第 14119 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-block-integral；資料={"masses":[[9,49],[12,49],[12,49],[16,49]],"values":[6,1,20,-6],"blocks":[0,1,1,2],"block_label":1}。請依 [block_label,indices,original_integral,conditional_integral,equal,boundary] 欄序回答。

• G001 — X機率law與conditioning partition綁同一有限空間。操作=condexp-block-integral；資料={"masses":[[9,49],[12,49],[12,49],[16,49]],"values":[6,1,20,-6],"blocks":[0,1,1,2],"block_label":1}。請依 [block_label,indices,original_integral,conditional_integral,equal,boundary] 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(9/49, 12/49, 12/49, 16/49)

conditional model whole mass sum = 1

conditional model:raw aligned masses=[[9, 49], [12, 49], [12, 49], [16, 49]]

normalized masses=[9/49, 12/49, 12/49, 16/49];nonnegative=是;total=1

same outcome indices=[0, 1, 2, 3];X values=[6, 1, 20, -6];block labels=[0, 1, 1, 2]

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

block0 mass := sum(9/49)

block0 mass = 9/49

block0 atom0 weighted X := mul(9/49, 6)

block0 atom0 weighted X = 54/49

block0 original weighted integral := sum(54/49)

block0 original weighted integral = 54/49

block0 positive-mass conditional mean := div(54/49, 9/49)

block0 positive-mass conditional mean = 6

block label=0;indices=[0];positive mass=9/49;weighted integral=54/49

block-constant conditional value=6

block1 mass := sum(12/49, 12/49)

block1 mass = 24/49

block1 atom1 weighted X := mul(12/49, 1)

block1 atom1 weighted X = 12/49

block1 atom2 weighted X := mul(12/49, 20)

block1 atom2 weighted X = 240/49

block1 original weighted integral := sum(12/49, 240/49)

block1 original weighted integral = 36/7

block1 positive-mass conditional mean := div(36/7, 24/49)

block1 positive-mass conditional mean = 21/2

block label=1;indices=[1, 2];positive mass=24/49;weighted integral=36/7

block-constant conditional value=21/2

block2 mass := sum(16/49)

block2 mass = 16/49

block2 atom3 weighted X := mul(16/49, -6)

block2 atom3 weighted X = -96/49

block2 original weighted integral := sum(-96/49)

block2 original weighted integral = -96/49

block2 positive-mass conditional mean := div(-96/49, 16/49)

block2 positive-mass conditional mean = -6

block label=2;indices=[3];positive mass=16/49;weighted integral=-96/49

block-constant conditional value=-6

selected block block1 conditional integral := mul(24/49, 21/2)

selected block block1 conditional integral = 36/7

selected block=1;indices=[1, 2];original integral=36/7;conditional integral=36/7

equal=是;this is the selected block obligation,not only a global average

逐步反向核對

• S021 · 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。
• S020 · premise_and_result_audit — 以精確有理數重算 selected block block1 conditional integral，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

(1, [1, 2], 36/7, 36/7, 是, positive_block)

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。