第 27202 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-application-boundary；資料={"mode":"non_iid","model":{"values_a":[-1,0],"probabilities_a":[[1,2],[1,2]],"values_b":[-1,0],"probabilities_b":[[1,7],[6,7]]}}。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-application-boundary；資料={"mode":"non_iid","model":{"values_a":[-1,0],"probabilities_a":[[1,2],[1,2]],"values_b":[-1,0],"probabilities_b":[[1,7],[6,7]]}}。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-application-boundary]

derived = [[mode, first_marginal_law, second_marginal_law, identical_marginals, this_IID_theorem_applicable, scope]]

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

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

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

CLT ORIGINAL first marginal ORIGINAL law mass sum1 = 1

source roles = [CLT ORIGINAL first marginal, [-1, 0], [[1, 2], [1, 2]]]

derived = [[[-1, 1/2], [0, 1/2]], 1]

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

CLT ORIGINAL first marginal ORIGINAL value times probability0 = -1/2

CLT ORIGINAL first marginal ORIGINAL ascending mean sum0 := sum(0, -1/2)

CLT ORIGINAL first marginal ORIGINAL ascending mean sum0 = -1/2

CLT ORIGINAL first marginal exact value square0 := mul(-1, -1)

CLT ORIGINAL first marginal exact value square0 = 1

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

CLT ORIGINAL first marginal ORIGINAL square times probability0 = 1/2

CLT ORIGINAL first marginal ORIGINAL ascending second moment sum0 := sum(0, 1/2)

CLT ORIGINAL first marginal ORIGINAL ascending second moment sum0 = 1/2

CLT ORIGINAL first marginal ORIGINAL value times probability1 := mul(0, 1/2)

CLT ORIGINAL first marginal ORIGINAL value times probability1 = 0

CLT ORIGINAL first marginal ORIGINAL ascending mean sum1 := sum(-1/2, 0)

CLT ORIGINAL first marginal ORIGINAL ascending mean sum1 = -1/2

CLT ORIGINAL first marginal exact value square1 := mul(0, 0)

CLT ORIGINAL first marginal exact value square1 = 0

CLT ORIGINAL first marginal ORIGINAL square times probability1 := mul(0, 1/2)

CLT ORIGINAL first marginal ORIGINAL square times probability1 = 0

CLT ORIGINAL first marginal ORIGINAL ascending second moment sum1 := sum(1/2, 0)

CLT ORIGINAL first marginal ORIGINAL ascending second moment sum1 = 1/2

CLT ORIGINAL first marginal ORIGINAL mean square := mul(-1/2, -1/2)

CLT ORIGINAL first marginal ORIGINAL mean square = 1/4

CLT ORIGINAL first marginal ORIGINAL second minus mean square := sub(1/2, 1/4)

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

source roles = [CLT ORIGINAL first marginal, [[-1, 1/2], [0, 1/2]]]

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

CLT ORIGINAL second marginal ORIGINAL law mass sum0 := sum(0, 1/7)

CLT ORIGINAL second marginal ORIGINAL law mass sum0 = 1/7

CLT ORIGINAL second marginal ORIGINAL law mass sum1 := sum(1/7, 6/7)

CLT ORIGINAL second marginal ORIGINAL law mass sum1 = 1

source roles = [CLT ORIGINAL second marginal, [-1, 0], [[1, 7], [6, 7]]]

derived = [[[-1, 1/7], [0, 6/7]], 1]

CLT ORIGINAL second marginal ORIGINAL value times probability0 := mul(-1, 1/7)

CLT ORIGINAL second marginal ORIGINAL value times probability0 = -1/7

CLT ORIGINAL second marginal ORIGINAL ascending mean sum0 := sum(0, -1/7)

CLT ORIGINAL second marginal ORIGINAL ascending mean sum0 = -1/7

CLT ORIGINAL second marginal exact value square0 := mul(-1, -1)

CLT ORIGINAL second marginal exact value square0 = 1

CLT ORIGINAL second marginal ORIGINAL square times probability0 := mul(1, 1/7)

CLT ORIGINAL second marginal ORIGINAL square times probability0 = 1/7

CLT ORIGINAL second marginal ORIGINAL ascending second moment sum0 := sum(0, 1/7)

CLT ORIGINAL second marginal ORIGINAL ascending second moment sum0 = 1/7

CLT ORIGINAL second marginal ORIGINAL value times probability1 := mul(0, 6/7)

CLT ORIGINAL second marginal ORIGINAL value times probability1 = 0

CLT ORIGINAL second marginal ORIGINAL ascending mean sum1 := sum(-1/7, 0)

CLT ORIGINAL second marginal ORIGINAL ascending mean sum1 = -1/7

CLT ORIGINAL second marginal exact value square1 := mul(0, 0)

CLT ORIGINAL second marginal exact value square1 = 0

CLT ORIGINAL second marginal ORIGINAL square times probability1 := mul(0, 6/7)

CLT ORIGINAL second marginal ORIGINAL square times probability1 = 0

CLT ORIGINAL second marginal ORIGINAL ascending second moment sum1 := sum(1/7, 0)

CLT ORIGINAL second marginal ORIGINAL ascending second moment sum1 = 1/7

CLT ORIGINAL second marginal ORIGINAL mean square := mul(-1/7, -1/7)

CLT ORIGINAL second marginal ORIGINAL mean square = 1/49

CLT ORIGINAL second marginal ORIGINAL second minus mean square := sub(1/7, 1/49)

CLT ORIGINAL second marginal ORIGINAL second minus mean square = 6/49

source roles = [CLT ORIGINAL second marginal, [[-1, 1/7], [0, 6/7]]]

derived = [-1/7, 1/7, 6/49]

source roles = [[[-1, 1/2], [0, 1/2]], [[-1, 1/7], [0, 6/7]]]

derived = [否, 否]

逐步反向核對

• S034 · 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。
• 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 ORIGINAL second marginal ORIGINAL second minus mean square，不以小數近似替代。
• S031 · premise_and_result_audit — 以精確有理數重算 CLT ORIGINAL second marginal ORIGINAL mean square，不以小數近似替代。
• S030 · premise_and_result_audit — 以精確有理數重算 CLT ORIGINAL second marginal ORIGINAL ascending second moment sum1，不以小數近似替代。
• S029 · premise_and_result_audit — 以精確有理數重算 CLT ORIGINAL second marginal ORIGINAL square times probability1，不以小數近似替代。
• S028 · premise_and_result_audit — 以精確有理數重算 CLT ORIGINAL second marginal exact value square1，不以小數近似替代。
• S027 · premise_and_result_audit — 以精確有理數重算 CLT ORIGINAL second marginal ORIGINAL ascending mean sum1，不以小數近似替代。
• S026 · premise_and_result_audit — 以精確有理數重算 CLT ORIGINAL second marginal ORIGINAL value times probability1，不以小數近似替代。
• S025 · premise_and_result_audit — 以精確有理數重算 CLT ORIGINAL second marginal ORIGINAL ascending second moment sum0，不以小數近似替代。
• S024 · premise_and_result_audit — 以精確有理數重算 CLT ORIGINAL second marginal ORIGINAL square times probability0，不以小數近似替代。
• S023 · premise_and_result_audit — 以精確有理數重算 CLT ORIGINAL second marginal exact value square0，不以小數近似替代。
• S022 · premise_and_result_audit — 以精確有理數重算 CLT ORIGINAL second marginal ORIGINAL ascending mean sum0，不以小數近似替代。
• S021 · premise_and_result_audit — 以精確有理數重算 CLT ORIGINAL second marginal ORIGINAL value times probability0，不以小數近似替代。
• S020 · 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。
• S019 · premise_and_result_audit — 以精確有理數重算 CLT ORIGINAL second marginal ORIGINAL law mass sum1，不以小數近似替代。
• S018 · premise_and_result_audit — 以精確有理數重算 CLT ORIGINAL second marginal ORIGINAL law mass sum0，不以小數近似替代。
• S017 · 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。
• S016 · premise_and_result_audit — 以精確有理數重算 CLT ORIGINAL first marginal ORIGINAL second minus 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 first marginal ORIGINAL mean square，不以小數近似替代。
• S014 · premise_and_result_audit — 以精確有理數重算 CLT ORIGINAL first marginal ORIGINAL ascending second moment sum1，不以小數近似替代。
• S013 · premise_and_result_audit — 以精確有理數重算 CLT ORIGINAL first marginal ORIGINAL square times probability1，不以小數近似替代。
• S012 · premise_and_result_audit — 以精確有理數重算 CLT ORIGINAL first marginal exact value square1，不以小數近似替代。
• S011 · premise_and_result_audit — 以精確有理數重算 CLT ORIGINAL first marginal ORIGINAL ascending mean sum1，不以小數近似替代。
• S010 · premise_and_result_audit — 以精確有理數重算 CLT ORIGINAL first marginal ORIGINAL value times probability1，不以小數近似替代。
• S009 · premise_and_result_audit — 以精確有理數重算 CLT ORIGINAL first marginal ORIGINAL ascending second moment sum0，不以小數近似替代。
• S008 · premise_and_result_audit — 以精確有理數重算 CLT ORIGINAL first marginal ORIGINAL square times probability0，不以小數近似替代。
• S007 · premise_and_result_audit — 以精確有理數重算 CLT ORIGINAL first marginal exact value square0，不以小數近似替代。
• S006 · premise_and_result_audit — 以精確有理數重算 CLT ORIGINAL first marginal ORIGINAL ascending mean sum0，不以小數近似替代。
• S005 · premise_and_result_audit — 以精確有理數重算 CLT ORIGINAL first marginal ORIGINAL value times probability0，不以小數近似替代。
• S004 · 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。
• S003 · premise_and_result_audit — 以精確有理數重算 CLT ORIGINAL first marginal ORIGINAL law mass sum1，不以小數近似替代。
• S002 · premise_and_result_audit — 以精確有理數重算 CLT ORIGINAL first marginal ORIGINAL law mass sum0，不以小數近似替代。
• 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

(non_iid, [[-1, 1/2], [0, 1/2]], [[-1, 1/7], [0, 6/7]], 否, 否, this_IID_CLT_not_applicable_not_a_general_counterexample)

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。