第 27666 題：有限值IID強大數律

原題與已知條件

無限IID process certificate、finite marginal law與almost-sure limit責任。操作=slln-process-law-qualification；資料={"values":[-2,-1],"probabilities":[[2,5],[3,5]],"process_certificate":"infinite_iid"}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。

• G001 — 無限IID process certificate、finite marginal law與almost-sure limit責任。操作=slln-process-law-qualification；資料={"values":[-2,-1],"probabilities":[[2,5],[3,5]],"process_certificate":"infinite_iid"}。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-strong-law-large-numbers-finite-valued-iid

逐步推導

source roles = [slln-process-law-qualification]

derived = [[law_rows, mean, variance, declared_process_certificate, named_theorem_conclusion, conditional_application_qualified]]

source roles = [infinite_iid]

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

SLLN ORIGINAL common marginal ORIGINAL mass sum0 := sum(0, 2/5)

SLLN ORIGINAL common marginal ORIGINAL mass sum0 = 2/5

SLLN ORIGINAL common marginal ORIGINAL mass sum1 := sum(2/5, 3/5)

SLLN ORIGINAL common marginal ORIGINAL mass sum1 = 1

SLLN ORIGINAL common marginal ORIGINAL weighted value0 := mul(-2, 2/5)

SLLN ORIGINAL common marginal ORIGINAL weighted value0 = -4/5

SLLN ORIGINAL common marginal ORIGINAL mean sum0 := sum(0, -4/5)

SLLN ORIGINAL common marginal ORIGINAL mean sum0 = -4/5

SLLN ORIGINAL common marginal exact unbounded value square0 := mul(-2, -2)

SLLN ORIGINAL common marginal exact unbounded value square0 = 4

SLLN ORIGINAL common marginal ORIGINAL weighted square0 := mul(4, 2/5)

SLLN ORIGINAL common marginal ORIGINAL weighted square0 = 8/5

SLLN ORIGINAL common marginal ORIGINAL second moment sum0 := sum(0, 8/5)

SLLN ORIGINAL common marginal ORIGINAL second moment sum0 = 8/5

SLLN ORIGINAL common marginal ORIGINAL weighted value1 := mul(-1, 3/5)

SLLN ORIGINAL common marginal ORIGINAL weighted value1 = -3/5

SLLN ORIGINAL common marginal ORIGINAL mean sum1 := sum(-4/5, -3/5)

SLLN ORIGINAL common marginal ORIGINAL mean sum1 = -7/5

SLLN ORIGINAL common marginal exact unbounded value square1 := mul(-1, -1)

SLLN ORIGINAL common marginal exact unbounded value square1 = 1

SLLN ORIGINAL common marginal ORIGINAL weighted square1 := mul(1, 3/5)

SLLN ORIGINAL common marginal ORIGINAL weighted square1 = 3/5

SLLN ORIGINAL common marginal ORIGINAL second moment sum1 := sum(8/5, 3/5)

SLLN ORIGINAL common marginal ORIGINAL second moment sum1 = 11/5

SLLN ORIGINAL common marginal ORIGINAL mean squared := mul(-7/5, -7/5)

SLLN ORIGINAL common marginal ORIGINAL mean squared = 49/25

SLLN ORIGINAL common marginal ORIGINAL second minus mean squared := sub(11/5, 49/25)

SLLN ORIGINAL common marginal ORIGINAL second minus mean squared = 6/25

source roles = [SLLN ORIGINAL common marginal, [-2, -1], [[2, 5], [3, 5]]]

derived = [[[-2, 2/5], [-1, 3/5]], 1, -7/5, 11/5, 6/25]

source roles = [[[-2, 2/5], [-1, 3/5]], -7/5, 6/25]

derived = [infinite_iid, P(lim_sample_mean=mu)=1, 是]

逐步反向核對

• S018 · premise_and_result_audit — 回查原始資料、操作數、完整範圍與本步前提；有限見證、條件式定理引用與形式證明分開。
• S017 · premise_and_result_audit — 回查原始資料、操作數、完整範圍與本步前提；有限見證、條件式定理引用與形式證明分開。
• S016 · premise_and_result_audit — 以精確有理數重算 SLLN ORIGINAL common marginal ORIGINAL second minus mean squared，不以小數近似替代。
• S015 · premise_and_result_audit — 以精確有理數重算 SLLN ORIGINAL common marginal ORIGINAL mean squared，不以小數近似替代。
• S014 · premise_and_result_audit — 以精確有理數重算 SLLN ORIGINAL common marginal ORIGINAL second moment sum1，不以小數近似替代。
• S013 · premise_and_result_audit — 以精確有理數重算 SLLN ORIGINAL common marginal ORIGINAL weighted square1，不以小數近似替代。
• S012 · premise_and_result_audit — 以精確有理數重算 SLLN ORIGINAL common marginal exact unbounded value square1，不以小數近似替代。
• S011 · premise_and_result_audit — 以精確有理數重算 SLLN ORIGINAL common marginal ORIGINAL mean sum1，不以小數近似替代。
• S010 · premise_and_result_audit — 以精確有理數重算 SLLN ORIGINAL common marginal ORIGINAL weighted value1，不以小數近似替代。
• S009 · premise_and_result_audit — 以精確有理數重算 SLLN ORIGINAL common marginal ORIGINAL second moment sum0，不以小數近似替代。
• S008 · premise_and_result_audit — 以精確有理數重算 SLLN ORIGINAL common marginal ORIGINAL weighted square0，不以小數近似替代。
• S007 · premise_and_result_audit — 以精確有理數重算 SLLN ORIGINAL common marginal exact unbounded value square0，不以小數近似替代。
• S006 · premise_and_result_audit — 以精確有理數重算 SLLN ORIGINAL common marginal ORIGINAL mean sum0，不以小數近似替代。
• S005 · premise_and_result_audit — 以精確有理數重算 SLLN ORIGINAL common marginal ORIGINAL weighted value0，不以小數近似替代。
• S004 · premise_and_result_audit — 以精確有理數重算 SLLN ORIGINAL common marginal ORIGINAL mass sum1，不以小數近似替代。
• S003 · premise_and_result_audit — 以精確有理數重算 SLLN ORIGINAL common marginal ORIGINAL mass sum0，不以小數近似替代。
• S002 · premise_and_result_audit — 回查原始資料、操作數、完整範圍與本步前提；有限見證、條件式定理引用與形式證明分開。
• S001 · premise_and_result_audit — 回查原始資料、操作數、完整範圍與本步前提；有限見證、條件式定理引用與形式證明分開。

結論

([[-2, 2/5], [-1, 3/5]], -7/5, 6/25, infinite_iid, P(lim_sample_mean=mu)=1, 是)

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。