
Use Bernoulli success probability p=1/2, sample size n=2, integer threshold=2, and independent coordinates for this finite sum. Give (n, success_probability, binomial_distribution, total_probability, threshold, cdf_at_threshold, normalized, standardized_threshold_description, finite_scope).
View answer and explanation
(2, 1/2, [[0, 1/4], [1, 1/2], [2, 1/4]], 1, 2, 1, Yes, z_threshold=(2-n*p)/sqrt(n*p*(1-p)), exact_lattice_not_normal_approximation_or_convergence_evidence)
Compute the exact probabilities for every integer count0 through n and the inclusive CDF at the threshold. Retain the symbolic standardized threshold; neither that expression nor a finite lattice table establishes an asymptotic limit or an approximation error. The exact result, retaining the requested order, is (2, 1/2, [[0, 1/4], [1, 1/2], [2, 1/4]], 1, 2, 1, Yes, z_threshold=(2-n*p)/sqrt(n*p*(1-p)), exact_lattice_not_normal_approximation_or_convergence_evidence).
Reviewed step-by-step derivation
Central limit theorem for finite-variance IID sequences · qf-mod49-clt-binomial-lattice · candidate
54. An Exact Binomial Lattice Is Not a Normal Approximation
Use Bernoulli success probability p=1/2, sample size n=3, integer threshold=0, and independent coordinates for this finite sum. Give (n, success_probability, binomial_distribution, total_probability, threshold, cdf_at_threshold, normalized, standardized_threshold_description, finite_scope).
View answer and explanation
(3, 1/2, [[0, 1/8], [1, 3/8], [2, 3/8], [3, 1/8]], 1, 0, 1/8, Yes, z_threshold=(0-n*p)/sqrt(n*p*(1-p)), exact_lattice_not_normal_approximation_or_convergence_evidence)
Compute the exact probabilities for every integer count0 through n and the inclusive CDF at the threshold. Retain the symbolic standardized threshold; neither that expression nor a finite lattice table establishes an asymptotic limit or an approximation error. The exact result, retaining the requested order, is (3, 1/2, [[0, 1/8], [1, 3/8], [2, 3/8], [3, 1/8]], 1, 0, 1/8, Yes, z_threshold=(0-n*p)/sqrt(n*p*(1-p)), exact_lattice_not_normal_approximation_or_convergence_evidence).
Reviewed step-by-step derivation
Central limit theorem for finite-variance IID sequences · qf-mod49-clt-binomial-lattice · candidate
55. An Exact Binomial Lattice Is Not a Normal Approximation
Use Bernoulli success probability p=1/2, sample size n=3, integer threshold=1, and independent coordinates for this finite sum. Give (n, success_probability, binomial_distribution, total_probability, threshold, cdf_at_threshold, normalized, standardized_threshold_description, finite_scope).
View answer and explanation
(3, 1/2, [[0, 1/8], [1, 3/8], [2, 3/8], [3, 1/8]], 1, 1, 1/2, Yes, z_threshold=(1-n*p)/sqrt(n*p*(1-p)), exact_lattice_not_normal_approximation_or_convergence_evidence)
Compute the exact probabilities for every integer count0 through n and the inclusive CDF at the threshold. Retain the symbolic standardized threshold; neither that expression nor a finite lattice table establishes an asymptotic limit or an approximation error. The exact result, retaining the requested order, is (3, 1/2, [[0, 1/8], [1, 3/8], [2, 3/8], [3, 1/8]], 1, 1, 1/2, Yes, z_threshold=(1-n*p)/sqrt(n*p*(1-p)), exact_lattice_not_normal_approximation_or_convergence_evidence).
Reviewed step-by-step derivation
Central limit theorem for finite-variance IID sequences · qf-mod49-clt-binomial-lattice · candidate
56. An Exact Binomial Lattice Is Not a Normal Approximation
Use Bernoulli success probability p=1/2, sample size n=3, integer threshold=2, and independent coordinates for this finite sum. Give (n, success_probability, binomial_distribution, total_probability, threshold, cdf_at_threshold, normalized, standardized_threshold_description, finite_scope).
View answer and explanation
(3, 1/2, [[0, 1/8], [1, 3/8], [2, 3/8], [3, 1/8]], 1, 2, 7/8, Yes, z_threshold=(2-n*p)/sqrt(n*p*(1-p)), exact_lattice_not_normal_approximation_or_convergence_evidence)
Compute the exact probabilities for every integer count0 through n and the inclusive CDF at the threshold. Retain the symbolic standardized threshold; neither that expression nor a finite lattice table establishes an asymptotic limit or an approximation error. The exact result, retaining the requested order, is (3, 1/2, [[0, 1/8], [1, 3/8], [2, 3/8], [3, 1/8]], 1, 2, 7/8, Yes, z_threshold=(2-n*p)/sqrt(n*p*(1-p)), exact_lattice_not_normal_approximation_or_convergence_evidence).
Reviewed step-by-step derivation
Central limit theorem for finite-variance IID sequences · qf-mod49-clt-binomial-lattice · candidate
57. An Exact Binomial Lattice Is Not a Normal Approximation
Use Bernoulli success probability p=1/2, sample size n=3, integer threshold=3, and independent coordinates for this finite sum. Give (n, success_probability, binomial_distribution, total_probability, threshold, cdf_at_threshold, normalized, standardized_threshold_description, finite_scope).
View answer and explanation
(3, 1/2, [[0, 1/8], [1, 3/8], [2, 3/8], [3, 1/8]], 1, 3, 1, Yes, z_threshold=(3-n*p)/sqrt(n*p*(1-p)), exact_lattice_not_normal_approximation_or_convergence_evidence)
Compute the exact probabilities for every integer count0 through n and the inclusive CDF at the threshold. Retain the symbolic standardized threshold; neither that expression nor a finite lattice table establishes an asymptotic limit or an approximation error. The exact result, retaining the requested order, is (3, 1/2, [[0, 1/8], [1, 3/8], [2, 3/8], [3, 1/8]], 1, 3, 1, Yes, z_threshold=(3-n*p)/sqrt(n*p*(1-p)), exact_lattice_not_normal_approximation_or_convergence_evidence).
Reviewed step-by-step derivation
Central limit theorem for finite-variance IID sequences · qf-mod49-clt-binomial-lattice · candidate
58. An Exact Binomial Lattice Is Not a Normal Approximation
Use Bernoulli success probability p=1/2, sample size n=4, integer threshold=0, and independent coordinates for this finite sum. Give (n, success_probability, binomial_distribution, total_probability, threshold, cdf_at_threshold, normalized, standardized_threshold_description, finite_scope).
View answer and explanation
(4, 1/2, [[0, 1/16], [1, 1/4], [2, 3/8], [3, 1/4], [4, 1/16]], 1, 0, 1/16, Yes, z_threshold=(0-n*p)/sqrt(n*p*(1-p)), exact_lattice_not_normal_approximation_or_convergence_evidence)
Compute the exact probabilities for every integer count0 through n and the inclusive CDF at the threshold. Retain the symbolic standardized threshold; neither that expression nor a finite lattice table establishes an asymptotic limit or an approximation error. The exact result, retaining the requested order, is (4, 1/2, [[0, 1/16], [1, 1/4], [2, 3/8], [3, 1/4], [4, 1/16]], 1, 0, 1/16, Yes, z_threshold=(0-n*p)/sqrt(n*p*(1-p)), exact_lattice_not_normal_approximation_or_convergence_evidence).
Reviewed step-by-step derivation
Central limit theorem for finite-variance IID sequences · qf-mod49-clt-binomial-lattice · candidate
59. An Exact Binomial Lattice Is Not a Normal Approximation
Use Bernoulli success probability p=1/2, sample size n=4, integer threshold=1, and independent coordinates for this finite sum. Give (n, success_probability, binomial_distribution, total_probability, threshold, cdf_at_threshold, normalized, standardized_threshold_description, finite_scope).
View answer and explanation
(4, 1/2, [[0, 1/16], [1, 1/4], [2, 3/8], [3, 1/4], [4, 1/16]], 1, 1, 5/16, Yes, z_threshold=(1-n*p)/sqrt(n*p*(1-p)), exact_lattice_not_normal_approximation_or_convergence_evidence)
Compute the exact probabilities for every integer count0 through n and the inclusive CDF at the threshold. Retain the symbolic standardized threshold; neither that expression nor a finite lattice table establishes an asymptotic limit or an approximation error. The exact result, retaining the requested order, is (4, 1/2, [[0, 1/16], [1, 1/4], [2, 3/8], [3, 1/4], [4, 1/16]], 1, 1, 5/16, Yes, z_threshold=(1-n*p)/sqrt(n*p*(1-p)), exact_lattice_not_normal_approximation_or_convergence_evidence).
Reviewed step-by-step derivation
Central limit theorem for finite-variance IID sequences · qf-mod49-clt-binomial-lattice · candidate
60. An Exact Binomial Lattice Is Not a Normal Approximation
Use Bernoulli success probability p=1/2, sample size n=4, integer threshold=2, and independent coordinates for this finite sum. Give (n, success_probability, binomial_distribution, total_probability, threshold, cdf_at_threshold, normalized, standardized_threshold_description, finite_scope).
View answer and explanation
(4, 1/2, [[0, 1/16], [1, 1/4], [2, 3/8], [3, 1/4], [4, 1/16]], 1, 2, 11/16, Yes, z_threshold=(2-n*p)/sqrt(n*p*(1-p)), exact_lattice_not_normal_approximation_or_convergence_evidence)
Compute the exact probabilities for every integer count0 through n and the inclusive CDF at the threshold. Retain the symbolic standardized threshold; neither that expression nor a finite lattice table establishes an asymptotic limit or an approximation error. The exact result, retaining the requested order, is (4, 1/2, [[0, 1/16], [1, 1/4], [2, 3/8], [3, 1/4], [4, 1/16]], 1, 2, 11/16, Yes, z_threshold=(2-n*p)/sqrt(n*p*(1-p)), exact_lattice_not_normal_approximation_or_convergence_evidence).
Reviewed step-by-step derivation
Central limit theorem for finite-variance IID sequences · qf-mod49-clt-binomial-lattice · candidate
61. An Exact Binomial Lattice Is Not a Normal Approximation
Use Bernoulli success probability p=1/2, sample size n=4, integer threshold=3, and independent coordinates for this finite sum. Give (n, success_probability, binomial_distribution, total_probability, threshold, cdf_at_threshold, normalized, standardized_threshold_description, finite_scope).
View answer and explanation
(4, 1/2, [[0, 1/16], [1, 1/4], [2, 3/8], [3, 1/4], [4, 1/16]], 1, 3, 15/16, Yes, z_threshold=(3-n*p)/sqrt(n*p*(1-p)), exact_lattice_not_normal_approximation_or_convergence_evidence)
Compute the exact probabilities for every integer count0 through n and the inclusive CDF at the threshold. Retain the symbolic standardized threshold; neither that expression nor a finite lattice table establishes an asymptotic limit or an approximation error. The exact result, retaining the requested order, is (4, 1/2, [[0, 1/16], [1, 1/4], [2, 3/8], [3, 1/4], [4, 1/16]], 1, 3, 15/16, Yes, z_threshold=(3-n*p)/sqrt(n*p*(1-p)), exact_lattice_not_normal_approximation_or_convergence_evidence).
Reviewed step-by-step derivation
Central limit theorem for finite-variance IID sequences · qf-mod49-clt-binomial-lattice · candidate
62. An Exact Binomial Lattice Is Not a Normal Approximation
Use Bernoulli success probability p=1/2, sample size n=4, integer threshold=4, and independent coordinates for this finite sum. Give (n, success_probability, binomial_distribution, total_probability, threshold, cdf_at_threshold, normalized, standardized_threshold_description, finite_scope).
View answer and explanation
(4, 1/2, [[0, 1/16], [1, 1/4], [2, 3/8], [3, 1/4], [4, 1/16]], 1, 4, 1, Yes, z_threshold=(4-n*p)/sqrt(n*p*(1-p)), exact_lattice_not_normal_approximation_or_convergence_evidence)
Compute the exact probabilities for every integer count0 through n and the inclusive CDF at the threshold. Retain the symbolic standardized threshold; neither that expression nor a finite lattice table establishes an asymptotic limit or an approximation error. The exact result, retaining the requested order, is (4, 1/2, [[0, 1/16], [1, 1/4], [2, 3/8], [3, 1/4], [4, 1/16]], 1, 4, 1, Yes, z_threshold=(4-n*p)/sqrt(n*p*(1-p)), exact_lattice_not_normal_approximation_or_convergence_evidence).
Reviewed step-by-step derivation
Central limit theorem for finite-variance IID sequences · qf-mod49-clt-binomial-lattice · candidate
63. An Exact Binomial Lattice Is Not a Normal Approximation
Use Bernoulli success probability p=1/2, sample size n=5, integer threshold=0, and independent coordinates for this finite sum. Give (n, success_probability, binomial_distribution, total_probability, threshold, cdf_at_threshold, normalized, standardized_threshold_description, finite_scope).
View answer and explanation
(5, 1/2, [[0, 1/32], [1, 5/32], [2, 5/16], [3, 5/16], [4, 5/32], [5, 1/32]], 1, 0, 1/32, Yes, z_threshold=(0-n*p)/sqrt(n*p*(1-p)), exact_lattice_not_normal_approximation_or_convergence_evidence)
Compute the exact probabilities for every integer count0 through n and the inclusive CDF at the threshold. Retain the symbolic standardized threshold; neither that expression nor a finite lattice table establishes an asymptotic limit or an approximation error. The exact result, retaining the requested order, is (5, 1/2, [[0, 1/32], [1, 5/32], [2, 5/16], [3, 5/16], [4, 5/32], [5, 1/32]], 1, 0, 1/32, Yes, z_threshold=(0-n*p)/sqrt(n*p*(1-p)), exact_lattice_not_normal_approximation_or_convergence_evidence).
Reviewed step-by-step derivation
Central limit theorem for finite-variance IID sequences · qf-mod49-clt-binomial-lattice · candidate
64. An Exact Binomial Lattice Is Not a Normal Approximation
Use Bernoulli success probability p=1/2, sample size n=5, integer threshold=1, and independent coordinates for this finite sum. Give (n, success_probability, binomial_distribution, total_probability, threshold, cdf_at_threshold, normalized, standardized_threshold_description, finite_scope).
View answer and explanation
(5, 1/2, [[0, 1/32], [1, 5/32], [2, 5/16], [3, 5/16], [4, 5/32], [5, 1/32]], 1, 1, 3/16, Yes, z_threshold=(1-n*p)/sqrt(n*p*(1-p)), exact_lattice_not_normal_approximation_or_convergence_evidence)
Compute the exact probabilities for every integer count0 through n and the inclusive CDF at the threshold. Retain the symbolic standardized threshold; neither that expression nor a finite lattice table establishes an asymptotic limit or an approximation error. The exact result, retaining the requested order, is (5, 1/2, [[0, 1/32], [1, 5/32], [2, 5/16], [3, 5/16], [4, 5/32], [5, 1/32]], 1, 1, 3/16, Yes, z_threshold=(1-n*p)/sqrt(n*p*(1-p)), exact_lattice_not_normal_approximation_or_convergence_evidence).
Reviewed step-by-step derivation
Central limit theorem for finite-variance IID sequences · qf-mod49-clt-binomial-lattice · candidate
65. An Exact Binomial Lattice Is Not a Normal Approximation
Use Bernoulli success probability p=1/2, sample size n=5, integer threshold=2, and independent coordinates for this finite sum. Give (n, success_probability, binomial_distribution, total_probability, threshold, cdf_at_threshold, normalized, standardized_threshold_description, finite_scope).
View answer and explanation
(5, 1/2, [[0, 1/32], [1, 5/32], [2, 5/16], [3, 5/16], [4, 5/32], [5, 1/32]], 1, 2, 1/2, Yes, z_threshold=(2-n*p)/sqrt(n*p*(1-p)), exact_lattice_not_normal_approximation_or_convergence_evidence)
Compute the exact probabilities for every integer count0 through n and the inclusive CDF at the threshold. Retain the symbolic standardized threshold; neither that expression nor a finite lattice table establishes an asymptotic limit or an approximation error. The exact result, retaining the requested order, is (5, 1/2, [[0, 1/32], [1, 5/32], [2, 5/16], [3, 5/16], [4, 5/32], [5, 1/32]], 1, 2, 1/2, Yes, z_threshold=(2-n*p)/sqrt(n*p*(1-p)), exact_lattice_not_normal_approximation_or_convergence_evidence).
Reviewed step-by-step derivation
Central limit theorem for finite-variance IID sequences · qf-mod49-clt-binomial-lattice · candidate
66. An Exact Binomial Lattice Is Not a Normal Approximation
Use Bernoulli success probability p=1/2, sample size n=5, integer threshold=3, and independent coordinates for this finite sum. Give (n, success_probability, binomial_distribution, total_probability, threshold, cdf_at_threshold, normalized, standardized_threshold_description, finite_scope).
View answer and explanation
(5, 1/2, [[0, 1/32], [1, 5/32], [2, 5/16], [3, 5/16], [4, 5/32], [5, 1/32]], 1, 3, 13/16, Yes, z_threshold=(3-n*p)/sqrt(n*p*(1-p)), exact_lattice_not_normal_approximation_or_convergence_evidence)
Compute the exact probabilities for every integer count0 through n and the inclusive CDF at the threshold. Retain the symbolic standardized threshold; neither that expression nor a finite lattice table establishes an asymptotic limit or an approximation error. The exact result, retaining the requested order, is (5, 1/2, [[0, 1/32], [1, 5/32], [2, 5/16], [3, 5/16], [4, 5/32], [5, 1/32]], 1, 3, 13/16, Yes, z_threshold=(3-n*p)/sqrt(n*p*(1-p)), exact_lattice_not_normal_approximation_or_convergence_evidence).
Reviewed step-by-step derivation
Central limit theorem for finite-variance IID sequences · qf-mod49-clt-binomial-lattice · candidate
67. An Exact Binomial Lattice Is Not a Normal Approximation
Use Bernoulli success probability p=1/2, sample size n=5, integer threshold=4, and independent coordinates for this finite sum. Give (n, success_probability, binomial_distribution, total_probability, threshold, cdf_at_threshold, normalized, standardized_threshold_description, finite_scope).
View answer and explanation
(5, 1/2, [[0, 1/32], [1, 5/32], [2, 5/16], [3, 5/16], [4, 5/32], [5, 1/32]], 1, 4, 31/32, Yes, z_threshold=(4-n*p)/sqrt(n*p*(1-p)), exact_lattice_not_normal_approximation_or_convergence_evidence)
Compute the exact probabilities for every integer count0 through n and the inclusive CDF at the threshold. Retain the symbolic standardized threshold; neither that expression nor a finite lattice table establishes an asymptotic limit or an approximation error. The exact result, retaining the requested order, is (5, 1/2, [[0, 1/32], [1, 5/32], [2, 5/16], [3, 5/16], [4, 5/32], [5, 1/32]], 1, 4, 31/32, Yes, z_threshold=(4-n*p)/sqrt(n*p*(1-p)), exact_lattice_not_normal_approximation_or_convergence_evidence).
Reviewed step-by-step derivation
Central limit theorem for finite-variance IID sequences · qf-mod49-clt-binomial-lattice · candidate
68. An Exact Binomial Lattice Is Not a Normal Approximation
Use Bernoulli success probability p=1/2, sample size n=5, integer threshold=5, and independent coordinates for this finite sum. Give (n, success_probability, binomial_distribution, total_probability, threshold, cdf_at_threshold, normalized, standardized_threshold_description, finite_scope).
View answer and explanation
(5, 1/2, [[0, 1/32], [1, 5/32], [2, 5/16], [3, 5/16], [4, 5/32], [5, 1/32]], 1, 5, 1, Yes, z_threshold=(5-n*p)/sqrt(n*p*(1-p)), exact_lattice_not_normal_approximation_or_convergence_evidence)
Compute the exact probabilities for every integer count0 through n and the inclusive CDF at the threshold. Retain the symbolic standardized threshold; neither that expression nor a finite lattice table establishes an asymptotic limit or an approximation error. The exact result, retaining the requested order, is (5, 1/2, [[0, 1/32], [1, 5/32], [2, 5/16], [3, 5/16], [4, 5/32], [5, 1/32]], 1, 5, 1, Yes, z_threshold=(5-n*p)/sqrt(n*p*(1-p)), exact_lattice_not_normal_approximation_or_convergence_evidence).
Reviewed step-by-step derivation
Central limit theorem for finite-variance IID sequences · qf-mod49-clt-binomial-lattice · candidate
69. An Exact Binomial Lattice Is Not a Normal Approximation
Use Bernoulli success probability p=1/2, sample size n=6, integer threshold=0, and independent coordinates for this finite sum. Give (n, success_probability, binomial_distribution, total_probability, threshold, cdf_at_threshold, normalized, standardized_threshold_description, finite_scope).
View answer and explanation
(6, 1/2, [[0, 1/64], [1, 3/32], [2, 15/64], [3, 5/16], [4, 15/64], [5, 3/32], [6, 1/64]], 1, 0, 1/64, Yes, z_threshold=(0-n*p)/sqrt(n*p*(1-p)), exact_lattice_not_normal_approximation_or_convergence_evidence)
Compute the exact probabilities for every integer count0 through n and the inclusive CDF at the threshold. Retain the symbolic standardized threshold; neither that expression nor a finite lattice table establishes an asymptotic limit or an approximation error. The exact result, retaining the requested order, is (6, 1/2, [[0, 1/64], [1, 3/32], [2, 15/64], [3, 5/16], [4, 15/64], [5, 3/32], [6, 1/64]], 1, 0, 1/64, Yes, z_threshold=(0-n*p)/sqrt(n*p*(1-p)), exact_lattice_not_normal_approximation_or_convergence_evidence).
Reviewed step-by-step derivation
Central limit theorem for finite-variance IID sequences · qf-mod49-clt-binomial-lattice · candidate
70. An Exact Binomial Lattice Is Not a Normal Approximation
Use Bernoulli success probability p=1/2, sample size n=6, integer threshold=1, and independent coordinates for this finite sum. Give (n, success_probability, binomial_distribution, total_probability, threshold, cdf_at_threshold, normalized, standardized_threshold_description, finite_scope).
View answer and explanation
(6, 1/2, [[0, 1/64], [1, 3/32], [2, 15/64], [3, 5/16], [4, 15/64], [5, 3/32], [6, 1/64]], 1, 1, 7/64, Yes, z_threshold=(1-n*p)/sqrt(n*p*(1-p)), exact_lattice_not_normal_approximation_or_convergence_evidence)
Compute the exact probabilities for every integer count0 through n and the inclusive CDF at the threshold. Retain the symbolic standardized threshold; neither that expression nor a finite lattice table establishes an asymptotic limit or an approximation error. The exact result, retaining the requested order, is (6, 1/2, [[0, 1/64], [1, 3/32], [2, 15/64], [3, 5/16], [4, 15/64], [5, 3/32], [6, 1/64]], 1, 1, 7/64, Yes, z_threshold=(1-n*p)/sqrt(n*p*(1-p)), exact_lattice_not_normal_approximation_or_convergence_evidence).
Reviewed step-by-step derivation
Central limit theorem for finite-variance IID sequences · qf-mod49-clt-binomial-lattice · candidate
71. An Exact Binomial Lattice Is Not a Normal Approximation
Use Bernoulli success probability p=1/2, sample size n=6, integer threshold=2, and independent coordinates for this finite sum. Give (n, success_probability, binomial_distribution, total_probability, threshold, cdf_at_threshold, normalized, standardized_threshold_description, finite_scope).
View answer and explanation
(6, 1/2, [[0, 1/64], [1, 3/32], [2, 15/64], [3, 5/16], [4, 15/64], [5, 3/32], [6, 1/64]], 1, 2, 11/32, Yes, z_threshold=(2-n*p)/sqrt(n*p*(1-p)), exact_lattice_not_normal_approximation_or_convergence_evidence)
Compute the exact probabilities for every integer count0 through n and the inclusive CDF at the threshold. Retain the symbolic standardized threshold; neither that expression nor a finite lattice table establishes an asymptotic limit or an approximation error. The exact result, retaining the requested order, is (6, 1/2, [[0, 1/64], [1, 3/32], [2, 15/64], [3, 5/16], [4, 15/64], [5, 3/32], [6, 1/64]], 1, 2, 11/32, Yes, z_threshold=(2-n*p)/sqrt(n*p*(1-p)), exact_lattice_not_normal_approximation_or_convergence_evidence).
Reviewed step-by-step derivation
Central limit theorem for finite-variance IID sequences · qf-mod49-clt-binomial-lattice · candidate
72. An Exact Binomial Lattice Is Not a Normal Approximation
Use Bernoulli success probability p=1/2, sample size n=6, integer threshold=3, and independent coordinates for this finite sum. Give (n, success_probability, binomial_distribution, total_probability, threshold, cdf_at_threshold, normalized, standardized_threshold_description, finite_scope).
View answer and explanation
(6, 1/2, [[0, 1/64], [1, 3/32], [2, 15/64], [3, 5/16], [4, 15/64], [5, 3/32], [6, 1/64]], 1, 3, 21/32, Yes, z_threshold=(3-n*p)/sqrt(n*p*(1-p)), exact_lattice_not_normal_approximation_or_convergence_evidence)
Compute the exact probabilities for every integer count0 through n and the inclusive CDF at the threshold. Retain the symbolic standardized threshold; neither that expression nor a finite lattice table establishes an asymptotic limit or an approximation error. The exact result, retaining the requested order, is (6, 1/2, [[0, 1/64], [1, 3/32], [2, 15/64], [3, 5/16], [4, 15/64], [5, 3/32], [6, 1/64]], 1, 3, 21/32, Yes, z_threshold=(3-n*p)/sqrt(n*p*(1-p)), exact_lattice_not_normal_approximation_or_convergence_evidence).
Reviewed step-by-step derivation
Central limit theorem for finite-variance IID sequences · qf-mod49-clt-binomial-lattice · candidate
73. An Exact Binomial Lattice Is Not a Normal Approximation
Use Bernoulli success probability p=1/2, sample size n=6, integer threshold=4, and independent coordinates for this finite sum. Give (n, success_probability, binomial_distribution, total_probability, threshold, cdf_at_threshold, normalized, standardized_threshold_description, finite_scope).
View answer and explanation
(6, 1/2, [[0, 1/64], [1, 3/32], [2, 15/64], [3, 5/16], [4, 15/64], [5, 3/32], [6, 1/64]], 1, 4, 57/64, Yes, z_threshold=(4-n*p)/sqrt(n*p*(1-p)), exact_lattice_not_normal_approximation_or_convergence_evidence)
Compute the exact probabilities for every integer count0 through n and the inclusive CDF at the threshold. Retain the symbolic standardized threshold; neither that expression nor a finite lattice table establishes an asymptotic limit or an approximation error. The exact result, retaining the requested order, is (6, 1/2, [[0, 1/64], [1, 3/32], [2, 15/64], [3, 5/16], [4, 15/64], [5, 3/32], [6, 1/64]], 1, 4, 57/64, Yes, z_threshold=(4-n*p)/sqrt(n*p*(1-p)), exact_lattice_not_normal_approximation_or_convergence_evidence).
Reviewed step-by-step derivation
Central limit theorem for finite-variance IID sequences · qf-mod49-clt-binomial-lattice · candidate
74. An Exact Binomial Lattice Is Not a Normal Approximation
Use Bernoulli success probability p=1/2, sample size n=6, integer threshold=5, and independent coordinates for this finite sum. Give (n, success_probability, binomial_distribution, total_probability, threshold, cdf_at_threshold, normalized, standardized_threshold_description, finite_scope).
View answer and explanation
(6, 1/2, [[0, 1/64], [1, 3/32], [2, 15/64], [3, 5/16], [4, 15/64], [5, 3/32], [6, 1/64]], 1, 5, 63/64, Yes, z_threshold=(5-n*p)/sqrt(n*p*(1-p)), exact_lattice_not_normal_approximation_or_convergence_evidence)
Compute the exact probabilities for every integer count0 through n and the inclusive CDF at the threshold. Retain the symbolic standardized threshold; neither that expression nor a finite lattice table establishes an asymptotic limit or an approximation error. The exact result, retaining the requested order, is (6, 1/2, [[0, 1/64], [1, 3/32], [2, 15/64], [3, 5/16], [4, 15/64], [5, 3/32], [6, 1/64]], 1, 5, 63/64, Yes, z_threshold=(5-n*p)/sqrt(n*p*(1-p)), exact_lattice_not_normal_approximation_or_convergence_evidence).
Reviewed step-by-step derivation
Central limit theorem for finite-variance IID sequences · qf-mod49-clt-binomial-lattice · candidate
75. An Exact Binomial Lattice Is Not a Normal Approximation
Use Bernoulli success probability p=1/2, sample size n=6, integer threshold=6, and independent coordinates for this finite sum. Give (n, success_probability, binomial_distribution, total_probability, threshold, cdf_at_threshold, normalized, standardized_threshold_description, finite_scope).
View answer and explanation
(6, 1/2, [[0, 1/64], [1, 3/32], [2, 15/64], [3, 5/16], [4, 15/64], [5, 3/32], [6, 1/64]], 1, 6, 1, Yes, z_threshold=(6-n*p)/sqrt(n*p*(1-p)), exact_lattice_not_normal_approximation_or_convergence_evidence)
Compute the exact probabilities for every integer count0 through n and the inclusive CDF at the threshold. Retain the symbolic standardized threshold; neither that expression nor a finite lattice table establishes an asymptotic limit or an approximation error. The exact result, retaining the requested order, is (6, 1/2, [[0, 1/64], [1, 3/32], [2, 15/64], [3, 5/16], [4, 15/64], [5, 3/32], [6, 1/64]], 1, 6, 1, Yes, z_threshold=(6-n*p)/sqrt(n*p*(1-p)), exact_lattice_not_normal_approximation_or_convergence_evidence).
Reviewed step-by-step derivation
Central limit theorem for finite-variance IID sequences · qf-mod49-clt-binomial-lattice · candidate
76. Declared Theorem Application, Failed Hypotheses and Finite Evidence Have Different Roles
Use mode=symbolic_application and complete model={"values":[-2,-1],"probabilities":[[1,2],[1,2]],"process_certificate":"infinite_iid","n":2}. Give (mode, declared_process_certificate, mean, variance, n, standardization_description, named_theorem_conclusion, conditional_application_qualified).
The five modes have8/6/6/7/7 fields. symbolic_application invokes the named theorem under declared hypotheses; this program does not prove it. Non-IID means this theorem does not apply, not that all CLTs fail.
View answer and explanation
(symbolic_application, infinite_iid, -3/2, 1/4, 2, Z_n=(S_n-n*mu)/(sigma*sqrt(n)), Z_n_converges_in_distribution_to_N(0,1), Yes)
symbolic_application invokes the named CLT conditionally on the declared infinite IID law and positive variance; it is not a proof performed here. Different marginals exclude this IID theorem only. Zero variance makes this normalization undefined. n*sigma is the wrong scaling. One finite histogram does not establish a distributional limit. Keep the five branch-specific field layouts. The exact result, retaining the requested order, is (symbolic_application, infinite_iid, -3/2, 1/4, 2, Z_n=(S_n-n*mu)/(sigma*sqrt(n)), Z_n_converges_in_distribution_to_N(0,1), Yes).
Reviewed step-by-step derivation
Central limit theorem for finite-variance IID sequences · qf-mod49-clt-application-boundary · candidate
77. Declared Theorem Application, Failed Hypotheses and Finite Evidence Have Different Roles
Use mode=non_iid and complete model={"values_a":[-1,0],"probabilities_a":[[1,2],[1,2]],"values_b":[-1,0],"probabilities_b":[[1,7],[6,7]]}. Give (mode, first_marginal_law, second_marginal_law, identical_marginals, this_IID_theorem_applicable, scope).
The five modes have8/6/6/7/7 fields. symbolic_application invokes the named theorem under declared hypotheses; this program does not prove it. Non-IID means this theorem does not apply, not that all CLTs fail.
View answer and explanation
(non_iid, [[-1, 1/2], [0, 1/2]], [[-1, 1/7], [0, 6/7]], No, No, this_IID_CLT_not_applicable_not_a_general_counterexample)
symbolic_application invokes the named CLT conditionally on the declared infinite IID law and positive variance; it is not a proof performed here. Different marginals exclude this IID theorem only. Zero variance makes this normalization undefined. n*sigma is the wrong scaling. One finite histogram does not establish a distributional limit. Keep the five branch-specific field layouts. The exact result, retaining the requested order, is (non_iid, [[-1, 1/2], [0, 1/2]], [[-1, 1/7], [0, 6/7]], No, No, this_IID_CLT_not_applicable_not_a_general_counterexample).
Reviewed step-by-step derivation
Central limit theorem for finite-variance IID sequences · qf-mod49-clt-application-boundary · candidate
78. Declared Theorem Application, Failed Hypotheses and Finite Evidence Have Different Roles
Use mode=zero_variance and complete model={"value":-1,"n":4}. Give (mode, constant_value, n, variance, standardization_obstruction, this_standardization_claim).
The five modes have8/6/6/7/7 fields. symbolic_application invokes the named theorem under declared hypotheses; this program does not prove it. Non-IID means this theorem does not apply, not that all CLTs fail.
View answer and explanation
(zero_variance, -1, 4, 0, sigma=0_makes_standardization_undefined, No)
symbolic_application invokes the named CLT conditionally on the declared infinite IID law and positive variance; it is not a proof performed here. Different marginals exclude this IID theorem only. Zero variance makes this normalization undefined. n*sigma is the wrong scaling. One finite histogram does not establish a distributional limit. Keep the five branch-specific field layouts. The exact result, retaining the requested order, is (zero_variance, -1, 4, 0, sigma=0_makes_standardization_undefined, No).
Reviewed step-by-step derivation
Central limit theorem for finite-variance IID sequences · qf-mod49-clt-application-boundary · candidate
79. Declared Theorem Application, Failed Hypotheses and Finite Evidence Have Different Roles
Use mode=wrong_scaling and complete model={"values":[1,2],"probabilities":[[1,2],[1,2]],"process_certificate":"infinite_iid","n":5}. Give (mode, mean, variance, n, proposed_denominator, required_denominator, proposed_scaling_claim).
The five modes have8/6/6/7/7 fields. symbolic_application invokes the named theorem under declared hypotheses; this program does not prove it. Non-IID means this theorem does not apply, not that all CLTs fail.
View answer and explanation
(wrong_scaling, 3/2, 1/4, 5, denominator=n*sigma, required=sigma*sqrt(n), No)
symbolic_application invokes the named CLT conditionally on the declared infinite IID law and positive variance; it is not a proof performed here. Different marginals exclude this IID theorem only. Zero variance makes this normalization undefined. n*sigma is the wrong scaling. One finite histogram does not establish a distributional limit. Keep the five branch-specific field layouts. The exact result, retaining the requested order, is (wrong_scaling, 3/2, 1/4, 5, denominator=n*sigma, required=sigma*sqrt(n), No).
Reviewed step-by-step derivation
Central limit theorem for finite-variance IID sequences · qf-mod49-clt-application-boundary · candidate
80. Declared Theorem Application, Failed Hypotheses and Finite Evidence Have Different Roles
Use mode=finite_histogram and complete model={"values":[2,3],"probabilities":[[1,2],[1,2]],"process_certificate":"infinite_iid","n":6}. Give (mode, mean, variance, n, evidence_scope, limit_proof_claim, boundary).
The five modes have8/6/6/7/7 fields. symbolic_application invokes the named theorem under declared hypotheses; this program does not prove it. Non-IID means this theorem does not apply, not that all CLTs fail.
View answer and explanation
(finite_histogram, 5/2, 1/4, 6, one_finite_histogram, No, finite_histogram_does_not_prove_distributional_limit)
symbolic_application invokes the named CLT conditionally on the declared infinite IID law and positive variance; it is not a proof performed here. Different marginals exclude this IID theorem only. Zero variance makes this normalization undefined. n*sigma is the wrong scaling. One finite histogram does not establish a distributional limit. Keep the five branch-specific field layouts. The exact result, retaining the requested order, is (finite_histogram, 5/2, 1/4, 6, one_finite_histogram, No, finite_histogram_does_not_prove_distributional_limit).
Reviewed step-by-step derivation
Central limit theorem for finite-variance IID sequences · qf-mod49-clt-application-boundary · candidate
81. Declared Theorem Application, Failed Hypotheses and Finite Evidence Have Different Roles
Use mode=symbolic_application and complete model={"values":[-2,-1],"probabilities":[[1,3],[2,3]],"process_certificate":"infinite_iid","n":2}. Give (mode, declared_process_certificate, mean, variance, n, standardization_description, named_theorem_conclusion, conditional_application_qualified).
The five modes have8/6/6/7/7 fields. symbolic_application invokes the named theorem under declared hypotheses; this program does not prove it. Non-IID means this theorem does not apply, not that all CLTs fail.
View answer and explanation
(symbolic_application, infinite_iid, -4/3, 2/9, 2, Z_n=(S_n-n*mu)/(sigma*sqrt(n)), Z_n_converges_in_distribution_to_N(0,1), Yes)
symbolic_application invokes the named CLT conditionally on the declared infinite IID law and positive variance; it is not a proof performed here. Different marginals exclude this IID theorem only. Zero variance makes this normalization undefined. n*sigma is the wrong scaling. One finite histogram does not establish a distributional limit. Keep the five branch-specific field layouts. The exact result, retaining the requested order, is (symbolic_application, infinite_iid, -4/3, 2/9, 2, Z_n=(S_n-n*mu)/(sigma*sqrt(n)), Z_n_converges_in_distribution_to_N(0,1), Yes).
Reviewed step-by-step derivation
Central limit theorem for finite-variance IID sequences · qf-mod49-clt-application-boundary · candidate
82. Declared Theorem Application, Failed Hypotheses and Finite Evidence Have Different Roles
Use mode=non_iid and complete model={"values_a":[-1,0],"probabilities_a":[[1,3],[2,3]],"values_b":[-1,0],"probabilities_b":[[1,7],[6,7]]}. Give (mode, first_marginal_law, second_marginal_law, identical_marginals, this_IID_theorem_applicable, scope).
The five modes have8/6/6/7/7 fields. symbolic_application invokes the named theorem under declared hypotheses; this program does not prove it. Non-IID means this theorem does not apply, not that all CLTs fail.
View answer and explanation
(non_iid, [[-1, 1/3], [0, 2/3]], [[-1, 1/7], [0, 6/7]], No, No, this_IID_CLT_not_applicable_not_a_general_counterexample)
symbolic_application invokes the named CLT conditionally on the declared infinite IID law and positive variance; it is not a proof performed here. Different marginals exclude this IID theorem only. Zero variance makes this normalization undefined. n*sigma is the wrong scaling. One finite histogram does not establish a distributional limit. Keep the five branch-specific field layouts. The exact result, retaining the requested order, is (non_iid, [[-1, 1/3], [0, 2/3]], [[-1, 1/7], [0, 6/7]], No, No, this_IID_CLT_not_applicable_not_a_general_counterexample).
Reviewed step-by-step derivation
Central limit theorem for finite-variance IID sequences · qf-mod49-clt-application-boundary · candidate
83. Declared Theorem Application, Failed Hypotheses and Finite Evidence Have Different Roles
Use mode=zero_variance and complete model={"value":-3,"n":4}. Give (mode, constant_value, n, variance, standardization_obstruction, this_standardization_claim).
The five modes have8/6/6/7/7 fields. symbolic_application invokes the named theorem under declared hypotheses; this program does not prove it. Non-IID means this theorem does not apply, not that all CLTs fail.
View answer and explanation
(zero_variance, -3, 4, 0, sigma=0_makes_standardization_undefined, No)
symbolic_application invokes the named CLT conditionally on the declared infinite IID law and positive variance; it is not a proof performed here. Different marginals exclude this IID theorem only. Zero variance makes this normalization undefined. n*sigma is the wrong scaling. One finite histogram does not establish a distributional limit. Keep the five branch-specific field layouts. The exact result, retaining the requested order, is (zero_variance, -3, 4, 0, sigma=0_makes_standardization_undefined, No).
Reviewed step-by-step derivation
Central limit theorem for finite-variance IID sequences · qf-mod49-clt-application-boundary · candidate
84. Declared Theorem Application, Failed Hypotheses and Finite Evidence Have Different Roles
Use mode=wrong_scaling and complete model={"values":[1,2],"probabilities":[[1,3],[2,3]],"process_certificate":"infinite_iid","n":5}. Give (mode, mean, variance, n, proposed_denominator, required_denominator, proposed_scaling_claim).
The five modes have8/6/6/7/7 fields. symbolic_application invokes the named theorem under declared hypotheses; this program does not prove it. Non-IID means this theorem does not apply, not that all CLTs fail.
View answer and explanation
(wrong_scaling, 5/3, 2/9, 5, denominator=n*sigma, required=sigma*sqrt(n), No)
symbolic_application invokes the named CLT conditionally on the declared infinite IID law and positive variance; it is not a proof performed here. Different marginals exclude this IID theorem only. Zero variance makes this normalization undefined. n*sigma is the wrong scaling. One finite histogram does not establish a distributional limit. Keep the five branch-specific field layouts. The exact result, retaining the requested order, is (wrong_scaling, 5/3, 2/9, 5, denominator=n*sigma, required=sigma*sqrt(n), No).
Reviewed step-by-step derivation
Central limit theorem for finite-variance IID sequences · qf-mod49-clt-application-boundary · candidate
85. Declared Theorem Application, Failed Hypotheses and Finite Evidence Have Different Roles
Use mode=finite_histogram and complete model={"values":[2,3],"probabilities":[[1,3],[2,3]],"process_certificate":"infinite_iid","n":6}. Give (mode, mean, variance, n, evidence_scope, limit_proof_claim, boundary).
The five modes have8/6/6/7/7 fields. symbolic_application invokes the named theorem under declared hypotheses; this program does not prove it. Non-IID means this theorem does not apply, not that all CLTs fail.
View answer and explanation
(finite_histogram, 8/3, 2/9, 6, one_finite_histogram, No, finite_histogram_does_not_prove_distributional_limit)
symbolic_application invokes the named CLT conditionally on the declared infinite IID law and positive variance; it is not a proof performed here. Different marginals exclude this IID theorem only. Zero variance makes this normalization undefined. n*sigma is the wrong scaling. One finite histogram does not establish a distributional limit. Keep the five branch-specific field layouts. The exact result, retaining the requested order, is (finite_histogram, 8/3, 2/9, 6, one_finite_histogram, No, finite_histogram_does_not_prove_distributional_limit).
Reviewed step-by-step derivation
Central limit theorem for finite-variance IID sequences · qf-mod49-clt-application-boundary · candidate
86. Declared Theorem Application, Failed Hypotheses and Finite Evidence Have Different Roles
Use mode=symbolic_application and complete model={"values":[-2,-1],"probabilities":[[1,4],[3,4]],"process_certificate":"infinite_iid","n":2}. Give (mode, declared_process_certificate, mean, variance, n, standardization_description, named_theorem_conclusion, conditional_application_qualified).
The five modes have8/6/6/7/7 fields. symbolic_application invokes the named theorem under declared hypotheses; this program does not prove it. Non-IID means this theorem does not apply, not that all CLTs fail.
View answer and explanation
(symbolic_application, infinite_iid, -5/4, 3/16, 2, Z_n=(S_n-n*mu)/(sigma*sqrt(n)), Z_n_converges_in_distribution_to_N(0,1), Yes)
symbolic_application invokes the named CLT conditionally on the declared infinite IID law and positive variance; it is not a proof performed here. Different marginals exclude this IID theorem only. Zero variance makes this normalization undefined. n*sigma is the wrong scaling. One finite histogram does not establish a distributional limit. Keep the five branch-specific field layouts. The exact result, retaining the requested order, is (symbolic_application, infinite_iid, -5/4, 3/16, 2, Z_n=(S_n-n*mu)/(sigma*sqrt(n)), Z_n_converges_in_distribution_to_N(0,1), Yes).
Reviewed step-by-step derivation
Central limit theorem for finite-variance IID sequences · qf-mod49-clt-application-boundary · candidate
87. Declared Theorem Application, Failed Hypotheses and Finite Evidence Have Different Roles
Use mode=non_iid and complete model={"values_a":[-1,0],"probabilities_a":[[1,4],[3,4]],"values_b":[-1,0],"probabilities_b":[[1,7],[6,7]]}. Give (mode, first_marginal_law, second_marginal_law, identical_marginals, this_IID_theorem_applicable, scope).
The five modes have8/6/6/7/7 fields. symbolic_application invokes the named theorem under declared hypotheses; this program does not prove it. Non-IID means this theorem does not apply, not that all CLTs fail.
View answer and explanation
(non_iid, [[-1, 1/4], [0, 3/4]], [[-1, 1/7], [0, 6/7]], No, No, this_IID_CLT_not_applicable_not_a_general_counterexample)
symbolic_application invokes the named CLT conditionally on the declared infinite IID law and positive variance; it is not a proof performed here. Different marginals exclude this IID theorem only. Zero variance makes this normalization undefined. n*sigma is the wrong scaling. One finite histogram does not establish a distributional limit. Keep the five branch-specific field layouts. The exact result, retaining the requested order, is (non_iid, [[-1, 1/4], [0, 3/4]], [[-1, 1/7], [0, 6/7]], No, No, this_IID_CLT_not_applicable_not_a_general_counterexample).
Reviewed step-by-step derivation
Central limit theorem for finite-variance IID sequences · qf-mod49-clt-application-boundary · candidate
88. Declared Theorem Application, Failed Hypotheses and Finite Evidence Have Different Roles
Use mode=zero_variance and complete model={"value":2,"n":4}. Give (mode, constant_value, n, variance, standardization_obstruction, this_standardization_claim).
The five modes have8/6/6/7/7 fields. symbolic_application invokes the named theorem under declared hypotheses; this program does not prove it. Non-IID means this theorem does not apply, not that all CLTs fail.
View answer and explanation
(zero_variance, 2, 4, 0, sigma=0_makes_standardization_undefined, No)
symbolic_application invokes the named CLT conditionally on the declared infinite IID law and positive variance; it is not a proof performed here. Different marginals exclude this IID theorem only. Zero variance makes this normalization undefined. n*sigma is the wrong scaling. One finite histogram does not establish a distributional limit. Keep the five branch-specific field layouts. The exact result, retaining the requested order, is (zero_variance, 2, 4, 0, sigma=0_makes_standardization_undefined, No).
Reviewed step-by-step derivation
Central limit theorem for finite-variance IID sequences · qf-mod49-clt-application-boundary · candidate
89. Declared Theorem Application, Failed Hypotheses and Finite Evidence Have Different Roles
Use mode=wrong_scaling and complete model={"values":[1,2],"probabilities":[[1,4],[3,4]],"process_certificate":"infinite_iid","n":5}. Give (mode, mean, variance, n, proposed_denominator, required_denominator, proposed_scaling_claim).
The five modes have8/6/6/7/7 fields. symbolic_application invokes the named theorem under declared hypotheses; this program does not prove it. Non-IID means this theorem does not apply, not that all CLTs fail.
View answer and explanation
(wrong_scaling, 7/4, 3/16, 5, denominator=n*sigma, required=sigma*sqrt(n), No)
symbolic_application invokes the named CLT conditionally on the declared infinite IID law and positive variance; it is not a proof performed here. Different marginals exclude this IID theorem only. Zero variance makes this normalization undefined. n*sigma is the wrong scaling. One finite histogram does not establish a distributional limit. Keep the five branch-specific field layouts. The exact result, retaining the requested order, is (wrong_scaling, 7/4, 3/16, 5, denominator=n*sigma, required=sigma*sqrt(n), No).
Reviewed step-by-step derivation
Central limit theorem for finite-variance IID sequences · qf-mod49-clt-application-boundary · candidate
90. Declared Theorem Application, Failed Hypotheses and Finite Evidence Have Different Roles
Use mode=finite_histogram and complete model={"values":[2,3],"probabilities":[[1,4],[3,4]],"process_certificate":"infinite_iid","n":6}. Give (mode, mean, variance, n, evidence_scope, limit_proof_claim, boundary).
The five modes have8/6/6/7/7 fields. symbolic_application invokes the named theorem under declared hypotheses; this program does not prove it. Non-IID means this theorem does not apply, not that all CLTs fail.
View answer and explanation
(finite_histogram, 11/4, 3/16, 6, one_finite_histogram, No, finite_histogram_does_not_prove_distributional_limit)
symbolic_application invokes the named CLT conditionally on the declared infinite IID law and positive variance; it is not a proof performed here. Different marginals exclude this IID theorem only. Zero variance makes this normalization undefined. n*sigma is the wrong scaling. One finite histogram does not establish a distributional limit. Keep the five branch-specific field layouts. The exact result, retaining the requested order, is (finite_histogram, 11/4, 3/16, 6, one_finite_histogram, No, finite_histogram_does_not_prove_distributional_limit).
Reviewed step-by-step derivation
Central limit theorem for finite-variance IID sequences · qf-mod49-clt-application-boundary · candidate
91. Declared Theorem Application, Failed Hypotheses and Finite Evidence Have Different Roles
Use mode=symbolic_application and complete model={"values":[-2,-1],"probabilities":[[2,5],[3,5]],"process_certificate":"infinite_iid","n":2}. Give (mode, declared_process_certificate, mean, variance, n, standardization_description, named_theorem_conclusion, conditional_application_qualified).
The five modes have8/6/6/7/7 fields. symbolic_application invokes the named theorem under declared hypotheses; this program does not prove it. Non-IID means this theorem does not apply, not that all CLTs fail.
View answer and explanation
(symbolic_application, infinite_iid, -7/5, 6/25, 2, Z_n=(S_n-n*mu)/(sigma*sqrt(n)), Z_n_converges_in_distribution_to_N(0,1), Yes)
symbolic_application invokes the named CLT conditionally on the declared infinite IID law and positive variance; it is not a proof performed here. Different marginals exclude this IID theorem only. Zero variance makes this normalization undefined. n*sigma is the wrong scaling. One finite histogram does not establish a distributional limit. Keep the five branch-specific field layouts. The exact result, retaining the requested order, is (symbolic_application, infinite_iid, -7/5, 6/25, 2, Z_n=(S_n-n*mu)/(sigma*sqrt(n)), Z_n_converges_in_distribution_to_N(0,1), Yes).
Reviewed step-by-step derivation
Central limit theorem for finite-variance IID sequences · qf-mod49-clt-application-boundary · candidate
92. Declared Theorem Application, Failed Hypotheses and Finite Evidence Have Different Roles
Use mode=non_iid and complete model={"values_a":[-1,0],"probabilities_a":[[2,5],[3,5]],"values_b":[-1,0],"probabilities_b":[[1,7],[6,7]]}. Give (mode, first_marginal_law, second_marginal_law, identical_marginals, this_IID_theorem_applicable, scope).
The five modes have8/6/6/7/7 fields. symbolic_application invokes the named theorem under declared hypotheses; this program does not prove it. Non-IID means this theorem does not apply, not that all CLTs fail.
View answer and explanation
(non_iid, [[-1, 2/5], [0, 3/5]], [[-1, 1/7], [0, 6/7]], No, No, this_IID_CLT_not_applicable_not_a_general_counterexample)
symbolic_application invokes the named CLT conditionally on the declared infinite IID law and positive variance; it is not a proof performed here. Different marginals exclude this IID theorem only. Zero variance makes this normalization undefined. n*sigma is the wrong scaling. One finite histogram does not establish a distributional limit. Keep the five branch-specific field layouts. The exact result, retaining the requested order, is (non_iid, [[-1, 2/5], [0, 3/5]], [[-1, 1/7], [0, 6/7]], No, No, this_IID_CLT_not_applicable_not_a_general_counterexample).
Reviewed step-by-step derivation
Central limit theorem for finite-variance IID sequences · qf-mod49-clt-application-boundary · candidate
93. Declared Theorem Application, Failed Hypotheses and Finite Evidence Have Different Roles
Use mode=zero_variance and complete model={"value":0,"n":4}. Give (mode, constant_value, n, variance, standardization_obstruction, this_standardization_claim).
The five modes have8/6/6/7/7 fields. symbolic_application invokes the named theorem under declared hypotheses; this program does not prove it. Non-IID means this theorem does not apply, not that all CLTs fail.
View answer and explanation
(zero_variance, 0, 4, 0, sigma=0_makes_standardization_undefined, No)
symbolic_application invokes the named CLT conditionally on the declared infinite IID law and positive variance; it is not a proof performed here. Different marginals exclude this IID theorem only. Zero variance makes this normalization undefined. n*sigma is the wrong scaling. One finite histogram does not establish a distributional limit. Keep the five branch-specific field layouts. The exact result, retaining the requested order, is (zero_variance, 0, 4, 0, sigma=0_makes_standardization_undefined, No).
Reviewed step-by-step derivation
Central limit theorem for finite-variance IID sequences · qf-mod49-clt-application-boundary · candidate
94. Declared Theorem Application, Failed Hypotheses and Finite Evidence Have Different Roles
Use mode=wrong_scaling and complete model={"values":[1,2],"probabilities":[[2,5],[3,5]],"process_certificate":"infinite_iid","n":5}. Give (mode, mean, variance, n, proposed_denominator, required_denominator, proposed_scaling_claim).
The five modes have8/6/6/7/7 fields. symbolic_application invokes the named theorem under declared hypotheses; this program does not prove it. Non-IID means this theorem does not apply, not that all CLTs fail.
View answer and explanation
(wrong_scaling, 8/5, 6/25, 5, denominator=n*sigma, required=sigma*sqrt(n), No)
symbolic_application invokes the named CLT conditionally on the declared infinite IID law and positive variance; it is not a proof performed here. Different marginals exclude this IID theorem only. Zero variance makes this normalization undefined. n*sigma is the wrong scaling. One finite histogram does not establish a distributional limit. Keep the five branch-specific field layouts. The exact result, retaining the requested order, is (wrong_scaling, 8/5, 6/25, 5, denominator=n*sigma, required=sigma*sqrt(n), No).
Reviewed step-by-step derivation
Central limit theorem for finite-variance IID sequences · qf-mod49-clt-application-boundary · candidate
95. Declared Theorem Application, Failed Hypotheses and Finite Evidence Have Different Roles
Use mode=finite_histogram and complete model={"values":[2,3],"probabilities":[[2,5],[3,5]],"process_certificate":"infinite_iid","n":6}. Give (mode, mean, variance, n, evidence_scope, limit_proof_claim, boundary).
The five modes have8/6/6/7/7 fields. symbolic_application invokes the named theorem under declared hypotheses; this program does not prove it. Non-IID means this theorem does not apply, not that all CLTs fail.
View answer and explanation
(finite_histogram, 13/5, 6/25, 6, one_finite_histogram, No, finite_histogram_does_not_prove_distributional_limit)
symbolic_application invokes the named CLT conditionally on the declared infinite IID law and positive variance; it is not a proof performed here. Different marginals exclude this IID theorem only. Zero variance makes this normalization undefined. n*sigma is the wrong scaling. One finite histogram does not establish a distributional limit. Keep the five branch-specific field layouts. The exact result, retaining the requested order, is (finite_histogram, 13/5, 6/25, 6, one_finite_histogram, No, finite_histogram_does_not_prove_distributional_limit).
Reviewed step-by-step derivation
Central limit theorem for finite-variance IID sequences · qf-mod49-clt-application-boundary · candidate
96. Declared Theorem Application, Failed Hypotheses and Finite Evidence Have Different Roles
Use mode=symbolic_application and complete model={"values":[-2,-1],"probabilities":[[1,5],[4,5]],"process_certificate":"infinite_iid","n":2}. Give (mode, declared_process_certificate, mean, variance, n, standardization_description, named_theorem_conclusion, conditional_application_qualified).
The five modes have8/6/6/7/7 fields. symbolic_application invokes the named theorem under declared hypotheses; this program does not prove it. Non-IID means this theorem does not apply, not that all CLTs fail.
View answer and explanation
(symbolic_application, infinite_iid, -6/5, 4/25, 2, Z_n=(S_n-n*mu)/(sigma*sqrt(n)), Z_n_converges_in_distribution_to_N(0,1), Yes)
symbolic_application invokes the named CLT conditionally on the declared infinite IID law and positive variance; it is not a proof performed here. Different marginals exclude this IID theorem only. Zero variance makes this normalization undefined. n*sigma is the wrong scaling. One finite histogram does not establish a distributional limit. Keep the five branch-specific field layouts. The exact result, retaining the requested order, is (symbolic_application, infinite_iid, -6/5, 4/25, 2, Z_n=(S_n-n*mu)/(sigma*sqrt(n)), Z_n_converges_in_distribution_to_N(0,1), Yes).
Reviewed step-by-step derivation
Central limit theorem for finite-variance IID sequences · qf-mod49-clt-application-boundary · candidate
97. Declared Theorem Application, Failed Hypotheses and Finite Evidence Have Different Roles
Use mode=non_iid and complete model={"values_a":[-1,0],"probabilities_a":[[1,5],[4,5]],"values_b":[-1,0],"probabilities_b":[[1,7],[6,7]]}. Give (mode, first_marginal_law, second_marginal_law, identical_marginals, this_IID_theorem_applicable, scope).
The five modes have8/6/6/7/7 fields. symbolic_application invokes the named theorem under declared hypotheses; this program does not prove it. Non-IID means this theorem does not apply, not that all CLTs fail.
View answer and explanation
(non_iid, [[-1, 1/5], [0, 4/5]], [[-1, 1/7], [0, 6/7]], No, No, this_IID_CLT_not_applicable_not_a_general_counterexample)
symbolic_application invokes the named CLT conditionally on the declared infinite IID law and positive variance; it is not a proof performed here. Different marginals exclude this IID theorem only. Zero variance makes this normalization undefined. n*sigma is the wrong scaling. One finite histogram does not establish a distributional limit. Keep the five branch-specific field layouts. The exact result, retaining the requested order, is (non_iid, [[-1, 1/5], [0, 4/5]], [[-1, 1/7], [0, 6/7]], No, No, this_IID_CLT_not_applicable_not_a_general_counterexample).
Reviewed step-by-step derivation
Central limit theorem for finite-variance IID sequences · qf-mod49-clt-application-boundary · candidate
98. Declared Theorem Application, Failed Hypotheses and Finite Evidence Have Different Roles
Use mode=zero_variance and complete model={"value":-2,"n":4}. Give (mode, constant_value, n, variance, standardization_obstruction, this_standardization_claim).
The five modes have8/6/6/7/7 fields. symbolic_application invokes the named theorem under declared hypotheses; this program does not prove it. Non-IID means this theorem does not apply, not that all CLTs fail.
View answer and explanation
(zero_variance, -2, 4, 0, sigma=0_makes_standardization_undefined, No)
symbolic_application invokes the named CLT conditionally on the declared infinite IID law and positive variance; it is not a proof performed here. Different marginals exclude this IID theorem only. Zero variance makes this normalization undefined. n*sigma is the wrong scaling. One finite histogram does not establish a distributional limit. Keep the five branch-specific field layouts. The exact result, retaining the requested order, is (zero_variance, -2, 4, 0, sigma=0_makes_standardization_undefined, No).
Reviewed step-by-step derivation
Central limit theorem for finite-variance IID sequences · qf-mod49-clt-application-boundary · candidate
99. Declared Theorem Application, Failed Hypotheses and Finite Evidence Have Different Roles
Use mode=wrong_scaling and complete model={"values":[1,2],"probabilities":[[1,5],[4,5]],"process_certificate":"infinite_iid","n":5}. Give (mode, mean, variance, n, proposed_denominator, required_denominator, proposed_scaling_claim).
The five modes have8/6/6/7/7 fields. symbolic_application invokes the named theorem under declared hypotheses; this program does not prove it. Non-IID means this theorem does not apply, not that all CLTs fail.
View answer and explanation
(wrong_scaling, 9/5, 4/25, 5, denominator=n*sigma, required=sigma*sqrt(n), No)
symbolic_application invokes the named CLT conditionally on the declared infinite IID law and positive variance; it is not a proof performed here. Different marginals exclude this IID theorem only. Zero variance makes this normalization undefined. n*sigma is the wrong scaling. One finite histogram does not establish a distributional limit. Keep the five branch-specific field layouts. The exact result, retaining the requested order, is (wrong_scaling, 9/5, 4/25, 5, denominator=n*sigma, required=sigma*sqrt(n), No).
Reviewed step-by-step derivation
Central limit theorem for finite-variance IID sequences · qf-mod49-clt-application-boundary · candidate
100. Declared Theorem Application, Failed Hypotheses and Finite Evidence Have Different Roles
Use mode=finite_histogram and complete model={"values":[2,3],"probabilities":[[1,5],[4,5]],"process_certificate":"infinite_iid","n":6}. Give (mode, mean, variance, n, evidence_scope, limit_proof_claim, boundary).
The five modes have8/6/6/7/7 fields. symbolic_application invokes the named theorem under declared hypotheses; this program does not prove it. Non-IID means this theorem does not apply, not that all CLTs fail.
View answer and explanation
(finite_histogram, 14/5, 4/25, 6, one_finite_histogram, No, finite_histogram_does_not_prove_distributional_limit)
symbolic_application invokes the named CLT conditionally on the declared infinite IID law and positive variance; it is not a proof performed here. Different marginals exclude this IID theorem only. Zero variance makes this normalization undefined. n*sigma is the wrong scaling. One finite histogram does not establish a distributional limit. Keep the five branch-specific field layouts. The exact result, retaining the requested order, is (finite_histogram, 14/5, 4/25, 6, one_finite_histogram, No, finite_histogram_does_not_prove_distributional_limit).
Reviewed step-by-step derivation
Central limit theorem for finite-variance IID sequences · qf-mod49-clt-application-boundary · candidate
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