第 31631 題：仿射變換的有限期望

原題與已知條件

非空有限 S={0, 1, 2, 3}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, -11), (1, -11), (2, -4), (3, 1004)}，各結果依列序 [0, 1, 2, 3] 的非負機率權重為 [1/12, 7/12, 4/12, 0/12]，權重總和恰為一。令 Z=-2X+(0)，求精確 E[Z]。

• G001 — 非空有限 S={0, 1, 2, 3}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, -11), (1, -11), (2, -4), (3, 1004)}，各結果依列序 [0, 1, 2, 3] 的非負機率權重為 [1/12, 7/12, 4/12, 0/12]，權重總和恰為一。令 Z=-2X+(0)，求精確 E[Z]。

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-expectation

逐步推導

source roles = [qf-expectation-affine, no-domain-field, affine_expectation, no-form-field]

derived = [[reduced_fraction_RETAIN1_STRING]]

Expectation ORIGINAL weight total through slot0 := sum(0, 1)

Expectation ORIGINAL weight total through slot0 = 1

Expectation ORIGINAL weight total through slot1 := sum(1, 7)

Expectation ORIGINAL weight total through slot1 = 8

Expectation ORIGINAL weight total through slot2 := sum(8, 4)

Expectation ORIGINAL weight total through slot2 = 12

Expectation ORIGINAL weight total through slot3 := sum(12, 0)

Expectation ORIGINAL weight total through slot3 = 12

source roles = [[0, 1, 2, 3], powerset(S), [1, 7, 4, 0], 12, [-11, -11, -4, 1004], not-supplied]

derived = [[0, 1, 2, 3], 12, 是]

Expectation affine ORIGINAL slot0 a times X := mul(-2, -11)

Expectation affine ORIGINAL slot0 a times X = 22

Expectation affine ORIGINAL slot0 plus b := sum(22, 0)

Expectation affine ORIGINAL slot0 plus b = 22

source roles = [0, 0, affine_expectation, -11, not-supplied, -2, 0, not-supplied]

derived = [22]

Expectation ORIGINAL slot0 weight times transformed value := mul(1, 22)

Expectation ORIGINAL slot0 weight times transformed value = 22

Expectation ORIGINAL weighted numerator through slot0 := sum(0, 22)

Expectation ORIGINAL weighted numerator through slot0 = 22

Expectation affine ORIGINAL slot1 a times X := mul(-2, -11)

Expectation affine ORIGINAL slot1 a times X = 22

Expectation affine ORIGINAL slot1 plus b := sum(22, 0)

Expectation affine ORIGINAL slot1 plus b = 22

source roles = [1, 1, affine_expectation, -11, not-supplied, -2, 0, not-supplied]

derived = [22]

Expectation ORIGINAL slot1 weight times transformed value := mul(7, 22)

Expectation ORIGINAL slot1 weight times transformed value = 154

Expectation ORIGINAL weighted numerator through slot1 := sum(22, 154)

Expectation ORIGINAL weighted numerator through slot1 = 176

Expectation affine ORIGINAL slot2 a times X := mul(-2, -4)

Expectation affine ORIGINAL slot2 a times X = 8

Expectation affine ORIGINAL slot2 plus b := sum(8, 0)

Expectation affine ORIGINAL slot2 plus b = 8

source roles = [2, 2, affine_expectation, -4, not-supplied, -2, 0, not-supplied]

derived = [8]

Expectation ORIGINAL slot2 weight times transformed value := mul(4, 8)

Expectation ORIGINAL slot2 weight times transformed value = 32

Expectation ORIGINAL weighted numerator through slot2 := sum(176, 32)

Expectation ORIGINAL weighted numerator through slot2 = 208

Expectation affine ORIGINAL slot3 a times X := mul(-2, 1004)

Expectation affine ORIGINAL slot3 a times X = -2008

Expectation affine ORIGINAL slot3 plus b := sum(-2008, 0)

Expectation affine ORIGINAL slot3 plus b = -2008

source roles = [3, 3, affine_expectation, 1004, not-supplied, -2, 0, not-supplied]

derived = [-2008]

Expectation ORIGINAL slot3 weight times transformed value := mul(0, -2008)

Expectation ORIGINAL slot3 weight times transformed value = 0

Expectation ORIGINAL weighted numerator through slot3 := sum(208, 0)

Expectation ORIGINAL weighted numerator through slot3 = 208

Expectation ORIGINAL weighted numerator over denominator := div(208, 12)

Expectation ORIGINAL weighted numerator over denominator = 52/3

source roles = [[0, 1, 2, 3], [1, 7, 4, 0], 12, affine_expectation, [22, 22, 8, -2008]]

derived = [[22, 154, 32, 0], 208, 52/3]

逐步反向核對

• S028 · premise_and_result_audit — 反向代回原輸入、欄位角色、論域及本步條件；一致性不表示結論能唯一恢復前提。有限核對、獨立審查、形式證明與發布分開標記。
• S027 · premise_and_result_audit — 以精確有理數重算 Expectation ORIGINAL weighted numerator over denominator，不以小數近似替代。
• S026 · premise_and_result_audit — 以精確有理數重算 Expectation ORIGINAL weighted numerator through slot3，不以小數近似替代。
• S025 · premise_and_result_audit — 以精確有理數重算 Expectation ORIGINAL slot3 weight times transformed value，不以小數近似替代。
• S024 · premise_and_result_audit — 反向代回原輸入、欄位角色、論域及本步條件；一致性不表示結論能唯一恢復前提。有限核對、獨立審查、形式證明與發布分開標記。
• S023 · premise_and_result_audit — 以精確有理數重算 Expectation affine ORIGINAL slot3 plus b，不以小數近似替代。
• S022 · premise_and_result_audit — 以精確有理數重算 Expectation affine ORIGINAL slot3 a times X，不以小數近似替代。
• S021 · premise_and_result_audit — 以精確有理數重算 Expectation ORIGINAL weighted numerator through slot2，不以小數近似替代。
• S020 · premise_and_result_audit — 以精確有理數重算 Expectation ORIGINAL slot2 weight times transformed value，不以小數近似替代。
• S019 · premise_and_result_audit — 反向代回原輸入、欄位角色、論域及本步條件；一致性不表示結論能唯一恢復前提。有限核對、獨立審查、形式證明與發布分開標記。
• S018 · premise_and_result_audit — 以精確有理數重算 Expectation affine ORIGINAL slot2 plus b，不以小數近似替代。
• S017 · premise_and_result_audit — 以精確有理數重算 Expectation affine ORIGINAL slot2 a times X，不以小數近似替代。
• S016 · premise_and_result_audit — 以精確有理數重算 Expectation ORIGINAL weighted numerator through slot1，不以小數近似替代。
• S015 · premise_and_result_audit — 以精確有理數重算 Expectation ORIGINAL slot1 weight times transformed value，不以小數近似替代。
• S014 · premise_and_result_audit — 反向代回原輸入、欄位角色、論域及本步條件；一致性不表示結論能唯一恢復前提。有限核對、獨立審查、形式證明與發布分開標記。
• S013 · premise_and_result_audit — 以精確有理數重算 Expectation affine ORIGINAL slot1 plus b，不以小數近似替代。
• S012 · premise_and_result_audit — 以精確有理數重算 Expectation affine ORIGINAL slot1 a times X，不以小數近似替代。
• S011 · premise_and_result_audit — 以精確有理數重算 Expectation ORIGINAL weighted numerator through slot0，不以小數近似替代。
• S010 · premise_and_result_audit — 以精確有理數重算 Expectation ORIGINAL slot0 weight times transformed value，不以小數近似替代。
• S009 · premise_and_result_audit — 反向代回原輸入、欄位角色、論域及本步條件；一致性不表示結論能唯一恢復前提。有限核對、獨立審查、形式證明與發布分開標記。
• S008 · premise_and_result_audit — 以精確有理數重算 Expectation affine ORIGINAL slot0 plus b，不以小數近似替代。
• S007 · premise_and_result_audit — 以精確有理數重算 Expectation affine ORIGINAL slot0 a times X，不以小數近似替代。
• S006 · premise_and_result_audit — 反向代回原輸入、欄位角色、論域及本步條件；一致性不表示結論能唯一恢復前提。有限核對、獨立審查、形式證明與發布分開標記。
• S005 · premise_and_result_audit — 以精確有理數重算 Expectation ORIGINAL weight total through slot3，不以小數近似替代。
• S004 · premise_and_result_audit — 以精確有理數重算 Expectation ORIGINAL weight total through slot2，不以小數近似替代。
• S003 · premise_and_result_audit — 以精確有理數重算 Expectation ORIGINAL weight total through slot1，不以小數近似替代。
• S002 · premise_and_result_audit — 以精確有理數重算 Expectation ORIGINAL weight total through slot0，不以小數近似替代。
• S001 · premise_and_result_audit — 反向代回原輸入、欄位角色、論域及本步條件；一致性不表示結論能唯一恢復前提。有限核對、獨立審查、形式證明與發布分開標記。

結論

52/3

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。