第 14199 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":[[6,49],[15,49],[8,49],[20,49]],"values":[11,-1,25,-11],"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":[[6,49],[15,49],[8,49],[20,49]],"values":[11,-1,25,-11],"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(6/49, 15/49, 8/49, 20/49)

conditional model whole mass sum = 1

conditional model:raw aligned masses=[[6, 49], [15, 49], [8, 49], [20, 49]]

normalized masses=[6/49, 15/49, 8/49, 20/49];nonnegative=是;total=1

same outcome indices=[0, 1, 2, 3];X values=[11, -1, 25, -11];block labels=[0, 0, 1, 1]

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

block0 mass := sum(6/49, 15/49)

block0 mass = 3/7

block0 atom0 weighted X := mul(6/49, 11)

block0 atom0 weighted X = 66/49

block0 atom1 weighted X := mul(15/49, -1)

block0 atom1 weighted X = -15/49

block0 original weighted integral := sum(66/49, -15/49)

block0 original weighted integral = 51/49

block0 positive-mass conditional mean := div(51/49, 3/7)

block0 positive-mass conditional mean = 17/7

block label=0;indices=[0, 1];positive mass=3/7;weighted integral=51/49

block-constant conditional value=17/7

block1 mass := sum(8/49, 20/49)

block1 mass = 4/7

block1 atom2 weighted X := mul(8/49, 25)

block1 atom2 weighted X = 200/49

block1 atom3 weighted X := mul(20/49, -11)

block1 atom3 weighted X = -220/49

block1 original weighted integral := sum(200/49, -220/49)

block1 original weighted integral = -20/49

block1 positive-mass conditional mean := div(-20/49, 4/7)

block1 positive-mass conditional mean = -5/7

block label=1;indices=[2, 3];positive mass=4/7;weighted integral=-20/49

block-constant conditional value=-5/7

total original expectation := sum(51/49, -20/49)

total original expectation = 31/49

total expectation block0 conditional integral := mul(3/7, 17/7)

total expectation block0 conditional integral = 51/49

total expectation block1 conditional integral := mul(4/7, -5/7)

total expectation block1 conditional integral = -20/49

total conditional expectation := sum(51/49, -20/49)

total conditional expectation = 31/49

all conditioning blocks partition the same carrier;E_X=31/49;E_conditional=31/49

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

(31/49, 31/49, 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。