第 14112 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":[[20,49],[15,49],[8,49],[6,49]],"values":[-1,1,13,1],"blocks":[0,0,1,1],"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":[[20,49],[15,49],[8,49],[6,49]],"values":[-1,1,13,1],"blocks":[0,0,1,1],"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(20/49, 15/49, 8/49, 6/49)

conditional model whole mass sum = 1

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

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

same outcome indices=[0, 1, 2, 3];X values=[-1, 1, 13, 1];block labels=[0, 0, 1, 1]

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

block0 mass := sum(20/49, 15/49)

block0 mass = 5/7

block0 atom0 weighted X := mul(20/49, -1)

block0 atom0 weighted X = -20/49

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

block0 atom1 weighted X = 15/49

block0 original weighted integral := sum(-20/49, 15/49)

block0 original weighted integral = -5/49

block0 positive-mass conditional mean := div(-5/49, 5/7)

block0 positive-mass conditional mean = -1/7

block label=0;indices=[0, 1];positive mass=5/7;weighted integral=-5/49

block-constant conditional value=-1/7

block1 mass := sum(8/49, 6/49)

block1 mass = 2/7

block1 atom2 weighted X := mul(8/49, 13)

block1 atom2 weighted X = 104/49

block1 atom3 weighted X := mul(6/49, 1)

block1 atom3 weighted X = 6/49

block1 original weighted integral := sum(104/49, 6/49)

block1 original weighted integral = 110/49

block1 positive-mass conditional mean := div(110/49, 2/7)

block1 positive-mass conditional mean = 55/7

block label=1;indices=[2, 3];positive mass=2/7;weighted integral=110/49

block-constant conditional value=55/7

selected block block1 conditional integral := mul(2/7, 55/7)

selected block block1 conditional integral = 110/49

selected block=1;indices=[2, 3];original integral=110/49;conditional integral=110/49

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

逐步反向核對

• S017 · 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。
• S016 · 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。
• 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

(1, [2, 3], 110/49, 110/49, 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。