第 15783 題：依分布收斂

原題與已知條件

Bernoulli symbolic tail與極限CDF。操作=convdist-pointwise-gap；資料={"p":[1,2],"c":[3,20],"shift":9,"n":8,"threshold":[1,2]}。請依 [n,threshold,absolute_gap,limit_CDF_continuity_point,symbolic_certificate] 欄序回答。

• G001 — Bernoulli symbolic tail與極限CDF。操作=convdist-pointwise-gap；資料={"p":[1,2],"c":[3,20],"shift":9,"n":8,"threshold":[1,2]}。請依 [n,threshold,absolute_gap,limit_CDF_continuity_point,symbolic_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-convergence-in-distribution

逐步推導

interior Bernoulli limit mass at0 := sub(1, 1/2)

interior Bernoulli limit mass at0 = 1/2

original limit parameter=[1, 2];reduced p=1/2;mass at0=1/2;mass at1=1/2

both endpoint masses positive;only limit-CDF jumps at0 and1;registered interior projection

Bernoulli first legal tail denominator := sum(1, 9)

Bernoulli first legal tail denominator = 10

Bernoulli first tail signed correction := div(3/20, 10)

Bernoulli first tail signed correction = 3/200

Bernoulli first legal parameter := sum(1/2, 3/200)

Bernoulli first legal parameter = 103/200

declared all-n family p_n=1/2+(3/20)/(n+9);n>=1;first parameter=103/200;limit=1/2

for every n>=1,lambda=(1+9)/(n+9) in(0,1];p_n=(1-lambda)*1/2+lambda*(103/200) in[0,1]

Bernoulli selected tail denominator := sum(8, 9)

Bernoulli selected tail denominator = 17

Bernoulli selected signed correction := div(3/20, 17)

Bernoulli selected signed correction = 3/340

Bernoulli selected exact parameter := sum(1/2, 3/340)

Bernoulli selected exact parameter = 173/340

selected distribution one minus Bernoulli parameter := sub(1, 173/340)

selected distribution one minus Bernoulli parameter = 167/340

selected distribution:Bernoulli parameter=173/340;same exact threshold=1/2;CDF branch=0<=t<1

selected distribution exact CDF value=167/340

limiting distribution one minus Bernoulli parameter := sub(1, 1/2)

limiting distribution one minus Bernoulli parameter = 1/2

limiting distribution:Bernoulli parameter=1/2;same exact threshold=1/2;CDF branch=0<=t<1

limiting distribution exact CDF value=1/2

Bernoulli selected signed CDF difference := sub(167/340, 1/2)

Bernoulli selected signed CDF difference = -3/340

Bernoulli negative CDF difference absolute value := mul(-1, -3/340)

Bernoulli negative CDF difference absolute value = 3/340

selected signed CDF difference=-3/340;same threshold=1/2

exact absolute gap=3/340;limit CDF continuity point=是

declared all-n parameters p_n=1/2+(3/20)/(n+9);same complete Bernoulli CDF formula

for every real t,gap=abs(3/20)/(n+9) on0<=t<1,and0 outside;for every epsilon>0 choose integer N>=1 with N+9>abs(3/20)/epsilon;then all n>=N give gap<epsilon

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 — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S015 · premise_and_result_audit — 以精確有理數重算 Bernoulli negative CDF difference absolute value，不以小數近似替代。
• S014 · premise_and_result_audit — 以精確有理數重算 Bernoulli selected signed CDF difference，不以小數近似替代。
• S013 · 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。
• S012 · premise_and_result_audit — 以精確有理數重算 limiting distribution one minus Bernoulli parameter，不以小數近似替代。
• S011 · 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。
• S010 · premise_and_result_audit — 以精確有理數重算 selected distribution one minus Bernoulli parameter，不以小數近似替代。
• S009 · premise_and_result_audit — 以精確有理數重算 Bernoulli selected exact parameter，不以小數近似替代。
• S008 · premise_and_result_audit — 以精確有理數重算 Bernoulli selected signed correction，不以小數近似替代。
• S007 · premise_and_result_audit — 以精確有理數重算 Bernoulli selected tail denominator，不以小數近似替代。
• S006 · 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。
• S005 · premise_and_result_audit — 以精確有理數重算 Bernoulli first legal parameter，不以小數近似替代。
• S004 · premise_and_result_audit — 以精確有理數重算 Bernoulli first tail signed correction，不以小數近似替代。
• S003 · premise_and_result_audit — 以精確有理數重算 Bernoulli first legal tail denominator，不以小數近似替代。
• 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 interior Bernoulli limit mass at0，不以小數近似替代。

結論

(8, 1/2, 3/340, 是, gap_tends_to_zero_at_every_limit_CDF_continuity_point)

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。