第 14179 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":[[18,49],[3,49],[24,49],[4,49]],"values":[-9,0,5,9],"blocks":[0,0,1,1]}。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":[[18,49],[3,49],[24,49],[4,49]],"values":[-9,0,5,9],"blocks":[0,0,1,1]}。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(18/49, 3/49, 24/49, 4/49)

conditional model whole mass sum = 1

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

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

same outcome indices=[0, 1, 2, 3];X values=[-9, 0, 5, 9];block labels=[0, 0, 1, 1]

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

block0 mass := sum(18/49, 3/49)

block0 mass = 3/7

block0 atom0 weighted X := mul(18/49, -9)

block0 atom0 weighted X = -162/49

block0 atom1 weighted X := mul(3/49, 0)

block0 atom1 weighted X = 0

block0 original weighted integral := sum(-162/49, 0)

block0 original weighted integral = -162/49

block0 positive-mass conditional mean := div(-162/49, 3/7)

block0 positive-mass conditional mean = -54/7

block label=0;indices=[0, 1];positive mass=3/7;weighted integral=-162/49

block-constant conditional value=-54/7

block1 mass := sum(24/49, 4/49)

block1 mass = 4/7

block1 atom2 weighted X := mul(24/49, 5)

block1 atom2 weighted X = 120/49

block1 atom3 weighted X := mul(4/49, 9)

block1 atom3 weighted X = 36/49

block1 original weighted integral := sum(120/49, 36/49)

block1 original weighted integral = 156/49

block1 positive-mass conditional mean := div(156/49, 4/7)

block1 positive-mass conditional mean = 39/7

block label=1;indices=[2, 3];positive mass=4/7;weighted integral=156/49

block-constant conditional value=39/7

total original expectation := sum(-162/49, 156/49)

total original expectation = -6/49

total expectation block0 conditional integral := mul(3/7, -54/7)

total expectation block0 conditional integral = -162/49

total expectation block1 conditional integral := mul(4/7, 39/7)

total expectation block1 conditional integral = 156/49

total conditional expectation := sum(-162/49, 156/49)

total conditional expectation = -6/49

all conditioning blocks partition the same carrier;E_X=-6/49;E_conditional=-6/49

equal=是;tower_to_trivial_sigma_algebra

逐步反向核對

• 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 conditional expectation，不以小數近似替代。
• S018 · 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。
• S017 · 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。
• S016 · premise_and_result_audit — 以精確有理數重算 total original expectation，不以小數近似替代。
• S015 · 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。
• S014 · 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。
• S013 · premise_and_result_audit — 以精確有理數重算 block1 original weighted integral，不以小數近似替代。
• S012 · premise_and_result_audit — 以精確有理數重算 block1 atom3 weighted X，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 mass，不以小數近似替代。
• S009 · 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。
• S008 · 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。
• S007 · premise_and_result_audit — 以精確有理數重算 block0 original weighted integral，不以小數近似替代。
• S006 · premise_and_result_audit — 以精確有理數重算 block0 atom1 weighted X，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

(-6/49, -6/49, 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。