OCME中英數學知識與推導English

標準常態分布基礎 · 脈衝 R04

本批已通過 Schema、去重、來源快照與獨立答案重算；仍未標記為人工審定。

本題庫已有對應的中英逐步詳解。原題的 candidate 與英文 ai_candidate 狀態保留；AI 覆核、有限計算與 Lean 形式證明分開標示。

batch-20261015-advance-20261114-r04-standard-normal-distribution-foundations · 2026-10-15 · 100 題

欄序為來源綁定的呈現補充；原始 JSONL 與數學答案不變。

欄序與來源綁定資料

1. 標準常態分布

standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-symbolic-density；資料={"x":[1,2]}。請依 [x, symbolic_density, density_law, normalization_scope] 欄序回答。

查看答案與解釋

(1/2, (2*pi)^(-1/2)*exp(-1/8), phi(x)=(2*pi)^(-1/2)*exp(-x^2/2), normalization_is_a_named_Gaussian_integral_fact_not_a_finite_evaluation)

Wave6第八層核對=(1/2, (2*pi)^(-1/2)*exp(-1/8), phi(x)=(2*pi)^(-1/2)*exp(-x^2/2), normalization_is_a_named_Gaussian_integral_fact_not_a_finite_evaluation)；依完整明示資料獨立重算；Phi(0)=1/2與Phi(-x)=1-Phi(x)只由已宣告normalization及symmetry推出；Gaussian integral、mean 0與variance 1是named theorem facts；finite quadrature不證它們或非零CDF exact value。

本題逐步詳解（AI 二次覆核）

標準常態分布 · qf-mod45-normaldist-symbolic-density · candidate

2. 標準常態分布

standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-symbolic-density；資料={"x":[-2,3]}。請依 [x, symbolic_density, density_law, normalization_scope] 欄序回答。

查看答案與解釋

(-2/3, (2*pi)^(-1/2)*exp(-2/9), phi(x)=(2*pi)^(-1/2)*exp(-x^2/2), normalization_is_a_named_Gaussian_integral_fact_not_a_finite_evaluation)

Wave6第八層核對=(-2/3, (2*pi)^(-1/2)*exp(-2/9), phi(x)=(2*pi)^(-1/2)*exp(-x^2/2), normalization_is_a_named_Gaussian_integral_fact_not_a_finite_evaluation)；依完整明示資料獨立重算；Phi(0)=1/2與Phi(-x)=1-Phi(x)只由已宣告normalization及symmetry推出；Gaussian integral、mean 0與variance 1是named theorem facts；finite quadrature不證它們或非零CDF exact value。

本題逐步詳解（AI 二次覆核）

標準常態分布 · qf-mod45-normaldist-symbolic-density · candidate

3. 標準常態分布

standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-symbolic-density；資料={"x":[3,4]}。請依 [x, symbolic_density, density_law, normalization_scope] 欄序回答。

查看答案與解釋

(3/4, (2*pi)^(-1/2)*exp(-9/32), phi(x)=(2*pi)^(-1/2)*exp(-x^2/2), normalization_is_a_named_Gaussian_integral_fact_not_a_finite_evaluation)

Wave6第八層核對=(3/4, (2*pi)^(-1/2)*exp(-9/32), phi(x)=(2*pi)^(-1/2)*exp(-x^2/2), normalization_is_a_named_Gaussian_integral_fact_not_a_finite_evaluation)；依完整明示資料獨立重算；Phi(0)=1/2與Phi(-x)=1-Phi(x)只由已宣告normalization及symmetry推出；Gaussian integral、mean 0與variance 1是named theorem facts；finite quadrature不證它們或非零CDF exact value。

本題逐步詳解（AI 二次覆核）

標準常態分布 · qf-mod45-normaldist-symbolic-density · candidate

4. 標準常態分布

standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-symbolic-density；資料={"x":[-4,5]}。請依 [x, symbolic_density, density_law, normalization_scope] 欄序回答。

查看答案與解釋

(-4/5, (2*pi)^(-1/2)*exp(-8/25), phi(x)=(2*pi)^(-1/2)*exp(-x^2/2), normalization_is_a_named_Gaussian_integral_fact_not_a_finite_evaluation)

Wave6第八層核對=(-4/5, (2*pi)^(-1/2)*exp(-8/25), phi(x)=(2*pi)^(-1/2)*exp(-x^2/2), normalization_is_a_named_Gaussian_integral_fact_not_a_finite_evaluation)；依完整明示資料獨立重算；Phi(0)=1/2與Phi(-x)=1-Phi(x)只由已宣告normalization及symmetry推出；Gaussian integral、mean 0與variance 1是named theorem facts；finite quadrature不證它們或非零CDF exact value。

本題逐步詳解（AI 二次覆核）

標準常態分布 · qf-mod45-normaldist-symbolic-density · candidate

5. 標準常態分布

standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-symbolic-density；資料={"x":[5,6]}。請依 [x, symbolic_density, density_law, normalization_scope] 欄序回答。

查看答案與解釋

(5/6, (2*pi)^(-1/2)*exp(-25/72), phi(x)=(2*pi)^(-1/2)*exp(-x^2/2), normalization_is_a_named_Gaussian_integral_fact_not_a_finite_evaluation)

Wave6第八層核對=(5/6, (2*pi)^(-1/2)*exp(-25/72), phi(x)=(2*pi)^(-1/2)*exp(-x^2/2), normalization_is_a_named_Gaussian_integral_fact_not_a_finite_evaluation)；依完整明示資料獨立重算；Phi(0)=1/2與Phi(-x)=1-Phi(x)只由已宣告normalization及symmetry推出；Gaussian integral、mean 0與variance 1是named theorem facts；finite quadrature不證它們或非零CDF exact value。

本題逐步詳解（AI 二次覆核）

標準常態分布 · qf-mod45-normaldist-symbolic-density · candidate

6. 標準常態分布

standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-symbolic-density；資料={"x":[-6,7]}。請依 [x, symbolic_density, density_law, normalization_scope] 欄序回答。

查看答案與解釋

(-6/7, (2*pi)^(-1/2)*exp(-18/49), phi(x)=(2*pi)^(-1/2)*exp(-x^2/2), normalization_is_a_named_Gaussian_integral_fact_not_a_finite_evaluation)

Wave6第八層核對=(-6/7, (2*pi)^(-1/2)*exp(-18/49), phi(x)=(2*pi)^(-1/2)*exp(-x^2/2), normalization_is_a_named_Gaussian_integral_fact_not_a_finite_evaluation)；依完整明示資料獨立重算；Phi(0)=1/2與Phi(-x)=1-Phi(x)只由已宣告normalization及symmetry推出；Gaussian integral、mean 0與variance 1是named theorem facts；finite quadrature不證它們或非零CDF exact value。

本題逐步詳解（AI 二次覆核）

標準常態分布 · qf-mod45-normaldist-symbolic-density · candidate

7. 標準常態分布

standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-symbolic-density；資料={"x":[7,8]}。請依 [x, symbolic_density, density_law, normalization_scope] 欄序回答。

查看答案與解釋

(7/8, (2*pi)^(-1/2)*exp(-49/128), phi(x)=(2*pi)^(-1/2)*exp(-x^2/2), normalization_is_a_named_Gaussian_integral_fact_not_a_finite_evaluation)

Wave6第八層核對=(7/8, (2*pi)^(-1/2)*exp(-49/128), phi(x)=(2*pi)^(-1/2)*exp(-x^2/2), normalization_is_a_named_Gaussian_integral_fact_not_a_finite_evaluation)；依完整明示資料獨立重算；Phi(0)=1/2與Phi(-x)=1-Phi(x)只由已宣告normalization及symmetry推出；Gaussian integral、mean 0與variance 1是named theorem facts；finite quadrature不證它們或非零CDF exact value。

本題逐步詳解（AI 二次覆核）

標準常態分布 · qf-mod45-normaldist-symbolic-density · candidate

8. 標準常態分布

standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-symbolic-density；資料={"x":[-8,9]}。請依 [x, symbolic_density, density_law, normalization_scope] 欄序回答。

查看答案與解釋

(-8/9, (2*pi)^(-1/2)*exp(-32/81), phi(x)=(2*pi)^(-1/2)*exp(-x^2/2), normalization_is_a_named_Gaussian_integral_fact_not_a_finite_evaluation)

Wave6第八層核對=(-8/9, (2*pi)^(-1/2)*exp(-32/81), phi(x)=(2*pi)^(-1/2)*exp(-x^2/2), normalization_is_a_named_Gaussian_integral_fact_not_a_finite_evaluation)；依完整明示資料獨立重算；Phi(0)=1/2與Phi(-x)=1-Phi(x)只由已宣告normalization及symmetry推出；Gaussian integral、mean 0與variance 1是named theorem facts；finite quadrature不證它們或非零CDF exact value。

本題逐步詳解（AI 二次覆核）

標準常態分布 · qf-mod45-normaldist-symbolic-density · candidate

9. 標準常態分布

standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-symbolic-density；資料={"x":[9,10]}。請依 [x, symbolic_density, density_law, normalization_scope] 欄序回答。

查看答案與解釋

(9/10, (2*pi)^(-1/2)*exp(-81/200), phi(x)=(2*pi)^(-1/2)*exp(-x^2/2), normalization_is_a_named_Gaussian_integral_fact_not_a_finite_evaluation)

Wave6第八層核對=(9/10, (2*pi)^(-1/2)*exp(-81/200), phi(x)=(2*pi)^(-1/2)*exp(-x^2/2), normalization_is_a_named_Gaussian_integral_fact_not_a_finite_evaluation)；依完整明示資料獨立重算；Phi(0)=1/2與Phi(-x)=1-Phi(x)只由已宣告normalization及symmetry推出；Gaussian integral、mean 0與variance 1是named theorem facts；finite quadrature不證它們或非零CDF exact value。

本題逐步詳解（AI 二次覆核）

標準常態分布 · qf-mod45-normaldist-symbolic-density · candidate

10. 標準常態分布

standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-symbolic-density；資料={"x":[-10,11]}。請依 [x, symbolic_density, density_law, normalization_scope] 欄序回答。

查看答案與解釋

(-10/11, (2*pi)^(-1/2)*exp(-50/121), phi(x)=(2*pi)^(-1/2)*exp(-x^2/2), normalization_is_a_named_Gaussian_integral_fact_not_a_finite_evaluation)

Wave6第八層核對=(-10/11, (2*pi)^(-1/2)*exp(-50/121), phi(x)=(2*pi)^(-1/2)*exp(-x^2/2), normalization_is_a_named_Gaussian_integral_fact_not_a_finite_evaluation)；依完整明示資料獨立重算；Phi(0)=1/2與Phi(-x)=1-Phi(x)只由已宣告normalization及symmetry推出；Gaussian integral、mean 0與variance 1是named theorem facts；finite quadrature不證它們或非零CDF exact value。

本題逐步詳解（AI 二次覆核）

標準常態分布 · qf-mod45-normaldist-symbolic-density · candidate

11. 標準常態分布

standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-symbolic-density；資料={"x":[11,12]}。請依 [x, symbolic_density, density_law, normalization_scope] 欄序回答。

查看答案與解釋

(11/12, (2*pi)^(-1/2)*exp(-121/288), phi(x)=(2*pi)^(-1/2)*exp(-x^2/2), normalization_is_a_named_Gaussian_integral_fact_not_a_finite_evaluation)

Wave6第八層核對=(11/12, (2*pi)^(-1/2)*exp(-121/288), phi(x)=(2*pi)^(-1/2)*exp(-x^2/2), normalization_is_a_named_Gaussian_integral_fact_not_a_finite_evaluation)；依完整明示資料獨立重算；Phi(0)=1/2與Phi(-x)=1-Phi(x)只由已宣告normalization及symmetry推出；Gaussian integral、mean 0與variance 1是named theorem facts；finite quadrature不證它們或非零CDF exact value。

本題逐步詳解（AI 二次覆核）

標準常態分布 · qf-mod45-normaldist-symbolic-density · candidate

12. 標準常態分布

standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-symbolic-density；資料={"x":[-12,13]}。請依 [x, symbolic_density, density_law, normalization_scope] 欄序回答。

查看答案與解釋

(-12/13, (2*pi)^(-1/2)*exp(-72/169), phi(x)=(2*pi)^(-1/2)*exp(-x^2/2), normalization_is_a_named_Gaussian_integral_fact_not_a_finite_evaluation)

Wave6第八層核對=(-12/13, (2*pi)^(-1/2)*exp(-72/169), phi(x)=(2*pi)^(-1/2)*exp(-x^2/2), normalization_is_a_named_Gaussian_integral_fact_not_a_finite_evaluation)；依完整明示資料獨立重算；Phi(0)=1/2與Phi(-x)=1-Phi(x)只由已宣告normalization及symmetry推出；Gaussian integral、mean 0與variance 1是named theorem facts；finite quadrature不證它們或非零CDF exact value。

本題逐步詳解（AI 二次覆核）

標準常態分布 · qf-mod45-normaldist-symbolic-density · candidate

13. 標準常態分布

standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-symbolic-density；資料={"x":[13,14]}。請依 [x, symbolic_density, density_law, normalization_scope] 欄序回答。

查看答案與解釋

(13/14, (2*pi)^(-1/2)*exp(-169/392), phi(x)=(2*pi)^(-1/2)*exp(-x^2/2), normalization_is_a_named_Gaussian_integral_fact_not_a_finite_evaluation)

Wave6第八層核對=(13/14, (2*pi)^(-1/2)*exp(-169/392), phi(x)=(2*pi)^(-1/2)*exp(-x^2/2), normalization_is_a_named_Gaussian_integral_fact_not_a_finite_evaluation)；依完整明示資料獨立重算；Phi(0)=1/2與Phi(-x)=1-Phi(x)只由已宣告normalization及symmetry推出；Gaussian integral、mean 0與variance 1是named theorem facts；finite quadrature不證它們或非零CDF exact value。

本題逐步詳解（AI 二次覆核）

標準常態分布 · qf-mod45-normaldist-symbolic-density · candidate

14. 標準常態分布

standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-symbolic-density；資料={"x":[-14,15]}。請依 [x, symbolic_density, density_law, normalization_scope] 欄序回答。

查看答案與解釋

(-14/15, (2*pi)^(-1/2)*exp(-98/225), phi(x)=(2*pi)^(-1/2)*exp(-x^2/2), normalization_is_a_named_Gaussian_integral_fact_not_a_finite_evaluation)

Wave6第八層核對=(-14/15, (2*pi)^(-1/2)*exp(-98/225), phi(x)=(2*pi)^(-1/2)*exp(-x^2/2), normalization_is_a_named_Gaussian_integral_fact_not_a_finite_evaluation)；依完整明示資料獨立重算；Phi(0)=1/2與Phi(-x)=1-Phi(x)只由已宣告normalization及symmetry推出；Gaussian integral、mean 0與variance 1是named theorem facts；finite quadrature不證它們或非零CDF exact value。

本題逐步詳解（AI 二次覆核）

標準常態分布 · qf-mod45-normaldist-symbolic-density · candidate

15. 標準常態分布

standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-symbolic-density；資料={"x":[15,16]}。請依 [x, symbolic_density, density_law, normalization_scope] 欄序回答。

查看答案與解釋

(15/16, (2*pi)^(-1/2)*exp(-225/512), phi(x)=(2*pi)^(-1/2)*exp(-x^2/2), normalization_is_a_named_Gaussian_integral_fact_not_a_finite_evaluation)

Wave6第八層核對=(15/16, (2*pi)^(-1/2)*exp(-225/512), phi(x)=(2*pi)^(-1/2)*exp(-x^2/2), normalization_is_a_named_Gaussian_integral_fact_not_a_finite_evaluation)；依完整明示資料獨立重算；Phi(0)=1/2與Phi(-x)=1-Phi(x)只由已宣告normalization及symmetry推出；Gaussian integral、mean 0與variance 1是named theorem facts；finite quadrature不證它們或非零CDF exact value。

本題逐步詳解（AI 二次覆核）

標準常態分布 · qf-mod45-normaldist-symbolic-density · candidate

16. 標準常態分布

standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-symbolic-density；資料={"x":[-16,17]}。請依 [x, symbolic_density, density_law, normalization_scope] 欄序回答。

查看答案與解釋

(-16/17, (2*pi)^(-1/2)*exp(-128/289), phi(x)=(2*pi)^(-1/2)*exp(-x^2/2), normalization_is_a_named_Gaussian_integral_fact_not_a_finite_evaluation)

Wave6第八層核對=(-16/17, (2*pi)^(-1/2)*exp(-128/289), phi(x)=(2*pi)^(-1/2)*exp(-x^2/2), normalization_is_a_named_Gaussian_integral_fact_not_a_finite_evaluation)；依完整明示資料獨立重算；Phi(0)=1/2與Phi(-x)=1-Phi(x)只由已宣告normalization及symmetry推出；Gaussian integral、mean 0與variance 1是named theorem facts；finite quadrature不證它們或非零CDF exact value。

本題逐步詳解（AI 二次覆核）

標準常態分布 · qf-mod45-normaldist-symbolic-density · candidate

17. 標準常態分布

standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-symbolic-density；資料={"x":[17,18]}。請依 [x, symbolic_density, density_law, normalization_scope] 欄序回答。

查看答案與解釋

(17/18, (2*pi)^(-1/2)*exp(-289/648), phi(x)=(2*pi)^(-1/2)*exp(-x^2/2), normalization_is_a_named_Gaussian_integral_fact_not_a_finite_evaluation)

Wave6第八層核對=(17/18, (2*pi)^(-1/2)*exp(-289/648), phi(x)=(2*pi)^(-1/2)*exp(-x^2/2), normalization_is_a_named_Gaussian_integral_fact_not_a_finite_evaluation)；依完整明示資料獨立重算；Phi(0)=1/2與Phi(-x)=1-Phi(x)只由已宣告normalization及symmetry推出；Gaussian integral、mean 0與variance 1是named theorem facts；finite quadrature不證它們或非零CDF exact value。

本題逐步詳解（AI 二次覆核）

標準常態分布 · qf-mod45-normaldist-symbolic-density · candidate

18. 標準常態分布

standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-symbolic-density；資料={"x":[-18,19]}。請依 [x, symbolic_density, density_law, normalization_scope] 欄序回答。

查看答案與解釋

(-18/19, (2*pi)^(-1/2)*exp(-162/361), phi(x)=(2*pi)^(-1/2)*exp(-x^2/2), normalization_is_a_named_Gaussian_integral_fact_not_a_finite_evaluation)

Wave6第八層核對=(-18/19, (2*pi)^(-1/2)*exp(-162/361), phi(x)=(2*pi)^(-1/2)*exp(-x^2/2), normalization_is_a_named_Gaussian_integral_fact_not_a_finite_evaluation)；依完整明示資料獨立重算；Phi(0)=1/2與Phi(-x)=1-Phi(x)只由已宣告normalization及symmetry推出；Gaussian integral、mean 0與variance 1是named theorem facts；finite quadrature不證它們或非零CDF exact value。

本題逐步詳解（AI 二次覆核）

標準常態分布 · qf-mod45-normaldist-symbolic-density · candidate

19. 標準常態分布

standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-symbolic-density；資料={"x":[19,20]}。請依 [x, symbolic_density, density_law, normalization_scope] 欄序回答。

查看答案與解釋

(19/20, (2*pi)^(-1/2)*exp(-361/800), phi(x)=(2*pi)^(-1/2)*exp(-x^2/2), normalization_is_a_named_Gaussian_integral_fact_not_a_finite_evaluation)

Wave6第八層核對=(19/20, (2*pi)^(-1/2)*exp(-361/800), phi(x)=(2*pi)^(-1/2)*exp(-x^2/2), normalization_is_a_named_Gaussian_integral_fact_not_a_finite_evaluation)；依完整明示資料獨立重算；Phi(0)=1/2與Phi(-x)=1-Phi(x)只由已宣告normalization及symmetry推出；Gaussian integral、mean 0與variance 1是named theorem facts；finite quadrature不證它們或非零CDF exact value。

本題逐步詳解（AI 二次覆核）

標準常態分布 · qf-mod45-normaldist-symbolic-density · candidate

20. 標準常態分布

standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-symbolic-density；資料={"x":[-20,21]}。請依 [x, symbolic_density, density_law, normalization_scope] 欄序回答。

查看答案與解釋

(-20/21, (2*pi)^(-1/2)*exp(-200/441), phi(x)=(2*pi)^(-1/2)*exp(-x^2/2), normalization_is_a_named_Gaussian_integral_fact_not_a_finite_evaluation)

Wave6第八層核對=(-20/21, (2*pi)^(-1/2)*exp(-200/441), phi(x)=(2*pi)^(-1/2)*exp(-x^2/2), normalization_is_a_named_Gaussian_integral_fact_not_a_finite_evaluation)；依完整明示資料獨立重算；Phi(0)=1/2與Phi(-x)=1-Phi(x)只由已宣告normalization及symmetry推出；Gaussian integral、mean 0與variance 1是named theorem facts；finite quadrature不證它們或非零CDF exact value。

本題逐步詳解（AI 二次覆核）

標準常態分布 · qf-mod45-normaldist-symbolic-density · candidate

21. 標準常態分布

standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-symbolic-density；資料={"x":[21,22]}。請依 [x, symbolic_density, density_law, normalization_scope] 欄序回答。

查看答案與解釋

(21/22, (2*pi)^(-1/2)*exp(-441/968), phi(x)=(2*pi)^(-1/2)*exp(-x^2/2), normalization_is_a_named_Gaussian_integral_fact_not_a_finite_evaluation)

Wave6第八層核對=(21/22, (2*pi)^(-1/2)*exp(-441/968), phi(x)=(2*pi)^(-1/2)*exp(-x^2/2), normalization_is_a_named_Gaussian_integral_fact_not_a_finite_evaluation)；依完整明示資料獨立重算；Phi(0)=1/2與Phi(-x)=1-Phi(x)只由已宣告normalization及symmetry推出；Gaussian integral、mean 0與variance 1是named theorem facts；finite quadrature不證它們或非零CDF exact value。

本題逐步詳解（AI 二次覆核）

標準常態分布 · qf-mod45-normaldist-symbolic-density · candidate

22. 標準常態分布

standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-symbolic-density；資料={"x":[-22,23]}。請依 [x, symbolic_density, density_law, normalization_scope] 欄序回答。

查看答案與解釋

(-22/23, (2*pi)^(-1/2)*exp(-242/529), phi(x)=(2*pi)^(-1/2)*exp(-x^2/2), normalization_is_a_named_Gaussian_integral_fact_not_a_finite_evaluation)

Wave6第八層核對=(-22/23, (2*pi)^(-1/2)*exp(-242/529), phi(x)=(2*pi)^(-1/2)*exp(-x^2/2), normalization_is_a_named_Gaussian_integral_fact_not_a_finite_evaluation)；依完整明示資料獨立重算；Phi(0)=1/2與Phi(-x)=1-Phi(x)只由已宣告normalization及symmetry推出；Gaussian integral、mean 0與variance 1是named theorem facts；finite quadrature不證它們或非零CDF exact value。

本題逐步詳解（AI 二次覆核）

標準常態分布 · qf-mod45-normaldist-symbolic-density · candidate

23. 標準常態分布

standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-symbolic-density；資料={"x":[23,24]}。請依 [x, symbolic_density, density_law, normalization_scope] 欄序回答。

查看答案與解釋

(23/24, (2*pi)^(-1/2)*exp(-529/1152), phi(x)=(2*pi)^(-1/2)*exp(-x^2/2), normalization_is_a_named_Gaussian_integral_fact_not_a_finite_evaluation)

Wave6第八層核對=(23/24, (2*pi)^(-1/2)*exp(-529/1152), phi(x)=(2*pi)^(-1/2)*exp(-x^2/2), normalization_is_a_named_Gaussian_integral_fact_not_a_finite_evaluation)；依完整明示資料獨立重算；Phi(0)=1/2與Phi(-x)=1-Phi(x)只由已宣告normalization及symmetry推出；Gaussian integral、mean 0與variance 1是named theorem facts；finite quadrature不證它們或非零CDF exact value。

本題逐步詳解（AI 二次覆核）

標準常態分布 · qf-mod45-normaldist-symbolic-density · candidate

24. 標準常態分布

standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-symbolic-density；資料={"x":[-24,25]}。請依 [x, symbolic_density, density_law, normalization_scope] 欄序回答。

查看答案與解釋

(-24/25, (2*pi)^(-1/2)*exp(-288/625), phi(x)=(2*pi)^(-1/2)*exp(-x^2/2), normalization_is_a_named_Gaussian_integral_fact_not_a_finite_evaluation)

Wave6第八層核對=(-24/25, (2*pi)^(-1/2)*exp(-288/625), phi(x)=(2*pi)^(-1/2)*exp(-x^2/2), normalization_is_a_named_Gaussian_integral_fact_not_a_finite_evaluation)；依完整明示資料獨立重算；Phi(0)=1/2與Phi(-x)=1-Phi(x)只由已宣告normalization及symmetry推出；Gaussian integral、mean 0與variance 1是named theorem facts；finite quadrature不證它們或非零CDF exact value。

本題逐步詳解（AI 二次覆核）

標準常態分布 · qf-mod45-normaldist-symbolic-density · candidate

25. 標準常態分布

standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-symbolic-density；資料={"x":[25,26]}。請依 [x, symbolic_density, density_law, normalization_scope] 欄序回答。

查看答案與解釋

(25/26, (2*pi)^(-1/2)*exp(-625/1352), phi(x)=(2*pi)^(-1/2)*exp(-x^2/2), normalization_is_a_named_Gaussian_integral_fact_not_a_finite_evaluation)

Wave6第八層核對=(25/26, (2*pi)^(-1/2)*exp(-625/1352), phi(x)=(2*pi)^(-1/2)*exp(-x^2/2), normalization_is_a_named_Gaussian_integral_fact_not_a_finite_evaluation)；依完整明示資料獨立重算；Phi(0)=1/2與Phi(-x)=1-Phi(x)只由已宣告normalization及symmetry推出；Gaussian integral、mean 0與variance 1是named theorem facts；finite quadrature不證它們或非零CDF exact value。

本題逐步詳解（AI 二次覆核）

標準常態分布 · qf-mod45-normaldist-symbolic-density · candidate

26. 標準常態分布

standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-evenness-ratio；資料={"x":[1,2]}。請依 [x, negative_x, density_at_x, density_at_negative_x, even, density_ratio, certificate] 欄序回答。

查看答案與解釋

(1/2, -1/2, (2*pi)^(-1/2)*exp(-1/8), (2*pi)^(-1/2)*exp(-1/8), 是, phi(x)/phi(0)=exp(-1/8), evenness_follows_from_x_squared)

Wave6第八層核對=(1/2, -1/2, (2*pi)^(-1/2)*exp(-1/8), (2*pi)^(-1/2)*exp(-1/8), 是, phi(x)/phi(0)=exp(-1/8), evenness_follows_from_x_squared)；依完整明示資料獨立重算；Phi(0)=1/2與Phi(-x)=1-Phi(x)只由已宣告normalization及symmetry推出；Gaussian integral、mean 0與variance 1是named theorem facts；finite quadrature不證它們或非零CDF exact value。

本題逐步詳解（AI 二次覆核）

標準常態分布 · qf-mod45-normaldist-evenness-ratio · candidate

27. 標準常態分布

standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-evenness-ratio；資料={"x":[-2,3]}。請依 [x, negative_x, density_at_x, density_at_negative_x, even, density_ratio, certificate] 欄序回答。

查看答案與解釋

(-2/3, 2/3, (2*pi)^(-1/2)*exp(-2/9), (2*pi)^(-1/2)*exp(-2/9), 是, phi(x)/phi(0)=exp(-2/9), evenness_follows_from_x_squared)

Wave6第八層核對=(-2/3, 2/3, (2*pi)^(-1/2)*exp(-2/9), (2*pi)^(-1/2)*exp(-2/9), 是, phi(x)/phi(0)=exp(-2/9), evenness_follows_from_x_squared)；依完整明示資料獨立重算；Phi(0)=1/2與Phi(-x)=1-Phi(x)只由已宣告normalization及symmetry推出；Gaussian integral、mean 0與variance 1是named theorem facts；finite quadrature不證它們或非零CDF exact value。

本題逐步詳解（AI 二次覆核）

標準常態分布 · qf-mod45-normaldist-evenness-ratio · candidate

28. 標準常態分布

standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-evenness-ratio；資料={"x":[3,4]}。請依 [x, negative_x, density_at_x, density_at_negative_x, even, density_ratio, certificate] 欄序回答。

查看答案與解釋

(3/4, -3/4, (2*pi)^(-1/2)*exp(-9/32), (2*pi)^(-1/2)*exp(-9/32), 是, phi(x)/phi(0)=exp(-9/32), evenness_follows_from_x_squared)

Wave6第八層核對=(3/4, -3/4, (2*pi)^(-1/2)*exp(-9/32), (2*pi)^(-1/2)*exp(-9/32), 是, phi(x)/phi(0)=exp(-9/32), evenness_follows_from_x_squared)；依完整明示資料獨立重算；Phi(0)=1/2與Phi(-x)=1-Phi(x)只由已宣告normalization及symmetry推出；Gaussian integral、mean 0與variance 1是named theorem facts；finite quadrature不證它們或非零CDF exact value。

本題逐步詳解（AI 二次覆核）

標準常態分布 · qf-mod45-normaldist-evenness-ratio · candidate

29. 標準常態分布

standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-evenness-ratio；資料={"x":[-4,5]}。請依 [x, negative_x, density_at_x, density_at_negative_x, even, density_ratio, certificate] 欄序回答。

查看答案與解釋

(-4/5, 4/5, (2*pi)^(-1/2)*exp(-8/25), (2*pi)^(-1/2)*exp(-8/25), 是, phi(x)/phi(0)=exp(-8/25), evenness_follows_from_x_squared)

Wave6第八層核對=(-4/5, 4/5, (2*pi)^(-1/2)*exp(-8/25), (2*pi)^(-1/2)*exp(-8/25), 是, phi(x)/phi(0)=exp(-8/25), evenness_follows_from_x_squared)；依完整明示資料獨立重算；Phi(0)=1/2與Phi(-x)=1-Phi(x)只由已宣告normalization及symmetry推出；Gaussian integral、mean 0與variance 1是named theorem facts；finite quadrature不證它們或非零CDF exact value。

本題逐步詳解（AI 二次覆核）

標準常態分布 · qf-mod45-normaldist-evenness-ratio · candidate

30. 標準常態分布

standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-evenness-ratio；資料={"x":[5,6]}。請依 [x, negative_x, density_at_x, density_at_negative_x, even, density_ratio, certificate] 欄序回答。

查看答案與解釋

(5/6, -5/6, (2*pi)^(-1/2)*exp(-25/72), (2*pi)^(-1/2)*exp(-25/72), 是, phi(x)/phi(0)=exp(-25/72), evenness_follows_from_x_squared)

Wave6第八層核對=(5/6, -5/6, (2*pi)^(-1/2)*exp(-25/72), (2*pi)^(-1/2)*exp(-25/72), 是, phi(x)/phi(0)=exp(-25/72), evenness_follows_from_x_squared)；依完整明示資料獨立重算；Phi(0)=1/2與Phi(-x)=1-Phi(x)只由已宣告normalization及symmetry推出；Gaussian integral、mean 0與variance 1是named theorem facts；finite quadrature不證它們或非零CDF exact value。

本題逐步詳解（AI 二次覆核）

標準常態分布 · qf-mod45-normaldist-evenness-ratio · candidate

31. 標準常態分布

standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-evenness-ratio；資料={"x":[-6,7]}。請依 [x, negative_x, density_at_x, density_at_negative_x, even, density_ratio, certificate] 欄序回答。

查看答案與解釋

(-6/7, 6/7, (2*pi)^(-1/2)*exp(-18/49), (2*pi)^(-1/2)*exp(-18/49), 是, phi(x)/phi(0)=exp(-18/49), evenness_follows_from_x_squared)

Wave6第八層核對=(-6/7, 6/7, (2*pi)^(-1/2)*exp(-18/49), (2*pi)^(-1/2)*exp(-18/49), 是, phi(x)/phi(0)=exp(-18/49), evenness_follows_from_x_squared)；依完整明示資料獨立重算；Phi(0)=1/2與Phi(-x)=1-Phi(x)只由已宣告normalization及symmetry推出；Gaussian integral、mean 0與variance 1是named theorem facts；finite quadrature不證它們或非零CDF exact value。

本題逐步詳解（AI 二次覆核）

標準常態分布 · qf-mod45-normaldist-evenness-ratio · candidate

32. 標準常態分布

standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-evenness-ratio；資料={"x":[7,8]}。請依 [x, negative_x, density_at_x, density_at_negative_x, even, density_ratio, certificate] 欄序回答。

查看答案與解釋

(7/8, -7/8, (2*pi)^(-1/2)*exp(-49/128), (2*pi)^(-1/2)*exp(-49/128), 是, phi(x)/phi(0)=exp(-49/128), evenness_follows_from_x_squared)

Wave6第八層核對=(7/8, -7/8, (2*pi)^(-1/2)*exp(-49/128), (2*pi)^(-1/2)*exp(-49/128), 是, phi(x)/phi(0)=exp(-49/128), evenness_follows_from_x_squared)；依完整明示資料獨立重算；Phi(0)=1/2與Phi(-x)=1-Phi(x)只由已宣告normalization及symmetry推出；Gaussian integral、mean 0與variance 1是named theorem facts；finite quadrature不證它們或非零CDF exact value。

本題逐步詳解（AI 二次覆核）

標準常態分布 · qf-mod45-normaldist-evenness-ratio · candidate

33. 標準常態分布

standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-evenness-ratio；資料={"x":[-8,9]}。請依 [x, negative_x, density_at_x, density_at_negative_x, even, density_ratio, certificate] 欄序回答。

查看答案與解釋

(-8/9, 8/9, (2*pi)^(-1/2)*exp(-32/81), (2*pi)^(-1/2)*exp(-32/81), 是, phi(x)/phi(0)=exp(-32/81), evenness_follows_from_x_squared)

Wave6第八層核對=(-8/9, 8/9, (2*pi)^(-1/2)*exp(-32/81), (2*pi)^(-1/2)*exp(-32/81), 是, phi(x)/phi(0)=exp(-32/81), evenness_follows_from_x_squared)；依完整明示資料獨立重算；Phi(0)=1/2與Phi(-x)=1-Phi(x)只由已宣告normalization及symmetry推出；Gaussian integral、mean 0與variance 1是named theorem facts；finite quadrature不證它們或非零CDF exact value。

本題逐步詳解（AI 二次覆核）

標準常態分布 · qf-mod45-normaldist-evenness-ratio · candidate

34. 標準常態分布

standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-evenness-ratio；資料={"x":[9,10]}。請依 [x, negative_x, density_at_x, density_at_negative_x, even, density_ratio, certificate] 欄序回答。

查看答案與解釋

(9/10, -9/10, (2*pi)^(-1/2)*exp(-81/200), (2*pi)^(-1/2)*exp(-81/200), 是, phi(x)/phi(0)=exp(-81/200), evenness_follows_from_x_squared)

Wave6第八層核對=(9/10, -9/10, (2*pi)^(-1/2)*exp(-81/200), (2*pi)^(-1/2)*exp(-81/200), 是, phi(x)/phi(0)=exp(-81/200), evenness_follows_from_x_squared)；依完整明示資料獨立重算；Phi(0)=1/2與Phi(-x)=1-Phi(x)只由已宣告normalization及symmetry推出；Gaussian integral、mean 0與variance 1是named theorem facts；finite quadrature不證它們或非零CDF exact value。

本題逐步詳解（AI 二次覆核）

標準常態分布 · qf-mod45-normaldist-evenness-ratio · candidate

35. 標準常態分布

standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-evenness-ratio；資料={"x":[-10,11]}。請依 [x, negative_x, density_at_x, density_at_negative_x, even, density_ratio, certificate] 欄序回答。

查看答案與解釋

(-10/11, 10/11, (2*pi)^(-1/2)*exp(-50/121), (2*pi)^(-1/2)*exp(-50/121), 是, phi(x)/phi(0)=exp(-50/121), evenness_follows_from_x_squared)

Wave6第八層核對=(-10/11, 10/11, (2*pi)^(-1/2)*exp(-50/121), (2*pi)^(-1/2)*exp(-50/121), 是, phi(x)/phi(0)=exp(-50/121), evenness_follows_from_x_squared)；依完整明示資料獨立重算；Phi(0)=1/2與Phi(-x)=1-Phi(x)只由已宣告normalization及symmetry推出；Gaussian integral、mean 0與variance 1是named theorem facts；finite quadrature不證它們或非零CDF exact value。

本題逐步詳解（AI 二次覆核）

標準常態分布 · qf-mod45-normaldist-evenness-ratio · candidate

36. 標準常態分布

standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-evenness-ratio；資料={"x":[11,12]}。請依 [x, negative_x, density_at_x, density_at_negative_x, even, density_ratio, certificate] 欄序回答。

查看答案與解釋

(11/12, -11/12, (2*pi)^(-1/2)*exp(-121/288), (2*pi)^(-1/2)*exp(-121/288), 是, phi(x)/phi(0)=exp(-121/288), evenness_follows_from_x_squared)

Wave6第八層核對=(11/12, -11/12, (2*pi)^(-1/2)*exp(-121/288), (2*pi)^(-1/2)*exp(-121/288), 是, phi(x)/phi(0)=exp(-121/288), evenness_follows_from_x_squared)；依完整明示資料獨立重算；Phi(0)=1/2與Phi(-x)=1-Phi(x)只由已宣告normalization及symmetry推出；Gaussian integral、mean 0與variance 1是named theorem facts；finite quadrature不證它們或非零CDF exact value。

本題逐步詳解（AI 二次覆核）

標準常態分布 · qf-mod45-normaldist-evenness-ratio · candidate

37. 標準常態分布

standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-evenness-ratio；資料={"x":[-12,13]}。請依 [x, negative_x, density_at_x, density_at_negative_x, even, density_ratio, certificate] 欄序回答。

查看答案與解釋

(-12/13, 12/13, (2*pi)^(-1/2)*exp(-72/169), (2*pi)^(-1/2)*exp(-72/169), 是, phi(x)/phi(0)=exp(-72/169), evenness_follows_from_x_squared)

Wave6第八層核對=(-12/13, 12/13, (2*pi)^(-1/2)*exp(-72/169), (2*pi)^(-1/2)*exp(-72/169), 是, phi(x)/phi(0)=exp(-72/169), evenness_follows_from_x_squared)；依完整明示資料獨立重算；Phi(0)=1/2與Phi(-x)=1-Phi(x)只由已宣告normalization及symmetry推出；Gaussian integral、mean 0與variance 1是named theorem facts；finite quadrature不證它們或非零CDF exact value。

本題逐步詳解（AI 二次覆核）

標準常態分布 · qf-mod45-normaldist-evenness-ratio · candidate

38. 標準常態分布

standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-evenness-ratio；資料={"x":[13,14]}。請依 [x, negative_x, density_at_x, density_at_negative_x, even, density_ratio, certificate] 欄序回答。

查看答案與解釋

(13/14, -13/14, (2*pi)^(-1/2)*exp(-169/392), (2*pi)^(-1/2)*exp(-169/392), 是, phi(x)/phi(0)=exp(-169/392), evenness_follows_from_x_squared)

Wave6第八層核對=(13/14, -13/14, (2*pi)^(-1/2)*exp(-169/392), (2*pi)^(-1/2)*exp(-169/392), 是, phi(x)/phi(0)=exp(-169/392), evenness_follows_from_x_squared)；依完整明示資料獨立重算；Phi(0)=1/2與Phi(-x)=1-Phi(x)只由已宣告normalization及symmetry推出；Gaussian integral、mean 0與variance 1是named theorem facts；finite quadrature不證它們或非零CDF exact value。

本題逐步詳解（AI 二次覆核）

標準常態分布 · qf-mod45-normaldist-evenness-ratio · candidate

39. 標準常態分布

standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-evenness-ratio；資料={"x":[-14,15]}。請依 [x, negative_x, density_at_x, density_at_negative_x, even, density_ratio, certificate] 欄序回答。

查看答案與解釋

(-14/15, 14/15, (2*pi)^(-1/2)*exp(-98/225), (2*pi)^(-1/2)*exp(-98/225), 是, phi(x)/phi(0)=exp(-98/225), evenness_follows_from_x_squared)

Wave6第八層核對=(-14/15, 14/15, (2*pi)^(-1/2)*exp(-98/225), (2*pi)^(-1/2)*exp(-98/225), 是, phi(x)/phi(0)=exp(-98/225), evenness_follows_from_x_squared)；依完整明示資料獨立重算；Phi(0)=1/2與Phi(-x)=1-Phi(x)只由已宣告normalization及symmetry推出；Gaussian integral、mean 0與variance 1是named theorem facts；finite quadrature不證它們或非零CDF exact value。

本題逐步詳解（AI 二次覆核）

標準常態分布 · qf-mod45-normaldist-evenness-ratio · candidate

40. 標準常態分布

standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-evenness-ratio；資料={"x":[15,16]}。請依 [x, negative_x, density_at_x, density_at_negative_x, even, density_ratio, certificate] 欄序回答。

查看答案與解釋

(15/16, -15/16, (2*pi)^(-1/2)*exp(-225/512), (2*pi)^(-1/2)*exp(-225/512), 是, phi(x)/phi(0)=exp(-225/512), evenness_follows_from_x_squared)

Wave6第八層核對=(15/16, -15/16, (2*pi)^(-1/2)*exp(-225/512), (2*pi)^(-1/2)*exp(-225/512), 是, phi(x)/phi(0)=exp(-225/512), evenness_follows_from_x_squared)；依完整明示資料獨立重算；Phi(0)=1/2與Phi(-x)=1-Phi(x)只由已宣告normalization及symmetry推出；Gaussian integral、mean 0與variance 1是named theorem facts；finite quadrature不證它們或非零CDF exact value。

本題逐步詳解（AI 二次覆核）

標準常態分布 · qf-mod45-normaldist-evenness-ratio · candidate

41. 標準常態分布

standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-evenness-ratio；資料={"x":[-16,17]}。請依 [x, negative_x, density_at_x, density_at_negative_x, even, density_ratio, certificate] 欄序回答。

查看答案與解釋

(-16/17, 16/17, (2*pi)^(-1/2)*exp(-128/289), (2*pi)^(-1/2)*exp(-128/289), 是, phi(x)/phi(0)=exp(-128/289), evenness_follows_from_x_squared)

Wave6第八層核對=(-16/17, 16/17, (2*pi)^(-1/2)*exp(-128/289), (2*pi)^(-1/2)*exp(-128/289), 是, phi(x)/phi(0)=exp(-128/289), evenness_follows_from_x_squared)；依完整明示資料獨立重算；Phi(0)=1/2與Phi(-x)=1-Phi(x)只由已宣告normalization及symmetry推出；Gaussian integral、mean 0與variance 1是named theorem facts；finite quadrature不證它們或非零CDF exact value。

本題逐步詳解（AI 二次覆核）

標準常態分布 · qf-mod45-normaldist-evenness-ratio · candidate

42. 標準常態分布

standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-evenness-ratio；資料={"x":[17,18]}。請依 [x, negative_x, density_at_x, density_at_negative_x, even, density_ratio, certificate] 欄序回答。

查看答案與解釋

(17/18, -17/18, (2*pi)^(-1/2)*exp(-289/648), (2*pi)^(-1/2)*exp(-289/648), 是, phi(x)/phi(0)=exp(-289/648), evenness_follows_from_x_squared)

Wave6第八層核對=(17/18, -17/18, (2*pi)^(-1/2)*exp(-289/648), (2*pi)^(-1/2)*exp(-289/648), 是, phi(x)/phi(0)=exp(-289/648), evenness_follows_from_x_squared)；依完整明示資料獨立重算；Phi(0)=1/2與Phi(-x)=1-Phi(x)只由已宣告normalization及symmetry推出；Gaussian integral、mean 0與variance 1是named theorem facts；finite quadrature不證它們或非零CDF exact value。

本題逐步詳解（AI 二次覆核）

標準常態分布 · qf-mod45-normaldist-evenness-ratio · candidate

43. 標準常態分布

standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-evenness-ratio；資料={"x":[-18,19]}。請依 [x, negative_x, density_at_x, density_at_negative_x, even, density_ratio, certificate] 欄序回答。

查看答案與解釋

(-18/19, 18/19, (2*pi)^(-1/2)*exp(-162/361), (2*pi)^(-1/2)*exp(-162/361), 是, phi(x)/phi(0)=exp(-162/361), evenness_follows_from_x_squared)

Wave6第八層核對=(-18/19, 18/19, (2*pi)^(-1/2)*exp(-162/361), (2*pi)^(-1/2)*exp(-162/361), 是, phi(x)/phi(0)=exp(-162/361), evenness_follows_from_x_squared)；依完整明示資料獨立重算；Phi(0)=1/2與Phi(-x)=1-Phi(x)只由已宣告normalization及symmetry推出；Gaussian integral、mean 0與variance 1是named theorem facts；finite quadrature不證它們或非零CDF exact value。

本題逐步詳解（AI 二次覆核）

標準常態分布 · qf-mod45-normaldist-evenness-ratio · candidate

44. 標準常態分布

standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-evenness-ratio；資料={"x":[19,20]}。請依 [x, negative_x, density_at_x, density_at_negative_x, even, density_ratio, certificate] 欄序回答。

查看答案與解釋

(19/20, -19/20, (2*pi)^(-1/2)*exp(-361/800), (2*pi)^(-1/2)*exp(-361/800), 是, phi(x)/phi(0)=exp(-361/800), evenness_follows_from_x_squared)

Wave6第八層核對=(19/20, -19/20, (2*pi)^(-1/2)*exp(-361/800), (2*pi)^(-1/2)*exp(-361/800), 是, phi(x)/phi(0)=exp(-361/800), evenness_follows_from_x_squared)；依完整明示資料獨立重算；Phi(0)=1/2與Phi(-x)=1-Phi(x)只由已宣告normalization及symmetry推出；Gaussian integral、mean 0與variance 1是named theorem facts；finite quadrature不證它們或非零CDF exact value。

本題逐步詳解（AI 二次覆核）

標準常態分布 · qf-mod45-normaldist-evenness-ratio · candidate

45. 標準常態分布

standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-evenness-ratio；資料={"x":[-20,21]}。請依 [x, negative_x, density_at_x, density_at_negative_x, even, density_ratio, certificate] 欄序回答。

查看答案與解釋

(-20/21, 20/21, (2*pi)^(-1/2)*exp(-200/441), (2*pi)^(-1/2)*exp(-200/441), 是, phi(x)/phi(0)=exp(-200/441), evenness_follows_from_x_squared)

Wave6第八層核對=(-20/21, 20/21, (2*pi)^(-1/2)*exp(-200/441), (2*pi)^(-1/2)*exp(-200/441), 是, phi(x)/phi(0)=exp(-200/441), evenness_follows_from_x_squared)；依完整明示資料獨立重算；Phi(0)=1/2與Phi(-x)=1-Phi(x)只由已宣告normalization及symmetry推出；Gaussian integral、mean 0與variance 1是named theorem facts；finite quadrature不證它們或非零CDF exact value。

本題逐步詳解（AI 二次覆核）

標準常態分布 · qf-mod45-normaldist-evenness-ratio · candidate

46. 標準常態分布

standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-evenness-ratio；資料={"x":[21,22]}。請依 [x, negative_x, density_at_x, density_at_negative_x, even, density_ratio, certificate] 欄序回答。

查看答案與解釋

(21/22, -21/22, (2*pi)^(-1/2)*exp(-441/968), (2*pi)^(-1/2)*exp(-441/968), 是, phi(x)/phi(0)=exp(-441/968), evenness_follows_from_x_squared)

Wave6第八層核對=(21/22, -21/22, (2*pi)^(-1/2)*exp(-441/968), (2*pi)^(-1/2)*exp(-441/968), 是, phi(x)/phi(0)=exp(-441/968), evenness_follows_from_x_squared)；依完整明示資料獨立重算；Phi(0)=1/2與Phi(-x)=1-Phi(x)只由已宣告normalization及symmetry推出；Gaussian integral、mean 0與variance 1是named theorem facts；finite quadrature不證它們或非零CDF exact value。

本題逐步詳解（AI 二次覆核）

標準常態分布 · qf-mod45-normaldist-evenness-ratio · candidate

47. 標準常態分布

standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-evenness-ratio；資料={"x":[-22,23]}。請依 [x, negative_x, density_at_x, density_at_negative_x, even, density_ratio, certificate] 欄序回答。

查看答案與解釋

(-22/23, 22/23, (2*pi)^(-1/2)*exp(-242/529), (2*pi)^(-1/2)*exp(-242/529), 是, phi(x)/phi(0)=exp(-242/529), evenness_follows_from_x_squared)

Wave6第八層核對=(-22/23, 22/23, (2*pi)^(-1/2)*exp(-242/529), (2*pi)^(-1/2)*exp(-242/529), 是, phi(x)/phi(0)=exp(-242/529), evenness_follows_from_x_squared)；依完整明示資料獨立重算；Phi(0)=1/2與Phi(-x)=1-Phi(x)只由已宣告normalization及symmetry推出；Gaussian integral、mean 0與variance 1是named theorem facts；finite quadrature不證它們或非零CDF exact value。

本題逐步詳解（AI 二次覆核）

標準常態分布 · qf-mod45-normaldist-evenness-ratio · candidate

48. 標準常態分布

standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-evenness-ratio；資料={"x":[23,24]}。請依 [x, negative_x, density_at_x, density_at_negative_x, even, density_ratio, certificate] 欄序回答。

查看答案與解釋

(23/24, -23/24, (2*pi)^(-1/2)*exp(-529/1152), (2*pi)^(-1/2)*exp(-529/1152), 是, phi(x)/phi(0)=exp(-529/1152), evenness_follows_from_x_squared)

Wave6第八層核對=(23/24, -23/24, (2*pi)^(-1/2)*exp(-529/1152), (2*pi)^(-1/2)*exp(-529/1152), 是, phi(x)/phi(0)=exp(-529/1152), evenness_follows_from_x_squared)；依完整明示資料獨立重算；Phi(0)=1/2與Phi(-x)=1-Phi(x)只由已宣告normalization及symmetry推出；Gaussian integral、mean 0與variance 1是named theorem facts；finite quadrature不證它們或非零CDF exact value。

本題逐步詳解（AI 二次覆核）

標準常態分布 · qf-mod45-normaldist-evenness-ratio · candidate

49. 標準常態分布

standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-evenness-ratio；資料={"x":[-24,25]}。請依 [x, negative_x, density_at_x, density_at_negative_x, even, density_ratio, certificate] 欄序回答。

查看答案與解釋

(-24/25, 24/25, (2*pi)^(-1/2)*exp(-288/625), (2*pi)^(-1/2)*exp(-288/625), 是, phi(x)/phi(0)=exp(-288/625), evenness_follows_from_x_squared)

Wave6第八層核對=(-24/25, 24/25, (2*pi)^(-1/2)*exp(-288/625), (2*pi)^(-1/2)*exp(-288/625), 是, phi(x)/phi(0)=exp(-288/625), evenness_follows_from_x_squared)；依完整明示資料獨立重算；Phi(0)=1/2與Phi(-x)=1-Phi(x)只由已宣告normalization及symmetry推出；Gaussian integral、mean 0與variance 1是named theorem facts；finite quadrature不證它們或非零CDF exact value。

本題逐步詳解（AI 二次覆核）

標準常態分布 · qf-mod45-normaldist-evenness-ratio · candidate

50. 標準常態分布

standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-evenness-ratio；資料={"x":[25,26]}。請依 [x, negative_x, density_at_x, density_at_negative_x, even, density_ratio, certificate] 欄序回答。

查看答案與解釋

(25/26, -25/26, (2*pi)^(-1/2)*exp(-625/1352), (2*pi)^(-1/2)*exp(-625/1352), 是, phi(x)/phi(0)=exp(-625/1352), evenness_follows_from_x_squared)

Wave6第八層核對=(25/26, -25/26, (2*pi)^(-1/2)*exp(-625/1352), (2*pi)^(-1/2)*exp(-625/1352), 是, phi(x)/phi(0)=exp(-625/1352), evenness_follows_from_x_squared)；依完整明示資料獨立重算；Phi(0)=1/2與Phi(-x)=1-Phi(x)只由已宣告normalization及symmetry推出；Gaussian integral、mean 0與variance 1是named theorem facts；finite quadrature不證它們或非零CDF exact value。

本題逐步詳解（AI 二次覆核）

標準常態分布 · qf-mod45-normaldist-evenness-ratio · candidate

51. 標準常態分布

standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-cdf-symmetry；資料={"x":-12}。請依 [x, CDF_symmetry_identity, CDF_at_zero, declared_symmetry_certificate, boundary] 欄序回答。

查看答案與解釋

(-12, Phi(12)=1-Phi(-12), Phi(0)=1/2, 是, declared_normalization_plus_even_density_implies_CDF_symmetry_but_no_nonzero_elementary_CDF_value)

Wave6第八層核對=(-12, Phi(12)=1-Phi(-12), Phi(0)=1/2, 是, declared_normalization_plus_even_density_implies_CDF_symmetry_but_no_nonzero_elementary_CDF_value)；依完整明示資料獨立重算；Phi(0)=1/2與Phi(-x)=1-Phi(x)只由已宣告normalization及symmetry推出；Gaussian integral、mean 0與variance 1是named theorem facts；finite quadrature不證它們或非零CDF exact value。

本題逐步詳解（AI 二次覆核）

標準常態分布 · qf-mod45-normaldist-cdf-symmetry · candidate

52. 標準常態分布

standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-cdf-symmetry；資料={"x":-11}。請依 [x, CDF_symmetry_identity, CDF_at_zero, declared_symmetry_certificate, boundary] 欄序回答。

查看答案與解釋

(-11, Phi(11)=1-Phi(-11), Phi(0)=1/2, 是, declared_normalization_plus_even_density_implies_CDF_symmetry_but_no_nonzero_elementary_CDF_value)

Wave6第八層核對=(-11, Phi(11)=1-Phi(-11), Phi(0)=1/2, 是, declared_normalization_plus_even_density_implies_CDF_symmetry_but_no_nonzero_elementary_CDF_value)；依完整明示資料獨立重算；Phi(0)=1/2與Phi(-x)=1-Phi(x)只由已宣告normalization及symmetry推出；Gaussian integral、mean 0與variance 1是named theorem facts；finite quadrature不證它們或非零CDF exact value。

本題逐步詳解（AI 二次覆核）

標準常態分布 · qf-mod45-normaldist-cdf-symmetry · candidate

53. 標準常態分布

standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-cdf-symmetry；資料={"x":-10}。請依 [x, CDF_symmetry_identity, CDF_at_zero, declared_symmetry_certificate, boundary] 欄序回答。

查看答案與解釋

(-10, Phi(10)=1-Phi(-10), Phi(0)=1/2, 是, declared_normalization_plus_even_density_implies_CDF_symmetry_but_no_nonzero_elementary_CDF_value)

Wave6第八層核對=(-10, Phi(10)=1-Phi(-10), Phi(0)=1/2, 是, declared_normalization_plus_even_density_implies_CDF_symmetry_but_no_nonzero_elementary_CDF_value)；依完整明示資料獨立重算；Phi(0)=1/2與Phi(-x)=1-Phi(x)只由已宣告normalization及symmetry推出；Gaussian integral、mean 0與variance 1是named theorem facts；finite quadrature不證它們或非零CDF exact value。

本題逐步詳解（AI 二次覆核）

標準常態分布 · qf-mod45-normaldist-cdf-symmetry · candidate

54. 標準常態分布

standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-cdf-symmetry；資料={"x":-9}。請依 [x, CDF_symmetry_identity, CDF_at_zero, declared_symmetry_certificate, boundary] 欄序回答。

查看答案與解釋

(-9, Phi(9)=1-Phi(-9), Phi(0)=1/2, 是, declared_normalization_plus_even_density_implies_CDF_symmetry_but_no_nonzero_elementary_CDF_value)

Wave6第八層核對=(-9, Phi(9)=1-Phi(-9), Phi(0)=1/2, 是, declared_normalization_plus_even_density_implies_CDF_symmetry_but_no_nonzero_elementary_CDF_value)；依完整明示資料獨立重算；Phi(0)=1/2與Phi(-x)=1-Phi(x)只由已宣告normalization及symmetry推出；Gaussian integral、mean 0與variance 1是named theorem facts；finite quadrature不證它們或非零CDF exact value。

本題逐步詳解（AI 二次覆核）

標準常態分布 · qf-mod45-normaldist-cdf-symmetry · candidate

55. 標準常態分布

standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-cdf-symmetry；資料={"x":-8}。請依 [x, CDF_symmetry_identity, CDF_at_zero, declared_symmetry_certificate, boundary] 欄序回答。

查看答案與解釋

(-8, Phi(8)=1-Phi(-8), Phi(0)=1/2, 是, declared_normalization_plus_even_density_implies_CDF_symmetry_but_no_nonzero_elementary_CDF_value)

Wave6第八層核對=(-8, Phi(8)=1-Phi(-8), Phi(0)=1/2, 是, declared_normalization_plus_even_density_implies_CDF_symmetry_but_no_nonzero_elementary_CDF_value)；依完整明示資料獨立重算；Phi(0)=1/2與Phi(-x)=1-Phi(x)只由已宣告normalization及symmetry推出；Gaussian integral、mean 0與variance 1是named theorem facts；finite quadrature不證它們或非零CDF exact value。

本題逐步詳解（AI 二次覆核）

標準常態分布 · qf-mod45-normaldist-cdf-symmetry · candidate

56. 標準常態分布

standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-cdf-symmetry；資料={"x":-7}。請依 [x, CDF_symmetry_identity, CDF_at_zero, declared_symmetry_certificate, boundary] 欄序回答。

查看答案與解釋

(-7, Phi(7)=1-Phi(-7), Phi(0)=1/2, 是, declared_normalization_plus_even_density_implies_CDF_symmetry_but_no_nonzero_elementary_CDF_value)

Wave6第八層核對=(-7, Phi(7)=1-Phi(-7), Phi(0)=1/2, 是, declared_normalization_plus_even_density_implies_CDF_symmetry_but_no_nonzero_elementary_CDF_value)；依完整明示資料獨立重算；Phi(0)=1/2與Phi(-x)=1-Phi(x)只由已宣告normalization及symmetry推出；Gaussian integral、mean 0與variance 1是named theorem facts；finite quadrature不證它們或非零CDF exact value。

本題逐步詳解（AI 二次覆核）

標準常態分布 · qf-mod45-normaldist-cdf-symmetry · candidate

57. 標準常態分布

standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-cdf-symmetry；資料={"x":-6}。請依 [x, CDF_symmetry_identity, CDF_at_zero, declared_symmetry_certificate, boundary] 欄序回答。

查看答案與解釋

(-6, Phi(6)=1-Phi(-6), Phi(0)=1/2, 是, declared_normalization_plus_even_density_implies_CDF_symmetry_but_no_nonzero_elementary_CDF_value)

Wave6第八層核對=(-6, Phi(6)=1-Phi(-6), Phi(0)=1/2, 是, declared_normalization_plus_even_density_implies_CDF_symmetry_but_no_nonzero_elementary_CDF_value)；依完整明示資料獨立重算；Phi(0)=1/2與Phi(-x)=1-Phi(x)只由已宣告normalization及symmetry推出；Gaussian integral、mean 0與variance 1是named theorem facts；finite quadrature不證它們或非零CDF exact value。

本題逐步詳解（AI 二次覆核）

標準常態分布 · qf-mod45-normaldist-cdf-symmetry · candidate

58. 標準常態分布

standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-cdf-symmetry；資料={"x":-5}。請依 [x, CDF_symmetry_identity, CDF_at_zero, declared_symmetry_certificate, boundary] 欄序回答。

查看答案與解釋

(-5, Phi(5)=1-Phi(-5), Phi(0)=1/2, 是, declared_normalization_plus_even_density_implies_CDF_symmetry_but_no_nonzero_elementary_CDF_value)

Wave6第八層核對=(-5, Phi(5)=1-Phi(-5), Phi(0)=1/2, 是, declared_normalization_plus_even_density_implies_CDF_symmetry_but_no_nonzero_elementary_CDF_value)；依完整明示資料獨立重算；Phi(0)=1/2與Phi(-x)=1-Phi(x)只由已宣告normalization及symmetry推出；Gaussian integral、mean 0與variance 1是named theorem facts；finite quadrature不證它們或非零CDF exact value。

本題逐步詳解（AI 二次覆核）

標準常態分布 · qf-mod45-normaldist-cdf-symmetry · candidate

59. 標準常態分布

standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-cdf-symmetry；資料={"x":-4}。請依 [x, CDF_symmetry_identity, CDF_at_zero, declared_symmetry_certificate, boundary] 欄序回答。

查看答案與解釋

(-4, Phi(4)=1-Phi(-4), Phi(0)=1/2, 是, declared_normalization_plus_even_density_implies_CDF_symmetry_but_no_nonzero_elementary_CDF_value)

Wave6第八層核對=(-4, Phi(4)=1-Phi(-4), Phi(0)=1/2, 是, declared_normalization_plus_even_density_implies_CDF_symmetry_but_no_nonzero_elementary_CDF_value)；依完整明示資料獨立重算；Phi(0)=1/2與Phi(-x)=1-Phi(x)只由已宣告normalization及symmetry推出；Gaussian integral、mean 0與variance 1是named theorem facts；finite quadrature不證它們或非零CDF exact value。

本題逐步詳解（AI 二次覆核）

標準常態分布 · qf-mod45-normaldist-cdf-symmetry · candidate

60. 標準常態分布

standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-cdf-symmetry；資料={"x":-3}。請依 [x, CDF_symmetry_identity, CDF_at_zero, declared_symmetry_certificate, boundary] 欄序回答。

查看答案與解釋

(-3, Phi(3)=1-Phi(-3), Phi(0)=1/2, 是, declared_normalization_plus_even_density_implies_CDF_symmetry_but_no_nonzero_elementary_CDF_value)

Wave6第八層核對=(-3, Phi(3)=1-Phi(-3), Phi(0)=1/2, 是, declared_normalization_plus_even_density_implies_CDF_symmetry_but_no_nonzero_elementary_CDF_value)；依完整明示資料獨立重算；Phi(0)=1/2與Phi(-x)=1-Phi(x)只由已宣告normalization及symmetry推出；Gaussian integral、mean 0與variance 1是named theorem facts；finite quadrature不證它們或非零CDF exact value。

本題逐步詳解（AI 二次覆核）

標準常態分布 · qf-mod45-normaldist-cdf-symmetry · candidate

61. 標準常態分布

standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-cdf-symmetry；資料={"x":-2}。請依 [x, CDF_symmetry_identity, CDF_at_zero, declared_symmetry_certificate, boundary] 欄序回答。

查看答案與解釋

(-2, Phi(2)=1-Phi(-2), Phi(0)=1/2, 是, declared_normalization_plus_even_density_implies_CDF_symmetry_but_no_nonzero_elementary_CDF_value)

Wave6第八層核對=(-2, Phi(2)=1-Phi(-2), Phi(0)=1/2, 是, declared_normalization_plus_even_density_implies_CDF_symmetry_but_no_nonzero_elementary_CDF_value)；依完整明示資料獨立重算；Phi(0)=1/2與Phi(-x)=1-Phi(x)只由已宣告normalization及symmetry推出；Gaussian integral、mean 0與variance 1是named theorem facts；finite quadrature不證它們或非零CDF exact value。

本題逐步詳解（AI 二次覆核）

標準常態分布 · qf-mod45-normaldist-cdf-symmetry · candidate

62. 標準常態分布

standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-cdf-symmetry；資料={"x":-1}。請依 [x, CDF_symmetry_identity, CDF_at_zero, declared_symmetry_certificate, boundary] 欄序回答。

查看答案與解釋

(-1, Phi(1)=1-Phi(-1), Phi(0)=1/2, 是, declared_normalization_plus_even_density_implies_CDF_symmetry_but_no_nonzero_elementary_CDF_value)

Wave6第八層核對=(-1, Phi(1)=1-Phi(-1), Phi(0)=1/2, 是, declared_normalization_plus_even_density_implies_CDF_symmetry_but_no_nonzero_elementary_CDF_value)；依完整明示資料獨立重算；Phi(0)=1/2與Phi(-x)=1-Phi(x)只由已宣告normalization及symmetry推出；Gaussian integral、mean 0與variance 1是named theorem facts；finite quadrature不證它們或非零CDF exact value。

本題逐步詳解（AI 二次覆核）

標準常態分布 · qf-mod45-normaldist-cdf-symmetry · candidate

63. 標準常態分布

standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-cdf-symmetry；資料={"x":0}。請依 [x, CDF_symmetry_identity, CDF_at_zero, declared_symmetry_certificate, boundary] 欄序回答。

查看答案與解釋

(0, Phi(0)=1-Phi(0), Phi(0)=1/2, 是, declared_normalization_plus_even_density_implies_CDF_symmetry_but_no_nonzero_elementary_CDF_value)

Wave6第八層核對=(0, Phi(0)=1-Phi(0), Phi(0)=1/2, 是, declared_normalization_plus_even_density_implies_CDF_symmetry_but_no_nonzero_elementary_CDF_value)；依完整明示資料獨立重算；Phi(0)=1/2與Phi(-x)=1-Phi(x)只由已宣告normalization及symmetry推出；Gaussian integral、mean 0與variance 1是named theorem facts；finite quadrature不證它們或非零CDF exact value。

本題逐步詳解（AI 二次覆核）

標準常態分布 · qf-mod45-normaldist-cdf-symmetry · candidate

64. 標準常態分布

standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-cdf-symmetry；資料={"x":1}。請依 [x, CDF_symmetry_identity, CDF_at_zero, declared_symmetry_certificate, boundary] 欄序回答。

查看答案與解釋

(1, Phi(-1)=1-Phi(1), Phi(0)=1/2, 是, declared_normalization_plus_even_density_implies_CDF_symmetry_but_no_nonzero_elementary_CDF_value)

Wave6第八層核對=(1, Phi(-1)=1-Phi(1), Phi(0)=1/2, 是, declared_normalization_plus_even_density_implies_CDF_symmetry_but_no_nonzero_elementary_CDF_value)；依完整明示資料獨立重算；Phi(0)=1/2與Phi(-x)=1-Phi(x)只由已宣告normalization及symmetry推出；Gaussian integral、mean 0與variance 1是named theorem facts；finite quadrature不證它們或非零CDF exact value。

本題逐步詳解（AI 二次覆核）

標準常態分布 · qf-mod45-normaldist-cdf-symmetry · candidate

65. 標準常態分布

standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-cdf-symmetry；資料={"x":2}。請依 [x, CDF_symmetry_identity, CDF_at_zero, declared_symmetry_certificate, boundary] 欄序回答。

查看答案與解釋

(2, Phi(-2)=1-Phi(2), Phi(0)=1/2, 是, declared_normalization_plus_even_density_implies_CDF_symmetry_but_no_nonzero_elementary_CDF_value)

Wave6第八層核對=(2, Phi(-2)=1-Phi(2), Phi(0)=1/2, 是, declared_normalization_plus_even_density_implies_CDF_symmetry_but_no_nonzero_elementary_CDF_value)；依完整明示資料獨立重算；Phi(0)=1/2與Phi(-x)=1-Phi(x)只由已宣告normalization及symmetry推出；Gaussian integral、mean 0與variance 1是named theorem facts；finite quadrature不證它們或非零CDF exact value。

本題逐步詳解（AI 二次覆核）

標準常態分布 · qf-mod45-normaldist-cdf-symmetry · candidate

66. 標準常態分布

standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-cdf-symmetry；資料={"x":3}。請依 [x, CDF_symmetry_identity, CDF_at_zero, declared_symmetry_certificate, boundary] 欄序回答。

查看答案與解釋

(3, Phi(-3)=1-Phi(3), Phi(0)=1/2, 是, declared_normalization_plus_even_density_implies_CDF_symmetry_but_no_nonzero_elementary_CDF_value)

Wave6第八層核對=(3, Phi(-3)=1-Phi(3), Phi(0)=1/2, 是, declared_normalization_plus_even_density_implies_CDF_symmetry_but_no_nonzero_elementary_CDF_value)；依完整明示資料獨立重算；Phi(0)=1/2與Phi(-x)=1-Phi(x)只由已宣告normalization及symmetry推出；Gaussian integral、mean 0與variance 1是named theorem facts；finite quadrature不證它們或非零CDF exact value。

本題逐步詳解（AI 二次覆核）

標準常態分布 · qf-mod45-normaldist-cdf-symmetry · candidate

67. 標準常態分布

standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-cdf-symmetry；資料={"x":4}。請依 [x, CDF_symmetry_identity, CDF_at_zero, declared_symmetry_certificate, boundary] 欄序回答。

查看答案與解釋

(4, Phi(-4)=1-Phi(4), Phi(0)=1/2, 是, declared_normalization_plus_even_density_implies_CDF_symmetry_but_no_nonzero_elementary_CDF_value)

Wave6第八層核對=(4, Phi(-4)=1-Phi(4), Phi(0)=1/2, 是, declared_normalization_plus_even_density_implies_CDF_symmetry_but_no_nonzero_elementary_CDF_value)；依完整明示資料獨立重算；Phi(0)=1/2與Phi(-x)=1-Phi(x)只由已宣告normalization及symmetry推出；Gaussian integral、mean 0與variance 1是named theorem facts；finite quadrature不證它們或非零CDF exact value。

本題逐步詳解（AI 二次覆核）

標準常態分布 · qf-mod45-normaldist-cdf-symmetry · candidate

68. 標準常態分布

standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-cdf-symmetry；資料={"x":5}。請依 [x, CDF_symmetry_identity, CDF_at_zero, declared_symmetry_certificate, boundary] 欄序回答。

查看答案與解釋

(5, Phi(-5)=1-Phi(5), Phi(0)=1/2, 是, declared_normalization_plus_even_density_implies_CDF_symmetry_but_no_nonzero_elementary_CDF_value)

Wave6第八層核對=(5, Phi(-5)=1-Phi(5), Phi(0)=1/2, 是, declared_normalization_plus_even_density_implies_CDF_symmetry_but_no_nonzero_elementary_CDF_value)；依完整明示資料獨立重算；Phi(0)=1/2與Phi(-x)=1-Phi(x)只由已宣告normalization及symmetry推出；Gaussian integral、mean 0與variance 1是named theorem facts；finite quadrature不證它們或非零CDF exact value。

本題逐步詳解（AI 二次覆核）

標準常態分布 · qf-mod45-normaldist-cdf-symmetry · candidate

69. 標準常態分布

standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-cdf-symmetry；資料={"x":6}。請依 [x, CDF_symmetry_identity, CDF_at_zero, declared_symmetry_certificate, boundary] 欄序回答。

查看答案與解釋

(6, Phi(-6)=1-Phi(6), Phi(0)=1/2, 是, declared_normalization_plus_even_density_implies_CDF_symmetry_but_no_nonzero_elementary_CDF_value)

Wave6第八層核對=(6, Phi(-6)=1-Phi(6), Phi(0)=1/2, 是, declared_normalization_plus_even_density_implies_CDF_symmetry_but_no_nonzero_elementary_CDF_value)；依完整明示資料獨立重算；Phi(0)=1/2與Phi(-x)=1-Phi(x)只由已宣告normalization及symmetry推出；Gaussian integral、mean 0與variance 1是named theorem facts；finite quadrature不證它們或非零CDF exact value。

本題逐步詳解（AI 二次覆核）

標準常態分布 · qf-mod45-normaldist-cdf-symmetry · candidate

70. 標準常態分布

standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-cdf-symmetry；資料={"x":7}。請依 [x, CDF_symmetry_identity, CDF_at_zero, declared_symmetry_certificate, boundary] 欄序回答。

查看答案與解釋

(7, Phi(-7)=1-Phi(7), Phi(0)=1/2, 是, declared_normalization_plus_even_density_implies_CDF_symmetry_but_no_nonzero_elementary_CDF_value)

Wave6第八層核對=(7, Phi(-7)=1-Phi(7), Phi(0)=1/2, 是, declared_normalization_plus_even_density_implies_CDF_symmetry_but_no_nonzero_elementary_CDF_value)；依完整明示資料獨立重算；Phi(0)=1/2與Phi(-x)=1-Phi(x)只由已宣告normalization及symmetry推出；Gaussian integral、mean 0與variance 1是named theorem facts；finite quadrature不證它們或非零CDF exact value。

本題逐步詳解（AI 二次覆核）

標準常態分布 · qf-mod45-normaldist-cdf-symmetry · candidate

71. 標準常態分布

standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-cdf-symmetry；資料={"x":8}。請依 [x, CDF_symmetry_identity, CDF_at_zero, declared_symmetry_certificate, boundary] 欄序回答。

查看答案與解釋

(8, Phi(-8)=1-Phi(8), Phi(0)=1/2, 是, declared_normalization_plus_even_density_implies_CDF_symmetry_but_no_nonzero_elementary_CDF_value)

Wave6第八層核對=(8, Phi(-8)=1-Phi(8), Phi(0)=1/2, 是, declared_normalization_plus_even_density_implies_CDF_symmetry_but_no_nonzero_elementary_CDF_value)；依完整明示資料獨立重算；Phi(0)=1/2與Phi(-x)=1-Phi(x)只由已宣告normalization及symmetry推出；Gaussian integral、mean 0與variance 1是named theorem facts；finite quadrature不證它們或非零CDF exact value。

本題逐步詳解（AI 二次覆核）

標準常態分布 · qf-mod45-normaldist-cdf-symmetry · candidate

72. 標準常態分布

standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-cdf-symmetry；資料={"x":9}。請依 [x, CDF_symmetry_identity, CDF_at_zero, declared_symmetry_certificate, boundary] 欄序回答。

查看答案與解釋

(9, Phi(-9)=1-Phi(9), Phi(0)=1/2, 是, declared_normalization_plus_even_density_implies_CDF_symmetry_but_no_nonzero_elementary_CDF_value)

Wave6第八層核對=(9, Phi(-9)=1-Phi(9), Phi(0)=1/2, 是, declared_normalization_plus_even_density_implies_CDF_symmetry_but_no_nonzero_elementary_CDF_value)；依完整明示資料獨立重算；Phi(0)=1/2與Phi(-x)=1-Phi(x)只由已宣告normalization及symmetry推出；Gaussian integral、mean 0與variance 1是named theorem facts；finite quadrature不證它們或非零CDF exact value。

本題逐步詳解（AI 二次覆核）

標準常態分布 · qf-mod45-normaldist-cdf-symmetry · candidate

73. 標準常態分布

standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-cdf-symmetry；資料={"x":10}。請依 [x, CDF_symmetry_identity, CDF_at_zero, declared_symmetry_certificate, boundary] 欄序回答。

查看答案與解釋

(10, Phi(-10)=1-Phi(10), Phi(0)=1/2, 是, declared_normalization_plus_even_density_implies_CDF_symmetry_but_no_nonzero_elementary_CDF_value)

Wave6第八層核對=(10, Phi(-10)=1-Phi(10), Phi(0)=1/2, 是, declared_normalization_plus_even_density_implies_CDF_symmetry_but_no_nonzero_elementary_CDF_value)；依完整明示資料獨立重算；Phi(0)=1/2與Phi(-x)=1-Phi(x)只由已宣告normalization及symmetry推出；Gaussian integral、mean 0與variance 1是named theorem facts；finite quadrature不證它們或非零CDF exact value。

本題逐步詳解（AI 二次覆核）

標準常態分布 · qf-mod45-normaldist-cdf-symmetry · candidate

74. 標準常態分布

standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-cdf-symmetry；資料={"x":11}。請依 [x, CDF_symmetry_identity, CDF_at_zero, declared_symmetry_certificate, boundary] 欄序回答。

查看答案與解釋

(11, Phi(-11)=1-Phi(11), Phi(0)=1/2, 是, declared_normalization_plus_even_density_implies_CDF_symmetry_but_no_nonzero_elementary_CDF_value)

Wave6第八層核對=(11, Phi(-11)=1-Phi(11), Phi(0)=1/2, 是, declared_normalization_plus_even_density_implies_CDF_symmetry_but_no_nonzero_elementary_CDF_value)；依完整明示資料獨立重算；Phi(0)=1/2與Phi(-x)=1-Phi(x)只由已宣告normalization及symmetry推出；Gaussian integral、mean 0與variance 1是named theorem facts；finite quadrature不證它們或非零CDF exact value。

本題逐步詳解（AI 二次覆核）

標準常態分布 · qf-mod45-normaldist-cdf-symmetry · candidate

75. 標準常態分布

standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-cdf-symmetry；資料={"x":12}。請依 [x, CDF_symmetry_identity, CDF_at_zero, declared_symmetry_certificate, boundary] 欄序回答。

查看答案與解釋

(12, Phi(-12)=1-Phi(12), Phi(0)=1/2, 是, declared_normalization_plus_even_density_implies_CDF_symmetry_but_no_nonzero_elementary_CDF_value)

Wave6第八層核對=(12, Phi(-12)=1-Phi(12), Phi(0)=1/2, 是, declared_normalization_plus_even_density_implies_CDF_symmetry_but_no_nonzero_elementary_CDF_value)；依完整明示資料獨立重算；Phi(0)=1/2與Phi(-x)=1-Phi(x)只由已宣告normalization及symmetry推出；Gaussian integral、mean 0與variance 1是named theorem facts；finite quadrature不證它們或非零CDF exact value。

本題逐步詳解（AI 二次覆核）

標準常態分布 · qf-mod45-normaldist-cdf-symmetry · candidate

76. 標準常態分布

standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-normalization-moment-boundary；資料={"window":1,"panels":10}。請依 [window, panels, finite_method_scope, finite_method_proof_claim, normalization_named_fact, mean_named_fact, variance_named_fact, boundary] 欄序回答。

此題只記window/panels與有限方法範圍，未執行或提供Gaussian integral完整證明；named facts不由有限quadrature接受，也不被否旗標否定。

查看答案與解釋

(1, 10, finite_quadrature_only, 否, named_fact:integral_R_phi=1, named_fact:E[Z]=0, named_fact:Var(Z)=1, finite_window_or_panel_count_does_not_prove_normalization_or_moments)

Wave6第八層核對=(1, 10, finite_quadrature_only, 否, named_fact:integral_R_phi=1, named_fact:E[Z]=0, named_fact:Var(Z)=1, finite_window_or_panel_count_does_not_prove_normalization_or_moments)；依完整明示資料獨立重算；Phi(0)=1/2與Phi(-x)=1-Phi(x)只由已宣告normalization及symmetry推出；Gaussian integral、mean 0與variance 1是named theorem facts；finite quadrature不證它們或非零CDF exact value。

本題逐步詳解（AI 二次覆核）

標準常態分布 · qf-mod45-normaldist-normalization-moment-boundary · candidate

77. 標準常態分布

standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-normalization-moment-boundary；資料={"window":2,"panels":11}。請依 [window, panels, finite_method_scope, finite_method_proof_claim, normalization_named_fact, mean_named_fact, variance_named_fact, boundary] 欄序回答。

此題只記window/panels與有限方法範圍，未執行或提供Gaussian integral完整證明；named facts不由有限quadrature接受，也不被否旗標否定。

查看答案與解釋

(2, 11, finite_quadrature_only, 否, named_fact:integral_R_phi=1, named_fact:E[Z]=0, named_fact:Var(Z)=1, finite_window_or_panel_count_does_not_prove_normalization_or_moments)

Wave6第八層核對=(2, 11, finite_quadrature_only, 否, named_fact:integral_R_phi=1, named_fact:E[Z]=0, named_fact:Var(Z)=1, finite_window_or_panel_count_does_not_prove_normalization_or_moments)；依完整明示資料獨立重算；Phi(0)=1/2與Phi(-x)=1-Phi(x)只由已宣告normalization及symmetry推出；Gaussian integral、mean 0與variance 1是named theorem facts；finite quadrature不證它們或非零CDF exact value。

本題逐步詳解（AI 二次覆核）

標準常態分布 · qf-mod45-normaldist-normalization-moment-boundary · candidate

78. 標準常態分布

standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-normalization-moment-boundary；資料={"window":3,"panels":12}。請依 [window, panels, finite_method_scope, finite_method_proof_claim, normalization_named_fact, mean_named_fact, variance_named_fact, boundary] 欄序回答。

此題只記window/panels與有限方法範圍，未執行或提供Gaussian integral完整證明；named facts不由有限quadrature接受，也不被否旗標否定。

查看答案與解釋

(3, 12, finite_quadrature_only, 否, named_fact:integral_R_phi=1, named_fact:E[Z]=0, named_fact:Var(Z)=1, finite_window_or_panel_count_does_not_prove_normalization_or_moments)

Wave6第八層核對=(3, 12, finite_quadrature_only, 否, named_fact:integral_R_phi=1, named_fact:E[Z]=0, named_fact:Var(Z)=1, finite_window_or_panel_count_does_not_prove_normalization_or_moments)；依完整明示資料獨立重算；Phi(0)=1/2與Phi(-x)=1-Phi(x)只由已宣告normalization及symmetry推出；Gaussian integral、mean 0與variance 1是named theorem facts；finite quadrature不證它們或非零CDF exact value。

本題逐步詳解（AI 二次覆核）

標準常態分布 · qf-mod45-normaldist-normalization-moment-boundary · candidate

79. 標準常態分布

standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-normalization-moment-boundary；資料={"window":4,"panels":13}。請依 [window, panels, finite_method_scope, finite_method_proof_claim, normalization_named_fact, mean_named_fact, variance_named_fact, boundary] 欄序回答。

此題只記window/panels與有限方法範圍，未執行或提供Gaussian integral完整證明；named facts不由有限quadrature接受，也不被否旗標否定。

查看答案與解釋

(4, 13, finite_quadrature_only, 否, named_fact:integral_R_phi=1, named_fact:E[Z]=0, named_fact:Var(Z)=1, finite_window_or_panel_count_does_not_prove_normalization_or_moments)

Wave6第八層核對=(4, 13, finite_quadrature_only, 否, named_fact:integral_R_phi=1, named_fact:E[Z]=0, named_fact:Var(Z)=1, finite_window_or_panel_count_does_not_prove_normalization_or_moments)；依完整明示資料獨立重算；Phi(0)=1/2與Phi(-x)=1-Phi(x)只由已宣告normalization及symmetry推出；Gaussian integral、mean 0與variance 1是named theorem facts；finite quadrature不證它們或非零CDF exact value。

本題逐步詳解（AI 二次覆核）

標準常態分布 · qf-mod45-normaldist-normalization-moment-boundary · candidate

80. 標準常態分布

standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-normalization-moment-boundary；資料={"window":5,"panels":14}。請依 [window, panels, finite_method_scope, finite_method_proof_claim, normalization_named_fact, mean_named_fact, variance_named_fact, boundary] 欄序回答。

此題只記window/panels與有限方法範圍，未執行或提供Gaussian integral完整證明；named facts不由有限quadrature接受，也不被否旗標否定。

查看答案與解釋

(5, 14, finite_quadrature_only, 否, named_fact:integral_R_phi=1, named_fact:E[Z]=0, named_fact:Var(Z)=1, finite_window_or_panel_count_does_not_prove_normalization_or_moments)

Wave6第八層核對=(5, 14, finite_quadrature_only, 否, named_fact:integral_R_phi=1, named_fact:E[Z]=0, named_fact:Var(Z)=1, finite_window_or_panel_count_does_not_prove_normalization_or_moments)；依完整明示資料獨立重算；Phi(0)=1/2與Phi(-x)=1-Phi(x)只由已宣告normalization及symmetry推出；Gaussian integral、mean 0與variance 1是named theorem facts；finite quadrature不證它們或非零CDF exact value。

本題逐步詳解（AI 二次覆核）

標準常態分布 · qf-mod45-normaldist-normalization-moment-boundary · candidate

81. 標準常態分布

standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-normalization-moment-boundary；資料={"window":6,"panels":15}。請依 [window, panels, finite_method_scope, finite_method_proof_claim, normalization_named_fact, mean_named_fact, variance_named_fact, boundary] 欄序回答。

此題只記window/panels與有限方法範圍，未執行或提供Gaussian integral完整證明；named facts不由有限quadrature接受，也不被否旗標否定。

查看答案與解釋

(6, 15, finite_quadrature_only, 否, named_fact:integral_R_phi=1, named_fact:E[Z]=0, named_fact:Var(Z)=1, finite_window_or_panel_count_does_not_prove_normalization_or_moments)

Wave6第八層核對=(6, 15, finite_quadrature_only, 否, named_fact:integral_R_phi=1, named_fact:E[Z]=0, named_fact:Var(Z)=1, finite_window_or_panel_count_does_not_prove_normalization_or_moments)；依完整明示資料獨立重算；Phi(0)=1/2與Phi(-x)=1-Phi(x)只由已宣告normalization及symmetry推出；Gaussian integral、mean 0與variance 1是named theorem facts；finite quadrature不證它們或非零CDF exact value。

本題逐步詳解（AI 二次覆核）

標準常態分布 · qf-mod45-normaldist-normalization-moment-boundary · candidate

82. 標準常態分布

standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-normalization-moment-boundary；資料={"window":7,"panels":16}。請依 [window, panels, finite_method_scope, finite_method_proof_claim, normalization_named_fact, mean_named_fact, variance_named_fact, boundary] 欄序回答。

此題只記window/panels與有限方法範圍，未執行或提供Gaussian integral完整證明；named facts不由有限quadrature接受，也不被否旗標否定。

查看答案與解釋

(7, 16, finite_quadrature_only, 否, named_fact:integral_R_phi=1, named_fact:E[Z]=0, named_fact:Var(Z)=1, finite_window_or_panel_count_does_not_prove_normalization_or_moments)

Wave6第八層核對=(7, 16, finite_quadrature_only, 否, named_fact:integral_R_phi=1, named_fact:E[Z]=0, named_fact:Var(Z)=1, finite_window_or_panel_count_does_not_prove_normalization_or_moments)；依完整明示資料獨立重算；Phi(0)=1/2與Phi(-x)=1-Phi(x)只由已宣告normalization及symmetry推出；Gaussian integral、mean 0與variance 1是named theorem facts；finite quadrature不證它們或非零CDF exact value。

本題逐步詳解（AI 二次覆核）

標準常態分布 · qf-mod45-normaldist-normalization-moment-boundary · candidate

83. 標準常態分布

standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-normalization-moment-boundary；資料={"window":8,"panels":17}。請依 [window, panels, finite_method_scope, finite_method_proof_claim, normalization_named_fact, mean_named_fact, variance_named_fact, boundary] 欄序回答。

此題只記window/panels與有限方法範圍，未執行或提供Gaussian integral完整證明；named facts不由有限quadrature接受，也不被否旗標否定。

查看答案與解釋

(8, 17, finite_quadrature_only, 否, named_fact:integral_R_phi=1, named_fact:E[Z]=0, named_fact:Var(Z)=1, finite_window_or_panel_count_does_not_prove_normalization_or_moments)

Wave6第八層核對=(8, 17, finite_quadrature_only, 否, named_fact:integral_R_phi=1, named_fact:E[Z]=0, named_fact:Var(Z)=1, finite_window_or_panel_count_does_not_prove_normalization_or_moments)；依完整明示資料獨立重算；Phi(0)=1/2與Phi(-x)=1-Phi(x)只由已宣告normalization及symmetry推出；Gaussian integral、mean 0與variance 1是named theorem facts；finite quadrature不證它們或非零CDF exact value。

本題逐步詳解（AI 二次覆核）

標準常態分布 · qf-mod45-normaldist-normalization-moment-boundary · candidate

84. 標準常態分布

standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-normalization-moment-boundary；資料={"window":9,"panels":18}。請依 [window, panels, finite_method_scope, finite_method_proof_claim, normalization_named_fact, mean_named_fact, variance_named_fact, boundary] 欄序回答。

此題只記window/panels與有限方法範圍，未執行或提供Gaussian integral完整證明；named facts不由有限quadrature接受，也不被否旗標否定。

查看答案與解釋

(9, 18, finite_quadrature_only, 否, named_fact:integral_R_phi=1, named_fact:E[Z]=0, named_fact:Var(Z)=1, finite_window_or_panel_count_does_not_prove_normalization_or_moments)

Wave6第八層核對=(9, 18, finite_quadrature_only, 否, named_fact:integral_R_phi=1, named_fact:E[Z]=0, named_fact:Var(Z)=1, finite_window_or_panel_count_does_not_prove_normalization_or_moments)；依完整明示資料獨立重算；Phi(0)=1/2與Phi(-x)=1-Phi(x)只由已宣告normalization及symmetry推出；Gaussian integral、mean 0與variance 1是named theorem facts；finite quadrature不證它們或非零CDF exact value。

本題逐步詳解（AI 二次覆核）

標準常態分布 · qf-mod45-normaldist-normalization-moment-boundary · candidate

85. 標準常態分布

standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-normalization-moment-boundary；資料={"window":10,"panels":19}。請依 [window, panels, finite_method_scope, finite_method_proof_claim, normalization_named_fact, mean_named_fact, variance_named_fact, boundary] 欄序回答。

此題只記window/panels與有限方法範圍，未執行或提供Gaussian integral完整證明；named facts不由有限quadrature接受，也不被否旗標否定。

查看答案與解釋

(10, 19, finite_quadrature_only, 否, named_fact:integral_R_phi=1, named_fact:E[Z]=0, named_fact:Var(Z)=1, finite_window_or_panel_count_does_not_prove_normalization_or_moments)

Wave6第八層核對=(10, 19, finite_quadrature_only, 否, named_fact:integral_R_phi=1, named_fact:E[Z]=0, named_fact:Var(Z)=1, finite_window_or_panel_count_does_not_prove_normalization_or_moments)；依完整明示資料獨立重算；Phi(0)=1/2與Phi(-x)=1-Phi(x)只由已宣告normalization及symmetry推出；Gaussian integral、mean 0與variance 1是named theorem facts；finite quadrature不證它們或非零CDF exact value。

本題逐步詳解（AI 二次覆核）

標準常態分布 · qf-mod45-normaldist-normalization-moment-boundary · candidate

86. 標準常態分布

standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-normalization-moment-boundary；資料={"window":11,"panels":20}。請依 [window, panels, finite_method_scope, finite_method_proof_claim, normalization_named_fact, mean_named_fact, variance_named_fact, boundary] 欄序回答。

此題只記window/panels與有限方法範圍，未執行或提供Gaussian integral完整證明；named facts不由有限quadrature接受，也不被否旗標否定。

查看答案與解釋

(11, 20, finite_quadrature_only, 否, named_fact:integral_R_phi=1, named_fact:E[Z]=0, named_fact:Var(Z)=1, finite_window_or_panel_count_does_not_prove_normalization_or_moments)

Wave6第八層核對=(11, 20, finite_quadrature_only, 否, named_fact:integral_R_phi=1, named_fact:E[Z]=0, named_fact:Var(Z)=1, finite_window_or_panel_count_does_not_prove_normalization_or_moments)；依完整明示資料獨立重算；Phi(0)=1/2與Phi(-x)=1-Phi(x)只由已宣告normalization及symmetry推出；Gaussian integral、mean 0與variance 1是named theorem facts；finite quadrature不證它們或非零CDF exact value。

本題逐步詳解（AI 二次覆核）

標準常態分布 · qf-mod45-normaldist-normalization-moment-boundary · candidate

87. 標準常態分布

standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-normalization-moment-boundary；資料={"window":12,"panels":21}。請依 [window, panels, finite_method_scope, finite_method_proof_claim, normalization_named_fact, mean_named_fact, variance_named_fact, boundary] 欄序回答。

此題只記window/panels與有限方法範圍，未執行或提供Gaussian integral完整證明；named facts不由有限quadrature接受，也不被否旗標否定。

查看答案與解釋

(12, 21, finite_quadrature_only, 否, named_fact:integral_R_phi=1, named_fact:E[Z]=0, named_fact:Var(Z)=1, finite_window_or_panel_count_does_not_prove_normalization_or_moments)

Wave6第八層核對=(12, 21, finite_quadrature_only, 否, named_fact:integral_R_phi=1, named_fact:E[Z]=0, named_fact:Var(Z)=1, finite_window_or_panel_count_does_not_prove_normalization_or_moments)；依完整明示資料獨立重算；Phi(0)=1/2與Phi(-x)=1-Phi(x)只由已宣告normalization及symmetry推出；Gaussian integral、mean 0與variance 1是named theorem facts；finite quadrature不證它們或非零CDF exact value。

本題逐步詳解（AI 二次覆核）

標準常態分布 · qf-mod45-normaldist-normalization-moment-boundary · candidate

88. 標準常態分布

standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-normalization-moment-boundary；資料={"window":13,"panels":22}。請依 [window, panels, finite_method_scope, finite_method_proof_claim, normalization_named_fact, mean_named_fact, variance_named_fact, boundary] 欄序回答。

此題只記window/panels與有限方法範圍，未執行或提供Gaussian integral完整證明；named facts不由有限quadrature接受，也不被否旗標否定。

查看答案與解釋

(13, 22, finite_quadrature_only, 否, named_fact:integral_R_phi=1, named_fact:E[Z]=0, named_fact:Var(Z)=1, finite_window_or_panel_count_does_not_prove_normalization_or_moments)

Wave6第八層核對=(13, 22, finite_quadrature_only, 否, named_fact:integral_R_phi=1, named_fact:E[Z]=0, named_fact:Var(Z)=1, finite_window_or_panel_count_does_not_prove_normalization_or_moments)；依完整明示資料獨立重算；Phi(0)=1/2與Phi(-x)=1-Phi(x)只由已宣告normalization及symmetry推出；Gaussian integral、mean 0與variance 1是named theorem facts；finite quadrature不證它們或非零CDF exact value。

本題逐步詳解（AI 二次覆核）

標準常態分布 · qf-mod45-normaldist-normalization-moment-boundary · candidate

89. 標準常態分布

standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-normalization-moment-boundary；資料={"window":14,"panels":23}。請依 [window, panels, finite_method_scope, finite_method_proof_claim, normalization_named_fact, mean_named_fact, variance_named_fact, boundary] 欄序回答。

此題只記window/panels與有限方法範圍，未執行或提供Gaussian integral完整證明；named facts不由有限quadrature接受，也不被否旗標否定。

查看答案與解釋

(14, 23, finite_quadrature_only, 否, named_fact:integral_R_phi=1, named_fact:E[Z]=0, named_fact:Var(Z)=1, finite_window_or_panel_count_does_not_prove_normalization_or_moments)

Wave6第八層核對=(14, 23, finite_quadrature_only, 否, named_fact:integral_R_phi=1, named_fact:E[Z]=0, named_fact:Var(Z)=1, finite_window_or_panel_count_does_not_prove_normalization_or_moments)；依完整明示資料獨立重算；Phi(0)=1/2與Phi(-x)=1-Phi(x)只由已宣告normalization及symmetry推出；Gaussian integral、mean 0與variance 1是named theorem facts；finite quadrature不證它們或非零CDF exact value。

本題逐步詳解（AI 二次覆核）

標準常態分布 · qf-mod45-normaldist-normalization-moment-boundary · candidate

90. 標準常態分布

standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-normalization-moment-boundary；資料={"window":15,"panels":24}。請依 [window, panels, finite_method_scope, finite_method_proof_claim, normalization_named_fact, mean_named_fact, variance_named_fact, boundary] 欄序回答。

此題只記window/panels與有限方法範圍，未執行或提供Gaussian integral完整證明；named facts不由有限quadrature接受，也不被否旗標否定。

查看答案與解釋

(15, 24, finite_quadrature_only, 否, named_fact:integral_R_phi=1, named_fact:E[Z]=0, named_fact:Var(Z)=1, finite_window_or_panel_count_does_not_prove_normalization_or_moments)

Wave6第八層核對=(15, 24, finite_quadrature_only, 否, named_fact:integral_R_phi=1, named_fact:E[Z]=0, named_fact:Var(Z)=1, finite_window_or_panel_count_does_not_prove_normalization_or_moments)；依完整明示資料獨立重算；Phi(0)=1/2與Phi(-x)=1-Phi(x)只由已宣告normalization及symmetry推出；Gaussian integral、mean 0與variance 1是named theorem facts；finite quadrature不證它們或非零CDF exact value。

本題逐步詳解（AI 二次覆核）

標準常態分布 · qf-mod45-normaldist-normalization-moment-boundary · candidate

91. 標準常態分布

standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-normalization-moment-boundary；資料={"window":16,"panels":25}。請依 [window, panels, finite_method_scope, finite_method_proof_claim, normalization_named_fact, mean_named_fact, variance_named_fact, boundary] 欄序回答。

此題只記window/panels與有限方法範圍，未執行或提供Gaussian integral完整證明；named facts不由有限quadrature接受，也不被否旗標否定。

查看答案與解釋

(16, 25, finite_quadrature_only, 否, named_fact:integral_R_phi=1, named_fact:E[Z]=0, named_fact:Var(Z)=1, finite_window_or_panel_count_does_not_prove_normalization_or_moments)

Wave6第八層核對=(16, 25, finite_quadrature_only, 否, named_fact:integral_R_phi=1, named_fact:E[Z]=0, named_fact:Var(Z)=1, finite_window_or_panel_count_does_not_prove_normalization_or_moments)；依完整明示資料獨立重算；Phi(0)=1/2與Phi(-x)=1-Phi(x)只由已宣告normalization及symmetry推出；Gaussian integral、mean 0與variance 1是named theorem facts；finite quadrature不證它們或非零CDF exact value。

本題逐步詳解（AI 二次覆核）

標準常態分布 · qf-mod45-normaldist-normalization-moment-boundary · candidate

92. 標準常態分布

standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-normalization-moment-boundary；資料={"window":17,"panels":26}。請依 [window, panels, finite_method_scope, finite_method_proof_claim, normalization_named_fact, mean_named_fact, variance_named_fact, boundary] 欄序回答。

此題只記window/panels與有限方法範圍，未執行或提供Gaussian integral完整證明；named facts不由有限quadrature接受，也不被否旗標否定。

查看答案與解釋

(17, 26, finite_quadrature_only, 否, named_fact:integral_R_phi=1, named_fact:E[Z]=0, named_fact:Var(Z)=1, finite_window_or_panel_count_does_not_prove_normalization_or_moments)

Wave6第八層核對=(17, 26, finite_quadrature_only, 否, named_fact:integral_R_phi=1, named_fact:E[Z]=0, named_fact:Var(Z)=1, finite_window_or_panel_count_does_not_prove_normalization_or_moments)；依完整明示資料獨立重算；Phi(0)=1/2與Phi(-x)=1-Phi(x)只由已宣告normalization及symmetry推出；Gaussian integral、mean 0與variance 1是named theorem facts；finite quadrature不證它們或非零CDF exact value。

本題逐步詳解（AI 二次覆核）

標準常態分布 · qf-mod45-normaldist-normalization-moment-boundary · candidate

93. 標準常態分布

standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-normalization-moment-boundary；資料={"window":18,"panels":27}。請依 [window, panels, finite_method_scope, finite_method_proof_claim, normalization_named_fact, mean_named_fact, variance_named_fact, boundary] 欄序回答。

此題只記window/panels與有限方法範圍，未執行或提供Gaussian integral完整證明；named facts不由有限quadrature接受，也不被否旗標否定。

查看答案與解釋

(18, 27, finite_quadrature_only, 否, named_fact:integral_R_phi=1, named_fact:E[Z]=0, named_fact:Var(Z)=1, finite_window_or_panel_count_does_not_prove_normalization_or_moments)

Wave6第八層核對=(18, 27, finite_quadrature_only, 否, named_fact:integral_R_phi=1, named_fact:E[Z]=0, named_fact:Var(Z)=1, finite_window_or_panel_count_does_not_prove_normalization_or_moments)；依完整明示資料獨立重算；Phi(0)=1/2與Phi(-x)=1-Phi(x)只由已宣告normalization及symmetry推出；Gaussian integral、mean 0與variance 1是named theorem facts；finite quadrature不證它們或非零CDF exact value。

本題逐步詳解（AI 二次覆核）

標準常態分布 · qf-mod45-normaldist-normalization-moment-boundary · candidate

94. 標準常態分布

standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-normalization-moment-boundary；資料={"window":19,"panels":28}。請依 [window, panels, finite_method_scope, finite_method_proof_claim, normalization_named_fact, mean_named_fact, variance_named_fact, boundary] 欄序回答。

此題只記window/panels與有限方法範圍，未執行或提供Gaussian integral完整證明；named facts不由有限quadrature接受，也不被否旗標否定。

查看答案與解釋

(19, 28, finite_quadrature_only, 否, named_fact:integral_R_phi=1, named_fact:E[Z]=0, named_fact:Var(Z)=1, finite_window_or_panel_count_does_not_prove_normalization_or_moments)

Wave6第八層核對=(19, 28, finite_quadrature_only, 否, named_fact:integral_R_phi=1, named_fact:E[Z]=0, named_fact:Var(Z)=1, finite_window_or_panel_count_does_not_prove_normalization_or_moments)；依完整明示資料獨立重算；Phi(0)=1/2與Phi(-x)=1-Phi(x)只由已宣告normalization及symmetry推出；Gaussian integral、mean 0與variance 1是named theorem facts；finite quadrature不證它們或非零CDF exact value。

本題逐步詳解（AI 二次覆核）

標準常態分布 · qf-mod45-normaldist-normalization-moment-boundary · candidate

95. 標準常態分布

standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-normalization-moment-boundary；資料={"window":20,"panels":29}。請依 [window, panels, finite_method_scope, finite_method_proof_claim, normalization_named_fact, mean_named_fact, variance_named_fact, boundary] 欄序回答。

此題只記window/panels與有限方法範圍，未執行或提供Gaussian integral完整證明；named facts不由有限quadrature接受，也不被否旗標否定。

查看答案與解釋

(20, 29, finite_quadrature_only, 否, named_fact:integral_R_phi=1, named_fact:E[Z]=0, named_fact:Var(Z)=1, finite_window_or_panel_count_does_not_prove_normalization_or_moments)

Wave6第八層核對=(20, 29, finite_quadrature_only, 否, named_fact:integral_R_phi=1, named_fact:E[Z]=0, named_fact:Var(Z)=1, finite_window_or_panel_count_does_not_prove_normalization_or_moments)；依完整明示資料獨立重算；Phi(0)=1/2與Phi(-x)=1-Phi(x)只由已宣告normalization及symmetry推出；Gaussian integral、mean 0與variance 1是named theorem facts；finite quadrature不證它們或非零CDF exact value。

本題逐步詳解（AI 二次覆核）

標準常態分布 · qf-mod45-normaldist-normalization-moment-boundary · candidate

96. 標準常態分布

standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-normalization-moment-boundary；資料={"window":21,"panels":30}。請依 [window, panels, finite_method_scope, finite_method_proof_claim, normalization_named_fact, mean_named_fact, variance_named_fact, boundary] 欄序回答。

此題只記window/panels與有限方法範圍，未執行或提供Gaussian integral完整證明；named facts不由有限quadrature接受，也不被否旗標否定。

查看答案與解釋

(21, 30, finite_quadrature_only, 否, named_fact:integral_R_phi=1, named_fact:E[Z]=0, named_fact:Var(Z)=1, finite_window_or_panel_count_does_not_prove_normalization_or_moments)

Wave6第八層核對=(21, 30, finite_quadrature_only, 否, named_fact:integral_R_phi=1, named_fact:E[Z]=0, named_fact:Var(Z)=1, finite_window_or_panel_count_does_not_prove_normalization_or_moments)；依完整明示資料獨立重算；Phi(0)=1/2與Phi(-x)=1-Phi(x)只由已宣告normalization及symmetry推出；Gaussian integral、mean 0與variance 1是named theorem facts；finite quadrature不證它們或非零CDF exact value。

本題逐步詳解（AI 二次覆核）

標準常態分布 · qf-mod45-normaldist-normalization-moment-boundary · candidate

97. 標準常態分布

standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-normalization-moment-boundary；資料={"window":22,"panels":31}。請依 [window, panels, finite_method_scope, finite_method_proof_claim, normalization_named_fact, mean_named_fact, variance_named_fact, boundary] 欄序回答。

此題只記window/panels與有限方法範圍，未執行或提供Gaussian integral完整證明；named facts不由有限quadrature接受，也不被否旗標否定。

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(22, 31, finite_quadrature_only, 否, named_fact:integral_R_phi=1, named_fact:E[Z]=0, named_fact:Var(Z)=1, finite_window_or_panel_count_does_not_prove_normalization_or_moments)

Wave6第八層核對=(22, 31, finite_quadrature_only, 否, named_fact:integral_R_phi=1, named_fact:E[Z]=0, named_fact:Var(Z)=1, finite_window_or_panel_count_does_not_prove_normalization_or_moments)；依完整明示資料獨立重算；Phi(0)=1/2與Phi(-x)=1-Phi(x)只由已宣告normalization及symmetry推出；Gaussian integral、mean 0與variance 1是named theorem facts；finite quadrature不證它們或非零CDF exact value。

本題逐步詳解（AI 二次覆核）

標準常態分布 · qf-mod45-normaldist-normalization-moment-boundary · candidate

98. 標準常態分布

standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-normalization-moment-boundary；資料={"window":23,"panels":32}。請依 [window, panels, finite_method_scope, finite_method_proof_claim, normalization_named_fact, mean_named_fact, variance_named_fact, boundary] 欄序回答。

此題只記window/panels與有限方法範圍，未執行或提供Gaussian integral完整證明；named facts不由有限quadrature接受，也不被否旗標否定。

查看答案與解釋

(23, 32, finite_quadrature_only, 否, named_fact:integral_R_phi=1, named_fact:E[Z]=0, named_fact:Var(Z)=1, finite_window_or_panel_count_does_not_prove_normalization_or_moments)

Wave6第八層核對=(23, 32, finite_quadrature_only, 否, named_fact:integral_R_phi=1, named_fact:E[Z]=0, named_fact:Var(Z)=1, finite_window_or_panel_count_does_not_prove_normalization_or_moments)；依完整明示資料獨立重算；Phi(0)=1/2與Phi(-x)=1-Phi(x)只由已宣告normalization及symmetry推出；Gaussian integral、mean 0與variance 1是named theorem facts；finite quadrature不證它們或非零CDF exact value。

本題逐步詳解（AI 二次覆核）

標準常態分布 · qf-mod45-normaldist-normalization-moment-boundary · candidate

99. 標準常態分布

standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-normalization-moment-boundary；資料={"window":24,"panels":33}。請依 [window, panels, finite_method_scope, finite_method_proof_claim, normalization_named_fact, mean_named_fact, variance_named_fact, boundary] 欄序回答。

此題只記window/panels與有限方法範圍，未執行或提供Gaussian integral完整證明；named facts不由有限quadrature接受，也不被否旗標否定。

查看答案與解釋

(24, 33, finite_quadrature_only, 否, named_fact:integral_R_phi=1, named_fact:E[Z]=0, named_fact:Var(Z)=1, finite_window_or_panel_count_does_not_prove_normalization_or_moments)

Wave6第八層核對=(24, 33, finite_quadrature_only, 否, named_fact:integral_R_phi=1, named_fact:E[Z]=0, named_fact:Var(Z)=1, finite_window_or_panel_count_does_not_prove_normalization_or_moments)；依完整明示資料獨立重算；Phi(0)=1/2與Phi(-x)=1-Phi(x)只由已宣告normalization及symmetry推出；Gaussian integral、mean 0與variance 1是named theorem facts；finite quadrature不證它們或非零CDF exact value。

本題逐步詳解（AI 二次覆核）

標準常態分布 · qf-mod45-normaldist-normalization-moment-boundary · candidate

100. 標準常態分布

standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-normalization-moment-boundary；資料={"window":25,"panels":34}。請依 [window, panels, finite_method_scope, finite_method_proof_claim, normalization_named_fact, mean_named_fact, variance_named_fact, boundary] 欄序回答。

此題只記window/panels與有限方法範圍，未執行或提供Gaussian integral完整證明；named facts不由有限quadrature接受，也不被否旗標否定。

查看答案與解釋

(25, 34, finite_quadrature_only, 否, named_fact:integral_R_phi=1, named_fact:E[Z]=0, named_fact:Var(Z)=1, finite_window_or_panel_count_does_not_prove_normalization_or_moments)

Wave6第八層核對=(25, 34, finite_quadrature_only, 否, named_fact:integral_R_phi=1, named_fact:E[Z]=0, named_fact:Var(Z)=1, finite_window_or_panel_count_does_not_prove_normalization_or_moments)；依完整明示資料獨立重算；Phi(0)=1/2與Phi(-x)=1-Phi(x)只由已宣告normalization及symmetry推出；Gaussian integral、mean 0與variance 1是named theorem facts；finite quadrature不證它們或非零CDF exact value。

本題逐步詳解（AI 二次覆核）

標準常態分布 · qf-mod45-normaldist-normalization-moment-boundary · candidate

原始題目 JSONL · 驗收報告