第 21957 題：全期望公式

原題與已知條件

同一有限probability space上的完整disjoint partition與integrable X。操作=totalexp-tower-identity；資料={"masses":[[1,16],[2,16],[13,16],[0,1]],"values":[6,7,9,105],"partition":[[0,1],[2,3]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。

• G001 — 同一有限probability space上的完整disjoint partition與integrable X。操作=totalexp-tower-identity；資料={"masses":[[1,16],[2,16],[13,16],[0,1]],"values":[6,7,9,105],"partition":[[0,1],[2,3]]}。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-law-of-total-expectation

逐步推導

source operation=totalexp-tower-identity;original snapshot and mathematics retained

explicit ordered source output fields=[conditional_values, tower_expectation, mean, equal, certificate]

Total expectation complete original mass := sum(1/16, 1/8, 13/16, 0)

Total expectation complete original mass = 1

Total expectation original atom0 weighted X := mul(1/16, 6)

Total expectation original atom0 weighted X = 3/8

Total expectation original atom1 weighted X := mul(1/8, 7)

Total expectation original atom1 weighted X = 7/8

Total expectation original atom2 weighted X := mul(13/16, 9)

Total expectation original atom2 weighted X = 117/16

Total expectation original atom3 weighted X := mul(0, 105)

Total expectation original atom3 weighted X = 0

Total expectation full original mean := sum(3/8, 7/8, 117/16, 0)

Total expectation full original mean = 137/16

ORIGINAL masses=[1/16, 1/8, 13/16, 0],SIGNED values X=[6, 7, 9, 105];ORIGINAL cell order=[[0, 1], [2, 3]]

flattened original atom labels=[0, 1, 2, 3] cover EVERY atom exactly ONCE including null atoms;total mass=1;integrable finite mean=137/16;NO independence premise

Total expectation cell0 mass := sum(1/16, 1/8)

Total expectation cell0 mass = 3/16

Total expectation cell0 numerator := sum(3/8, 7/8)

Total expectation cell0 numerator = 5/4

Total expectation cell0 positive-mass conditional mean := div(5/4, 3/16)

Total expectation cell0 positive-mass conditional mean = 20/3

ORIGINAL cell index=0,atom labels=[0, 1];mass=3/16;weighted numerator=5/4

conditional quotient=20/3;positive denominator qualifies division

Total expectation cell1 mass := sum(13/16, 0)

Total expectation cell1 mass = 13/16

Total expectation cell1 numerator := sum(117/16, 0)

Total expectation cell1 numerator = 117/16

Total expectation cell1 positive-mass conditional mean := div(117/16, 13/16)

Total expectation cell1 positive-mass conditional mean = 9

ORIGINAL cell index=1,atom labels=[2, 3];mass=13/16;weighted numerator=117/16

conditional quotient=9;positive denominator qualifies division

Total expectation positive cell0 weighted mean := mul(3/16, 20/3)

Total expectation positive cell0 weighted mean = 5/4

Total expectation positive cell1 weighted mean := mul(13/16, 9)

Total expectation positive cell1 weighted mean = 117/16

Total expectation sum positive-cell weighted means := sum(5/4, 117/16)

Total expectation sum positive-cell weighted means = 137/16

ALL original cell rows[index,atoms,mass,conditional_or_null,numerator]=[[0, [0, 1], 3/16, 20/3, 5/4], [1, [2, 3], 13/16, 9, 117/16]]

positive-cell weighted sum=137/16;ORIGINAL global mean=137/16;equal=是;disjoint complete cells regroup EVERY original p_i*X_i exactly ONCE,null contributions0

Total expectation tower atom0 weighted conditional version := mul(1/16, 20/3)

Total expectation tower atom0 weighted conditional version = 5/12

Total expectation tower atom1 weighted conditional version := mul(1/8, 20/3)

Total expectation tower atom1 weighted conditional version = 5/6

Total expectation tower atom2 weighted conditional version := mul(13/16, 9)

Total expectation tower atom2 weighted conditional version = 117/16

Total expectation tower atom3 weighted conditional version := mul(0, 9)

Total expectation tower atom3 weighted conditional version = 0

Total expectation full tower expectation := sum(5/12, 5/6, 117/16, 0)

Total expectation full tower expectation = 137/16

ORIGINAL atom-order conditional version=[20/3, 20/3, 9, 9];ORIGINAL cells=[[0, 1], [2, 3]]

version constant on each cell,so measurable for sigma(partition);each cell integral equals original X cell integral;null-cell version CHOSEN0,not quotient;E(E(X|G))=137/16=E(X)=137/16

逐步反向核對

• S026 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S025 · premise_and_result_audit — 以精確有理數重算 Total expectation full tower expectation，不以小數近似替代。
• S024 · premise_and_result_audit — 以精確有理數重算 Total expectation tower atom3 weighted conditional version，不以小數近似替代。
• S023 · premise_and_result_audit — 以精確有理數重算 Total expectation tower atom2 weighted conditional version，不以小數近似替代。
• S022 · premise_and_result_audit — 以精確有理數重算 Total expectation tower atom1 weighted conditional version，不以小數近似替代。
• S021 · premise_and_result_audit — 以精確有理數重算 Total expectation tower atom0 weighted conditional version，不以小數近似替代。
• S020 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S019 · premise_and_result_audit — 以精確有理數重算 Total expectation sum positive-cell weighted means，不以小數近似替代。
• S018 · premise_and_result_audit — 以精確有理數重算 Total expectation positive cell1 weighted mean，不以小數近似替代。
• S017 · premise_and_result_audit — 以精確有理數重算 Total expectation positive cell0 weighted mean，不以小數近似替代。
• S016 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S015 · premise_and_result_audit — 以精確有理數重算 Total expectation cell1 positive-mass conditional mean，不以小數近似替代。
• S014 · premise_and_result_audit — 以精確有理數重算 Total expectation cell1 numerator，不以小數近似替代。
• S013 · premise_and_result_audit — 以精確有理數重算 Total expectation cell1 mass，不以小數近似替代。
• S012 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S011 · premise_and_result_audit — 以精確有理數重算 Total expectation cell0 positive-mass conditional mean，不以小數近似替代。
• S010 · premise_and_result_audit — 以精確有理數重算 Total expectation cell0 numerator，不以小數近似替代。
• S009 · premise_and_result_audit — 以精確有理數重算 Total expectation cell0 mass，不以小數近似替代。
• S008 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S007 · premise_and_result_audit — 以精確有理數重算 Total expectation full original mean，不以小數近似替代。
• S006 · premise_and_result_audit — 以精確有理數重算 Total expectation original atom3 weighted X，不以小數近似替代。
• S005 · premise_and_result_audit — 以精確有理數重算 Total expectation original atom2 weighted X，不以小數近似替代。
• S004 · premise_and_result_audit — 以精確有理數重算 Total expectation original atom1 weighted X，不以小數近似替代。
• S003 · premise_and_result_audit — 以精確有理數重算 Total expectation original atom0 weighted X，不以小數近似替代。
• S002 · premise_and_result_audit — 以精確有理數重算 Total expectation complete original mass，不以小數近似替代。
• S001 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。

結論

([20/3, 20/3, 9, 9], 137/16, 137/16, 是, conditional_expectation_constant_on_partition_cells)

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。