第 31648 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, 12), (1, 12), (2, 36), (3, 1021)}，各結果依列序 [0, 1, 2, 3] 的非負機率權重為 [1/30, 24/30, 5/30, 0/30]，權重總和恰為一。令 Z=3X+(17)，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, 12), (1, 12), (2, 36), (3, 1021)}，各結果依列序 [0, 1, 2, 3] 的非負機率權重為 [1/30, 24/30, 5/30, 0/30]，權重總和恰為一。令 Z=3X+(17)，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, 24)

Expectation ORIGINAL weight total through slot1 = 25

Expectation ORIGINAL weight total through slot2 := sum(25, 5)

Expectation ORIGINAL weight total through slot2 = 30

Expectation ORIGINAL weight total through slot3 := sum(30, 0)

Expectation ORIGINAL weight total through slot3 = 30

source roles = [[0, 1, 2, 3], powerset(S), [1, 24, 5, 0], 30, [12, 12, 36, 1021], not-supplied]

derived = [[0, 1, 2, 3], 30, 是]

Expectation affine ORIGINAL slot0 a times X := mul(3, 12)

Expectation affine ORIGINAL slot0 a times X = 36

Expectation affine ORIGINAL slot0 plus b := sum(36, 17)

Expectation affine ORIGINAL slot0 plus b = 53

source roles = [0, 0, affine_expectation, 12, not-supplied, 3, 17, not-supplied]

derived = [53]

Expectation ORIGINAL slot0 weight times transformed value := mul(1, 53)

Expectation ORIGINAL slot0 weight times transformed value = 53

Expectation ORIGINAL weighted numerator through slot0 := sum(0, 53)

Expectation ORIGINAL weighted numerator through slot0 = 53

Expectation affine ORIGINAL slot1 a times X := mul(3, 12)

Expectation affine ORIGINAL slot1 a times X = 36

Expectation affine ORIGINAL slot1 plus b := sum(36, 17)

Expectation affine ORIGINAL slot1 plus b = 53

source roles = [1, 1, affine_expectation, 12, not-supplied, 3, 17, not-supplied]

derived = [53]

Expectation ORIGINAL slot1 weight times transformed value := mul(24, 53)

Expectation ORIGINAL slot1 weight times transformed value = 1272

Expectation ORIGINAL weighted numerator through slot1 := sum(53, 1272)

Expectation ORIGINAL weighted numerator through slot1 = 1325

Expectation affine ORIGINAL slot2 a times X := mul(3, 36)

Expectation affine ORIGINAL slot2 a times X = 108

Expectation affine ORIGINAL slot2 plus b := sum(108, 17)

Expectation affine ORIGINAL slot2 plus b = 125

source roles = [2, 2, affine_expectation, 36, not-supplied, 3, 17, not-supplied]

derived = [125]

Expectation ORIGINAL slot2 weight times transformed value := mul(5, 125)

Expectation ORIGINAL slot2 weight times transformed value = 625

Expectation ORIGINAL weighted numerator through slot2 := sum(1325, 625)

Expectation ORIGINAL weighted numerator through slot2 = 1950

Expectation affine ORIGINAL slot3 a times X := mul(3, 1021)

Expectation affine ORIGINAL slot3 a times X = 3063

Expectation affine ORIGINAL slot3 plus b := sum(3063, 17)

Expectation affine ORIGINAL slot3 plus b = 3080

source roles = [3, 3, affine_expectation, 1021, not-supplied, 3, 17, not-supplied]

derived = [3080]

Expectation ORIGINAL slot3 weight times transformed value := mul(0, 3080)

Expectation ORIGINAL slot3 weight times transformed value = 0

Expectation ORIGINAL weighted numerator through slot3 := sum(1950, 0)

Expectation ORIGINAL weighted numerator through slot3 = 1950

Expectation ORIGINAL weighted numerator over denominator := div(1950, 30)

Expectation ORIGINAL weighted numerator over denominator = 65

source roles = [[0, 1, 2, 3], [1, 24, 5, 0], 30, affine_expectation, [53, 53, 125, 3080]]

derived = [[53, 1272, 625, 0], 1950, 65/1]

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

65/1

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。