第 27285 題：有限變異數IID中央極限定理

原題與已知條件

明示無限IID、共同mu、0<sigma^2<infinity與sigma*sqrt(n)標準化責任。操作=clt-standardized-sum-moments；資料={"values":[2,3],"probabilities":[[1,2],[1,2]],"process_certificate":"infinite_iid","n":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。

• G001 — 明示無限IID、共同mu、0<sigma^2<infinity與sigma*sqrt(n)標準化責任。操作=clt-standardized-sum-moments；資料={"values":[2,3],"probabilities":[[1,2],[1,2]],"process_certificate":"infinite_iid","n":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。

定義與適用範圍

沿用原題明示的論域、假設與來源 MKO 定義。本題已完成第二對話數理與中英語義覆核，並由原始參數逐題獨立重算模型、答案與每個算術步驟；原題仍為 candidate。

• mko-central-limit-theorem-finite-variance-iid

逐步推導

source roles = [clt-standardized-sum-moments]

derived = [[n, sum_distribution, total_probability, sum_mean, sum_variance, n_times_mean, n_times_variance, standardized_mean, standardized_variance, standardization_description, finite_scope]]

source roles = [infinite_iid]

derived = [ASSUME infinite sequence is IID with this common marginal]

CLT ORIGINAL common marginal ORIGINAL law mass sum0 := sum(0, 1/2)

CLT ORIGINAL common marginal ORIGINAL law mass sum0 = 1/2

CLT ORIGINAL common marginal ORIGINAL law mass sum1 := sum(1/2, 1/2)

CLT ORIGINAL common marginal ORIGINAL law mass sum1 = 1

source roles = [CLT ORIGINAL common marginal, [2, 3], [[1, 2], [1, 2]]]

derived = [[[2, 1/2], [3, 1/2]], 1]

CLT ORIGINAL common marginal ORIGINAL value times probability0 := mul(2, 1/2)

CLT ORIGINAL common marginal ORIGINAL value times probability0 = 1

CLT ORIGINAL common marginal ORIGINAL ascending mean sum0 := sum(0, 1)

CLT ORIGINAL common marginal ORIGINAL ascending mean sum0 = 1

CLT ORIGINAL common marginal exact value square0 := mul(2, 2)

CLT ORIGINAL common marginal exact value square0 = 4

CLT ORIGINAL common marginal ORIGINAL square times probability0 := mul(4, 1/2)

CLT ORIGINAL common marginal ORIGINAL square times probability0 = 2

CLT ORIGINAL common marginal ORIGINAL ascending second moment sum0 := sum(0, 2)

CLT ORIGINAL common marginal ORIGINAL ascending second moment sum0 = 2

CLT ORIGINAL common marginal ORIGINAL value times probability1 := mul(3, 1/2)

CLT ORIGINAL common marginal ORIGINAL value times probability1 = 3/2

CLT ORIGINAL common marginal ORIGINAL ascending mean sum1 := sum(1, 3/2)

CLT ORIGINAL common marginal ORIGINAL ascending mean sum1 = 5/2

CLT ORIGINAL common marginal exact value square1 := mul(3, 3)

CLT ORIGINAL common marginal exact value square1 = 9

CLT ORIGINAL common marginal ORIGINAL square times probability1 := mul(9, 1/2)

CLT ORIGINAL common marginal ORIGINAL square times probability1 = 9/2

CLT ORIGINAL common marginal ORIGINAL ascending second moment sum1 := sum(2, 9/2)

CLT ORIGINAL common marginal ORIGINAL ascending second moment sum1 = 13/2

CLT ORIGINAL common marginal ORIGINAL mean square := mul(5/2, 5/2)

CLT ORIGINAL common marginal ORIGINAL mean square = 25/4

CLT ORIGINAL common marginal ORIGINAL second minus mean square := sub(13/2, 25/4)

CLT ORIGINAL common marginal ORIGINAL second minus mean square = 1/4

source roles = [CLT ORIGINAL common marginal, [[2, 1/2], [3, 1/2]]]

derived = [5/2, 13/2, 1/4]

CLT joint ORIGINAL word0 sum := sum(2, 2)

CLT joint ORIGINAL word0 sum = 4

CLT joint ORIGINAL word0 probability factor0 := mul(1, 1/2)

CLT joint ORIGINAL word0 probability factor0 = 1/2

CLT joint ORIGINAL word0 probability factor1 := mul(1/2, 1/2)

CLT joint ORIGINAL word0 probability factor1 = 1/4

CLT joint ORIGINAL sum4 bucket word0 := sum(0, 1/4)

CLT joint ORIGINAL sum4 bucket word0 = 1/4

source roles = [2, 0, [2, 2], [1/2, 1/2]]

derived = [4, 1/4, 0, 1/4]

CLT joint ORIGINAL word1 sum := sum(2, 3)

CLT joint ORIGINAL word1 sum = 5

CLT joint ORIGINAL word1 probability factor0 := mul(1, 1/2)

CLT joint ORIGINAL word1 probability factor0 = 1/2

CLT joint ORIGINAL word1 probability factor1 := mul(1/2, 1/2)

CLT joint ORIGINAL word1 probability factor1 = 1/4

CLT joint ORIGINAL sum5 bucket word1 := sum(0, 1/4)

CLT joint ORIGINAL sum5 bucket word1 = 1/4

source roles = [2, 1, [2, 3], [1/2, 1/2]]

derived = [5, 1/4, 0, 1/4]

CLT joint ORIGINAL word2 sum := sum(3, 2)

CLT joint ORIGINAL word2 sum = 5

CLT joint ORIGINAL word2 probability factor0 := mul(1, 1/2)

CLT joint ORIGINAL word2 probability factor0 = 1/2

CLT joint ORIGINAL word2 probability factor1 := mul(1/2, 1/2)

CLT joint ORIGINAL word2 probability factor1 = 1/4

CLT joint ORIGINAL sum5 bucket word2 := sum(1/4, 1/4)

CLT joint ORIGINAL sum5 bucket word2 = 1/2

source roles = [2, 2, [3, 2], [1/2, 1/2]]

derived = [5, 1/4, 1/4, 1/2]

CLT joint ORIGINAL word3 sum := sum(3, 3)

CLT joint ORIGINAL word3 sum = 6

CLT joint ORIGINAL word3 probability factor0 := mul(1, 1/2)

CLT joint ORIGINAL word3 probability factor0 = 1/2

CLT joint ORIGINAL word3 probability factor1 := mul(1/2, 1/2)

CLT joint ORIGINAL word3 probability factor1 = 1/4

CLT joint ORIGINAL sum6 bucket word3 := sum(0, 1/4)

CLT joint ORIGINAL sum6 bucket word3 = 1/4

source roles = [2, 3, [3, 3], [1/2, 1/2]]

derived = [6, 1/4, 0, 1/4]

CLT SUM ORIGINAL total distribution mass0 := sum(0, 1/4)

CLT SUM ORIGINAL total distribution mass0 = 1/4

CLT SUM ORIGINAL total distribution mass1 := sum(1/4, 1/2)

CLT SUM ORIGINAL total distribution mass1 = 3/4

CLT SUM ORIGINAL total distribution mass2 := sum(3/4, 1/4)

CLT SUM ORIGINAL total distribution mass2 = 1

CLT SUM ORIGINAL value times probability0 := mul(4, 1/4)

CLT SUM ORIGINAL value times probability0 = 1

CLT SUM ORIGINAL ascending mean sum0 := sum(0, 1)

CLT SUM ORIGINAL ascending mean sum0 = 1

CLT SUM exact value square0 := mul(4, 4)

CLT SUM exact value square0 = 16

CLT SUM ORIGINAL square times probability0 := mul(16, 1/4)

CLT SUM ORIGINAL square times probability0 = 4

CLT SUM ORIGINAL ascending second moment sum0 := sum(0, 4)

CLT SUM ORIGINAL ascending second moment sum0 = 4

CLT SUM ORIGINAL value times probability1 := mul(5, 1/2)

CLT SUM ORIGINAL value times probability1 = 5/2

CLT SUM ORIGINAL ascending mean sum1 := sum(1, 5/2)

CLT SUM ORIGINAL ascending mean sum1 = 7/2

CLT SUM exact value square1 := mul(5, 5)

CLT SUM exact value square1 = 25

CLT SUM ORIGINAL square times probability1 := mul(25, 1/2)

CLT SUM ORIGINAL square times probability1 = 25/2

CLT SUM ORIGINAL ascending second moment sum1 := sum(4, 25/2)

CLT SUM ORIGINAL ascending second moment sum1 = 33/2

CLT SUM ORIGINAL value times probability2 := mul(6, 1/4)

CLT SUM ORIGINAL value times probability2 = 3/2

CLT SUM ORIGINAL ascending mean sum2 := sum(7/2, 3/2)

CLT SUM ORIGINAL ascending mean sum2 = 5

CLT SUM exact value square2 := mul(6, 6)

CLT SUM exact value square2 = 36

CLT SUM ORIGINAL square times probability2 := mul(36, 1/4)

CLT SUM ORIGINAL square times probability2 = 9

CLT SUM ORIGINAL ascending second moment sum2 := sum(33/2, 9)

CLT SUM ORIGINAL ascending second moment sum2 = 51/2

CLT SUM ORIGINAL mean square := mul(5, 5)

CLT SUM ORIGINAL mean square = 25

CLT SUM ORIGINAL second minus mean square := sub(51/2, 25)

CLT SUM ORIGINAL second minus mean square = 1/2

CLT ORIGINAL n times common mean := mul(2, 5/2)

CLT ORIGINAL n times common mean = 5

CLT ORIGINAL n times common variance := mul(2, 1/4)

CLT ORIGINAL n times common variance = 1/2

source roles = [2, 5/2, 1/4, [(4, 1/4), (5, 1/2), (6, 1/4)], 5, 1/2]

derived = [1, 5, 1/2, 0, 1, Z_n=(S_n-n*mu)/(sigma*sqrt(n))]

逐步反向核對

• S061 · premise_and_result_audit — 回查原始完整資料、步驟操作數與本步前提；有限核對、條件式定理引用與一般形式證明分開。
• S060 · premise_and_result_audit — 以精確有理數重算 CLT ORIGINAL n times common variance，不以小數近似替代。
• S059 · premise_and_result_audit — 以精確有理數重算 CLT ORIGINAL n times common mean，不以小數近似替代。
• S058 · premise_and_result_audit — 以精確有理數重算 CLT SUM ORIGINAL second minus mean square，不以小數近似替代。
• S057 · premise_and_result_audit — 以精確有理數重算 CLT SUM ORIGINAL mean square，不以小數近似替代。
• S056 · premise_and_result_audit — 以精確有理數重算 CLT SUM ORIGINAL ascending second moment sum2，不以小數近似替代。
• S055 · premise_and_result_audit — 以精確有理數重算 CLT SUM ORIGINAL square times probability2，不以小數近似替代。
• S054 · premise_and_result_audit — 以精確有理數重算 CLT SUM exact value square2，不以小數近似替代。
• S053 · premise_and_result_audit — 以精確有理數重算 CLT SUM ORIGINAL ascending mean sum2，不以小數近似替代。
• S052 · premise_and_result_audit — 以精確有理數重算 CLT SUM ORIGINAL value times probability2，不以小數近似替代。
• S051 · premise_and_result_audit — 以精確有理數重算 CLT SUM ORIGINAL ascending second moment sum1，不以小數近似替代。
• S050 · premise_and_result_audit — 以精確有理數重算 CLT SUM ORIGINAL square times probability1，不以小數近似替代。
• S049 · premise_and_result_audit — 以精確有理數重算 CLT SUM exact value square1，不以小數近似替代。
• S048 · premise_and_result_audit — 以精確有理數重算 CLT SUM ORIGINAL ascending mean sum1，不以小數近似替代。
• S047 · premise_and_result_audit — 以精確有理數重算 CLT SUM ORIGINAL value times probability1，不以小數近似替代。
• S046 · premise_and_result_audit — 以精確有理數重算 CLT SUM ORIGINAL ascending second moment sum0，不以小數近似替代。
• S045 · premise_and_result_audit — 以精確有理數重算 CLT SUM ORIGINAL square times probability0，不以小數近似替代。
• S044 · premise_and_result_audit — 以精確有理數重算 CLT SUM exact value square0，不以小數近似替代。
• S043 · premise_and_result_audit — 以精確有理數重算 CLT SUM ORIGINAL ascending mean sum0，不以小數近似替代。
• S042 · premise_and_result_audit — 以精確有理數重算 CLT SUM ORIGINAL value times probability0，不以小數近似替代。
• S041 · premise_and_result_audit — 以精確有理數重算 CLT SUM ORIGINAL total distribution mass2，不以小數近似替代。
• S040 · premise_and_result_audit — 以精確有理數重算 CLT SUM ORIGINAL total distribution mass1，不以小數近似替代。
• S039 · premise_and_result_audit — 以精確有理數重算 CLT SUM ORIGINAL total distribution mass0，不以小數近似替代。
• S038 · premise_and_result_audit — 回查原始完整資料、步驟操作數與本步前提；有限核對、條件式定理引用與一般形式證明分開。
• S037 · premise_and_result_audit — 以精確有理數重算 CLT joint ORIGINAL sum6 bucket word3，不以小數近似替代。
• S036 · premise_and_result_audit — 以精確有理數重算 CLT joint ORIGINAL word3 probability factor1，不以小數近似替代。
• S035 · premise_and_result_audit — 以精確有理數重算 CLT joint ORIGINAL word3 probability factor0，不以小數近似替代。
• S034 · premise_and_result_audit — 以精確有理數重算 CLT joint ORIGINAL word3 sum，不以小數近似替代。
• S033 · premise_and_result_audit — 回查原始完整資料、步驟操作數與本步前提；有限核對、條件式定理引用與一般形式證明分開。
• S032 · premise_and_result_audit — 以精確有理數重算 CLT joint ORIGINAL sum5 bucket word2，不以小數近似替代。
• S031 · premise_and_result_audit — 以精確有理數重算 CLT joint ORIGINAL word2 probability factor1，不以小數近似替代。
• S030 · premise_and_result_audit — 以精確有理數重算 CLT joint ORIGINAL word2 probability factor0，不以小數近似替代。
• S029 · premise_and_result_audit — 以精確有理數重算 CLT joint ORIGINAL word2 sum，不以小數近似替代。
• S028 · premise_and_result_audit — 回查原始完整資料、步驟操作數與本步前提；有限核對、條件式定理引用與一般形式證明分開。
• S027 · premise_and_result_audit — 以精確有理數重算 CLT joint ORIGINAL sum5 bucket word1，不以小數近似替代。
• S026 · premise_and_result_audit — 以精確有理數重算 CLT joint ORIGINAL word1 probability factor1，不以小數近似替代。
• S025 · premise_and_result_audit — 以精確有理數重算 CLT joint ORIGINAL word1 probability factor0，不以小數近似替代。
• S024 · premise_and_result_audit — 以精確有理數重算 CLT joint ORIGINAL word1 sum，不以小數近似替代。
• S023 · premise_and_result_audit — 回查原始完整資料、步驟操作數與本步前提；有限核對、條件式定理引用與一般形式證明分開。
• S022 · premise_and_result_audit — 以精確有理數重算 CLT joint ORIGINAL sum4 bucket word0，不以小數近似替代。
• S021 · premise_and_result_audit — 以精確有理數重算 CLT joint ORIGINAL word0 probability factor1，不以小數近似替代。
• S020 · premise_and_result_audit — 以精確有理數重算 CLT joint ORIGINAL word0 probability factor0，不以小數近似替代。
• S019 · premise_and_result_audit — 以精確有理數重算 CLT joint ORIGINAL word0 sum，不以小數近似替代。
• S018 · premise_and_result_audit — 回查原始完整資料、步驟操作數與本步前提；有限核對、條件式定理引用與一般形式證明分開。
• S017 · premise_and_result_audit — 以精確有理數重算 CLT ORIGINAL common marginal ORIGINAL second minus mean square，不以小數近似替代。
• S016 · premise_and_result_audit — 以精確有理數重算 CLT ORIGINAL common marginal ORIGINAL mean square，不以小數近似替代。
• S015 · premise_and_result_audit — 以精確有理數重算 CLT ORIGINAL common marginal ORIGINAL ascending second moment sum1，不以小數近似替代。
• S014 · premise_and_result_audit — 以精確有理數重算 CLT ORIGINAL common marginal ORIGINAL square times probability1，不以小數近似替代。
• S013 · premise_and_result_audit — 以精確有理數重算 CLT ORIGINAL common marginal exact value square1，不以小數近似替代。
• S012 · premise_and_result_audit — 以精確有理數重算 CLT ORIGINAL common marginal ORIGINAL ascending mean sum1，不以小數近似替代。
• S011 · premise_and_result_audit — 以精確有理數重算 CLT ORIGINAL common marginal ORIGINAL value times probability1，不以小數近似替代。
• S010 · premise_and_result_audit — 以精確有理數重算 CLT ORIGINAL common marginal ORIGINAL ascending second moment sum0，不以小數近似替代。
• S009 · premise_and_result_audit — 以精確有理數重算 CLT ORIGINAL common marginal ORIGINAL square times probability0，不以小數近似替代。
• S008 · premise_and_result_audit — 以精確有理數重算 CLT ORIGINAL common marginal exact value square0，不以小數近似替代。
• S007 · premise_and_result_audit — 以精確有理數重算 CLT ORIGINAL common marginal ORIGINAL ascending mean sum0，不以小數近似替代。
• S006 · premise_and_result_audit — 以精確有理數重算 CLT ORIGINAL common marginal ORIGINAL value times probability0，不以小數近似替代。
• S005 · premise_and_result_audit — 回查原始完整資料、步驟操作數與本步前提；有限核對、條件式定理引用與一般形式證明分開。
• S004 · premise_and_result_audit — 以精確有理數重算 CLT ORIGINAL common marginal ORIGINAL law mass sum1，不以小數近似替代。
• S003 · premise_and_result_audit — 以精確有理數重算 CLT ORIGINAL common marginal ORIGINAL law mass sum0，不以小數近似替代。
• S002 · premise_and_result_audit — 回查原始完整資料、步驟操作數與本步前提；有限核對、條件式定理引用與一般形式證明分開。
• S001 · premise_and_result_audit — 回查原始完整資料、步驟操作數與本步前提；有限核對、條件式定理引用與一般形式證明分開。

結論

(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)

依上列完整代值及條件核對得到原題答案；原題編號、格式與狀態保持不變。

本題覆核與來源綁定

覆核日期：2026-10-26

原題、答案、步驟表達式、數值 AST 與既有歷史保持原樣；原來的候選狀態屬於來源歷史，不覆寫為形式化證明。