第 14178 題：有限條件期望

原題與已知條件

X機率law與conditioning partition綁同一有限空間。操作=condexp-total-expectation；資料={"masses":[[24,49],[4,49],[18,49],[3,49]],"values":[-10,-1,4,10],"blocks":[0,1,1,2]}。請依 [E_X,E_conditional,equal,certificate] 欄序回答。

• G001 — X機率law與conditioning partition綁同一有限空間。操作=condexp-total-expectation；資料={"masses":[[24,49],[4,49],[18,49],[3,49]],"values":[-10,-1,4,10],"blocks":[0,1,1,2]}。請依 [E_X,E_conditional,equal,certificate] 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-conditional-expectation

逐步推導

conditional model whole mass sum := sum(24/49, 4/49, 18/49, 3/49)

conditional model whole mass sum = 1

conditional model:raw aligned masses=[[24, 49], [4, 49], [18, 49], [3, 49]]

normalized masses=[24/49, 4/49, 18/49, 3/49];nonnegative=是;total=1

same outcome indices=[0, 1, 2, 3];X values=[-10, -1, 4, 10];block labels=[0, 1, 1, 2]

distinct conditioning labels in increasing order=[0, 1, 2]

block0 mass := sum(24/49)

block0 mass = 24/49

block0 atom0 weighted X := mul(24/49, -10)

block0 atom0 weighted X = -240/49

block0 original weighted integral := sum(-240/49)

block0 original weighted integral = -240/49

block0 positive-mass conditional mean := div(-240/49, 24/49)

block0 positive-mass conditional mean = -10

block label=0;indices=[0];positive mass=24/49;weighted integral=-240/49

block-constant conditional value=-10

block1 mass := sum(4/49, 18/49)

block1 mass = 22/49

block1 atom1 weighted X := mul(4/49, -1)

block1 atom1 weighted X = -4/49

block1 atom2 weighted X := mul(18/49, 4)

block1 atom2 weighted X = 72/49

block1 original weighted integral := sum(-4/49, 72/49)

block1 original weighted integral = 68/49

block1 positive-mass conditional mean := div(68/49, 22/49)

block1 positive-mass conditional mean = 34/11

block label=1;indices=[1, 2];positive mass=22/49;weighted integral=68/49

block-constant conditional value=34/11

block2 mass := sum(3/49)

block2 mass = 3/49

block2 atom3 weighted X := mul(3/49, 10)

block2 atom3 weighted X = 30/49

block2 original weighted integral := sum(30/49)

block2 original weighted integral = 30/49

block2 positive-mass conditional mean := div(30/49, 3/49)

block2 positive-mass conditional mean = 10

block label=2;indices=[3];positive mass=3/49;weighted integral=30/49

block-constant conditional value=10

total original expectation := sum(-240/49, 68/49, 30/49)

total original expectation = -142/49

total expectation block0 conditional integral := mul(24/49, -10)

total expectation block0 conditional integral = -240/49

total expectation block1 conditional integral := mul(22/49, 34/11)

total expectation block1 conditional integral = 68/49

total expectation block2 conditional integral := mul(3/49, 10)

total expectation block2 conditional integral = 30/49

total conditional expectation := sum(-240/49, 68/49, 30/49)

total conditional expectation = -142/49

all conditioning blocks partition the same carrier;E_X=-142/49;E_conditional=-142/49

equal=是;tower_to_trivial_sigma_algebra

逐步反向核對

• S025 · 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。
• S024 · premise_and_result_audit — 以精確有理數重算 total conditional expectation，不以小數近似替代。
• S023 · premise_and_result_audit — 以精確有理數重算 total expectation block2 conditional integral，不以小數近似替代。
• S022 · premise_and_result_audit — 以精確有理數重算 total expectation block1 conditional integral，不以小數近似替代。
• S021 · premise_and_result_audit — 以精確有理數重算 total expectation block0 conditional integral，不以小數近似替代。
• S020 · premise_and_result_audit — 以精確有理數重算 total original expectation，不以小數近似替代。
• S019 · 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。
• S018 · premise_and_result_audit — 以精確有理數重算 block2 positive-mass conditional mean，不以小數近似替代。
• S017 · premise_and_result_audit — 以精確有理數重算 block2 original weighted integral，不以小數近似替代。
• S016 · premise_and_result_audit — 以精確有理數重算 block2 atom3 weighted X，不以小數近似替代。
• S015 · premise_and_result_audit — 以精確有理數重算 block2 mass，不以小數近似替代。
• S014 · 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。
• S013 · premise_and_result_audit — 以精確有理數重算 block1 positive-mass conditional mean，不以小數近似替代。
• S012 · premise_and_result_audit — 以精確有理數重算 block1 original weighted integral，不以小數近似替代。
• S011 · premise_and_result_audit — 以精確有理數重算 block1 atom2 weighted X，不以小數近似替代。
• S010 · premise_and_result_audit — 以精確有理數重算 block1 atom1 weighted X，不以小數近似替代。
• S009 · premise_and_result_audit — 以精確有理數重算 block1 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 block0 positive-mass conditional mean，不以小數近似替代。
• S006 · premise_and_result_audit — 以精確有理數重算 block0 original weighted integral，不以小數近似替代。
• S005 · premise_and_result_audit — 以精確有理數重算 block0 atom0 weighted X，不以小數近似替代。
• S004 · premise_and_result_audit — 以精確有理數重算 block0 mass，不以小數近似替代。
• S003 · 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。
• S002 · 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。
• S001 · premise_and_result_audit — 以精確有理數重算 conditional model whole mass sum，不以小數近似替代。

結論

(-142/49, -142/49, 是, tower_to_trivial_sigma_algebra)

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。