第 27292 題：有限變異數IID中央極限定理

原題與已知條件

明示無限IID、共同mu、0<sigma^2<infinity與sigma*sqrt(n)標準化責任。操作=clt-standardized-sum-moments；資料={"values":[-1,0],"probabilities":[[1,2],[1,2]],"process_certificate":"infinite_iid","n":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。

• G001 — 明示無限IID、共同mu、0<sigma^2<infinity與sigma*sqrt(n)標準化責任。操作=clt-standardized-sum-moments；資料={"values":[-1,0],"probabilities":[[1,2],[1,2]],"process_certificate":"infinite_iid","n":4}。請依 exact_ordered_tuple_declared_by_interface 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-central-limit-theorem-finite-variance-iid

逐步推導

source roles = [clt-standardized-sum-moments]

derived = [[n, sum_distribution, total_probability, sum_mean, sum_variance, n_times_mean, n_times_variance, standardized_mean, standardized_variance, standardization_description, finite_scope]]

source roles = [infinite_iid]

derived = [ASSUME infinite sequence is IID with this common marginal]

CLT ORIGINAL common marginal ORIGINAL law mass sum0 := sum(0, 1/2)

CLT ORIGINAL common marginal ORIGINAL law mass sum0 = 1/2

CLT ORIGINAL common marginal ORIGINAL law mass sum1 := sum(1/2, 1/2)

CLT ORIGINAL common marginal ORIGINAL law mass sum1 = 1

source roles = [CLT ORIGINAL common marginal, [-1, 0], [[1, 2], [1, 2]]]

derived = [[[-1, 1/2], [0, 1/2]], 1]

CLT ORIGINAL common marginal ORIGINAL value times probability0 := mul(-1, 1/2)

CLT ORIGINAL common marginal ORIGINAL value times probability0 = -1/2

CLT ORIGINAL common marginal ORIGINAL ascending mean sum0 := sum(0, -1/2)

CLT ORIGINAL common marginal ORIGINAL ascending mean sum0 = -1/2

CLT ORIGINAL common marginal exact value square0 := mul(-1, -1)

CLT ORIGINAL common marginal exact value square0 = 1

CLT ORIGINAL common marginal ORIGINAL square times probability0 := mul(1, 1/2)

CLT ORIGINAL common marginal ORIGINAL square times probability0 = 1/2

CLT ORIGINAL common marginal ORIGINAL ascending second moment sum0 := sum(0, 1/2)

CLT ORIGINAL common marginal ORIGINAL ascending second moment sum0 = 1/2

CLT ORIGINAL common marginal ORIGINAL value times probability1 := mul(0, 1/2)

CLT ORIGINAL common marginal ORIGINAL value times probability1 = 0

CLT ORIGINAL common marginal ORIGINAL ascending mean sum1 := sum(-1/2, 0)

CLT ORIGINAL common marginal ORIGINAL ascending mean sum1 = -1/2

CLT ORIGINAL common marginal exact value square1 := mul(0, 0)

CLT ORIGINAL common marginal exact value square1 = 0

CLT ORIGINAL common marginal ORIGINAL square times probability1 := mul(0, 1/2)

CLT ORIGINAL common marginal ORIGINAL square times probability1 = 0

CLT ORIGINAL common marginal ORIGINAL ascending second moment sum1 := sum(1/2, 0)

CLT ORIGINAL common marginal ORIGINAL ascending second moment sum1 = 1/2

CLT ORIGINAL common marginal ORIGINAL mean square := mul(-1/2, -1/2)

CLT ORIGINAL common marginal ORIGINAL mean square = 1/4

CLT ORIGINAL common marginal ORIGINAL second minus mean square := sub(1/2, 1/4)

CLT ORIGINAL common marginal ORIGINAL second minus mean square = 1/4

source roles = [CLT ORIGINAL common marginal, [[-1, 1/2], [0, 1/2]]]

derived = [-1/2, 1/2, 1/4]

CLT joint ORIGINAL word0 sum := sum(-1, -1, -1, -1)

CLT joint ORIGINAL word0 sum = -4

CLT joint ORIGINAL word0 probability factor0 := mul(1, 1/2)

CLT joint ORIGINAL word0 probability factor0 = 1/2

CLT joint ORIGINAL word0 probability factor1 := mul(1/2, 1/2)

CLT joint ORIGINAL word0 probability factor1 = 1/4

CLT joint ORIGINAL word0 probability factor2 := mul(1/4, 1/2)

CLT joint ORIGINAL word0 probability factor2 = 1/8

CLT joint ORIGINAL word0 probability factor3 := mul(1/8, 1/2)

CLT joint ORIGINAL word0 probability factor3 = 1/16

CLT joint ORIGINAL sum-4 bucket word0 := sum(0, 1/16)

CLT joint ORIGINAL sum-4 bucket word0 = 1/16

source roles = [4, 0, [-1, -1, -1, -1], [1/2, 1/2, 1/2, 1/2]]

derived = [-4, 1/16, 0, 1/16]

CLT joint ORIGINAL word1 sum := sum(-1, -1, -1, 0)

CLT joint ORIGINAL word1 sum = -3

CLT joint ORIGINAL word1 probability factor0 := mul(1, 1/2)

CLT joint ORIGINAL word1 probability factor0 = 1/2

CLT joint ORIGINAL word1 probability factor1 := mul(1/2, 1/2)

CLT joint ORIGINAL word1 probability factor1 = 1/4

CLT joint ORIGINAL word1 probability factor2 := mul(1/4, 1/2)

CLT joint ORIGINAL word1 probability factor2 = 1/8

CLT joint ORIGINAL word1 probability factor3 := mul(1/8, 1/2)

CLT joint ORIGINAL word1 probability factor3 = 1/16

CLT joint ORIGINAL sum-3 bucket word1 := sum(0, 1/16)

CLT joint ORIGINAL sum-3 bucket word1 = 1/16

source roles = [4, 1, [-1, -1, -1, 0], [1/2, 1/2, 1/2, 1/2]]

derived = [-3, 1/16, 0, 1/16]

CLT joint ORIGINAL word2 sum := sum(-1, -1, 0, -1)

CLT joint ORIGINAL word2 sum = -3

CLT joint ORIGINAL word2 probability factor0 := mul(1, 1/2)

CLT joint ORIGINAL word2 probability factor0 = 1/2

CLT joint ORIGINAL word2 probability factor1 := mul(1/2, 1/2)

CLT joint ORIGINAL word2 probability factor1 = 1/4

CLT joint ORIGINAL word2 probability factor2 := mul(1/4, 1/2)

CLT joint ORIGINAL word2 probability factor2 = 1/8

CLT joint ORIGINAL word2 probability factor3 := mul(1/8, 1/2)

CLT joint ORIGINAL word2 probability factor3 = 1/16

CLT joint ORIGINAL sum-3 bucket word2 := sum(1/16, 1/16)

CLT joint ORIGINAL sum-3 bucket word2 = 1/8

source roles = [4, 2, [-1, -1, 0, -1], [1/2, 1/2, 1/2, 1/2]]

derived = [-3, 1/16, 1/16, 1/8]

CLT joint ORIGINAL word3 sum := sum(-1, -1, 0, 0)

CLT joint ORIGINAL word3 sum = -2

CLT joint ORIGINAL word3 probability factor0 := mul(1, 1/2)

CLT joint ORIGINAL word3 probability factor0 = 1/2

CLT joint ORIGINAL word3 probability factor1 := mul(1/2, 1/2)

CLT joint ORIGINAL word3 probability factor1 = 1/4

CLT joint ORIGINAL word3 probability factor2 := mul(1/4, 1/2)

CLT joint ORIGINAL word3 probability factor2 = 1/8

CLT joint ORIGINAL word3 probability factor3 := mul(1/8, 1/2)

CLT joint ORIGINAL word3 probability factor3 = 1/16

CLT joint ORIGINAL sum-2 bucket word3 := sum(0, 1/16)

CLT joint ORIGINAL sum-2 bucket word3 = 1/16

source roles = [4, 3, [-1, -1, 0, 0], [1/2, 1/2, 1/2, 1/2]]

derived = [-2, 1/16, 0, 1/16]

CLT joint ORIGINAL word4 sum := sum(-1, 0, -1, -1)

CLT joint ORIGINAL word4 sum = -3

CLT joint ORIGINAL word4 probability factor0 := mul(1, 1/2)

CLT joint ORIGINAL word4 probability factor0 = 1/2

CLT joint ORIGINAL word4 probability factor1 := mul(1/2, 1/2)

CLT joint ORIGINAL word4 probability factor1 = 1/4

CLT joint ORIGINAL word4 probability factor2 := mul(1/4, 1/2)

CLT joint ORIGINAL word4 probability factor2 = 1/8

CLT joint ORIGINAL word4 probability factor3 := mul(1/8, 1/2)

CLT joint ORIGINAL word4 probability factor3 = 1/16

CLT joint ORIGINAL sum-3 bucket word4 := sum(1/8, 1/16)

CLT joint ORIGINAL sum-3 bucket word4 = 3/16

source roles = [4, 4, [-1, 0, -1, -1], [1/2, 1/2, 1/2, 1/2]]

derived = [-3, 1/16, 1/8, 3/16]

CLT joint ORIGINAL word5 sum := sum(-1, 0, -1, 0)

CLT joint ORIGINAL word5 sum = -2

CLT joint ORIGINAL word5 probability factor0 := mul(1, 1/2)

CLT joint ORIGINAL word5 probability factor0 = 1/2

CLT joint ORIGINAL word5 probability factor1 := mul(1/2, 1/2)

CLT joint ORIGINAL word5 probability factor1 = 1/4

CLT joint ORIGINAL word5 probability factor2 := mul(1/4, 1/2)

CLT joint ORIGINAL word5 probability factor2 = 1/8

CLT joint ORIGINAL word5 probability factor3 := mul(1/8, 1/2)

CLT joint ORIGINAL word5 probability factor3 = 1/16

CLT joint ORIGINAL sum-2 bucket word5 := sum(1/16, 1/16)

CLT joint ORIGINAL sum-2 bucket word5 = 1/8

source roles = [4, 5, [-1, 0, -1, 0], [1/2, 1/2, 1/2, 1/2]]

derived = [-2, 1/16, 1/16, 1/8]

CLT joint ORIGINAL word6 sum := sum(-1, 0, 0, -1)

CLT joint ORIGINAL word6 sum = -2

CLT joint ORIGINAL word6 probability factor0 := mul(1, 1/2)

CLT joint ORIGINAL word6 probability factor0 = 1/2

CLT joint ORIGINAL word6 probability factor1 := mul(1/2, 1/2)

CLT joint ORIGINAL word6 probability factor1 = 1/4

CLT joint ORIGINAL word6 probability factor2 := mul(1/4, 1/2)

CLT joint ORIGINAL word6 probability factor2 = 1/8

CLT joint ORIGINAL word6 probability factor3 := mul(1/8, 1/2)

CLT joint ORIGINAL word6 probability factor3 = 1/16

CLT joint ORIGINAL sum-2 bucket word6 := sum(1/8, 1/16)

CLT joint ORIGINAL sum-2 bucket word6 = 3/16

source roles = [4, 6, [-1, 0, 0, -1], [1/2, 1/2, 1/2, 1/2]]

derived = [-2, 1/16, 1/8, 3/16]

CLT joint ORIGINAL word7 sum := sum(-1, 0, 0, 0)

CLT joint ORIGINAL word7 sum = -1

CLT joint ORIGINAL word7 probability factor0 := mul(1, 1/2)

CLT joint ORIGINAL word7 probability factor0 = 1/2

CLT joint ORIGINAL word7 probability factor1 := mul(1/2, 1/2)

CLT joint ORIGINAL word7 probability factor1 = 1/4

CLT joint ORIGINAL word7 probability factor2 := mul(1/4, 1/2)

CLT joint ORIGINAL word7 probability factor2 = 1/8

CLT joint ORIGINAL word7 probability factor3 := mul(1/8, 1/2)

CLT joint ORIGINAL word7 probability factor3 = 1/16

CLT joint ORIGINAL sum-1 bucket word7 := sum(0, 1/16)

CLT joint ORIGINAL sum-1 bucket word7 = 1/16

source roles = [4, 7, [-1, 0, 0, 0], [1/2, 1/2, 1/2, 1/2]]

derived = [-1, 1/16, 0, 1/16]

CLT joint ORIGINAL word8 sum := sum(0, -1, -1, -1)

CLT joint ORIGINAL word8 sum = -3

CLT joint ORIGINAL word8 probability factor0 := mul(1, 1/2)

CLT joint ORIGINAL word8 probability factor0 = 1/2

CLT joint ORIGINAL word8 probability factor1 := mul(1/2, 1/2)

CLT joint ORIGINAL word8 probability factor1 = 1/4

CLT joint ORIGINAL word8 probability factor2 := mul(1/4, 1/2)

CLT joint ORIGINAL word8 probability factor2 = 1/8

CLT joint ORIGINAL word8 probability factor3 := mul(1/8, 1/2)

CLT joint ORIGINAL word8 probability factor3 = 1/16

CLT joint ORIGINAL sum-3 bucket word8 := sum(3/16, 1/16)

CLT joint ORIGINAL sum-3 bucket word8 = 1/4

source roles = [4, 8, [0, -1, -1, -1], [1/2, 1/2, 1/2, 1/2]]

derived = [-3, 1/16, 3/16, 1/4]

CLT joint ORIGINAL word9 sum := sum(0, -1, -1, 0)

CLT joint ORIGINAL word9 sum = -2

CLT joint ORIGINAL word9 probability factor0 := mul(1, 1/2)

CLT joint ORIGINAL word9 probability factor0 = 1/2

CLT joint ORIGINAL word9 probability factor1 := mul(1/2, 1/2)

CLT joint ORIGINAL word9 probability factor1 = 1/4

CLT joint ORIGINAL word9 probability factor2 := mul(1/4, 1/2)

CLT joint ORIGINAL word9 probability factor2 = 1/8

CLT joint ORIGINAL word9 probability factor3 := mul(1/8, 1/2)

CLT joint ORIGINAL word9 probability factor3 = 1/16

CLT joint ORIGINAL sum-2 bucket word9 := sum(3/16, 1/16)

CLT joint ORIGINAL sum-2 bucket word9 = 1/4

source roles = [4, 9, [0, -1, -1, 0], [1/2, 1/2, 1/2, 1/2]]

derived = [-2, 1/16, 3/16, 1/4]

CLT joint ORIGINAL word10 sum := sum(0, -1, 0, -1)

CLT joint ORIGINAL word10 sum = -2

CLT joint ORIGINAL word10 probability factor0 := mul(1, 1/2)

CLT joint ORIGINAL word10 probability factor0 = 1/2

CLT joint ORIGINAL word10 probability factor1 := mul(1/2, 1/2)

CLT joint ORIGINAL word10 probability factor1 = 1/4

CLT joint ORIGINAL word10 probability factor2 := mul(1/4, 1/2)

CLT joint ORIGINAL word10 probability factor2 = 1/8

CLT joint ORIGINAL word10 probability factor3 := mul(1/8, 1/2)

CLT joint ORIGINAL word10 probability factor3 = 1/16

CLT joint ORIGINAL sum-2 bucket word10 := sum(1/4, 1/16)

CLT joint ORIGINAL sum-2 bucket word10 = 5/16

source roles = [4, 10, [0, -1, 0, -1], [1/2, 1/2, 1/2, 1/2]]

derived = [-2, 1/16, 1/4, 5/16]

CLT joint ORIGINAL word11 sum := sum(0, -1, 0, 0)

CLT joint ORIGINAL word11 sum = -1

CLT joint ORIGINAL word11 probability factor0 := mul(1, 1/2)

CLT joint ORIGINAL word11 probability factor0 = 1/2

CLT joint ORIGINAL word11 probability factor1 := mul(1/2, 1/2)

CLT joint ORIGINAL word11 probability factor1 = 1/4

CLT joint ORIGINAL word11 probability factor2 := mul(1/4, 1/2)

CLT joint ORIGINAL word11 probability factor2 = 1/8

CLT joint ORIGINAL word11 probability factor3 := mul(1/8, 1/2)

CLT joint ORIGINAL word11 probability factor3 = 1/16

CLT joint ORIGINAL sum-1 bucket word11 := sum(1/16, 1/16)

CLT joint ORIGINAL sum-1 bucket word11 = 1/8

source roles = [4, 11, [0, -1, 0, 0], [1/2, 1/2, 1/2, 1/2]]

derived = [-1, 1/16, 1/16, 1/8]

CLT joint ORIGINAL word12 sum := sum(0, 0, -1, -1)

CLT joint ORIGINAL word12 sum = -2

CLT joint ORIGINAL word12 probability factor0 := mul(1, 1/2)

CLT joint ORIGINAL word12 probability factor0 = 1/2

CLT joint ORIGINAL word12 probability factor1 := mul(1/2, 1/2)

CLT joint ORIGINAL word12 probability factor1 = 1/4

CLT joint ORIGINAL word12 probability factor2 := mul(1/4, 1/2)

CLT joint ORIGINAL word12 probability factor2 = 1/8

CLT joint ORIGINAL word12 probability factor3 := mul(1/8, 1/2)

CLT joint ORIGINAL word12 probability factor3 = 1/16

CLT joint ORIGINAL sum-2 bucket word12 := sum(5/16, 1/16)

CLT joint ORIGINAL sum-2 bucket word12 = 3/8

source roles = [4, 12, [0, 0, -1, -1], [1/2, 1/2, 1/2, 1/2]]

derived = [-2, 1/16, 5/16, 3/8]

CLT joint ORIGINAL word13 sum := sum(0, 0, -1, 0)

CLT joint ORIGINAL word13 sum = -1

CLT joint ORIGINAL word13 probability factor0 := mul(1, 1/2)

CLT joint ORIGINAL word13 probability factor0 = 1/2

CLT joint ORIGINAL word13 probability factor1 := mul(1/2, 1/2)

CLT joint ORIGINAL word13 probability factor1 = 1/4

CLT joint ORIGINAL word13 probability factor2 := mul(1/4, 1/2)

CLT joint ORIGINAL word13 probability factor2 = 1/8

CLT joint ORIGINAL word13 probability factor3 := mul(1/8, 1/2)

CLT joint ORIGINAL word13 probability factor3 = 1/16

CLT joint ORIGINAL sum-1 bucket word13 := sum(1/8, 1/16)

CLT joint ORIGINAL sum-1 bucket word13 = 3/16

source roles = [4, 13, [0, 0, -1, 0], [1/2, 1/2, 1/2, 1/2]]

derived = [-1, 1/16, 1/8, 3/16]

CLT joint ORIGINAL word14 sum := sum(0, 0, 0, -1)

CLT joint ORIGINAL word14 sum = -1

CLT joint ORIGINAL word14 probability factor0 := mul(1, 1/2)

CLT joint ORIGINAL word14 probability factor0 = 1/2

CLT joint ORIGINAL word14 probability factor1 := mul(1/2, 1/2)

CLT joint ORIGINAL word14 probability factor1 = 1/4

CLT joint ORIGINAL word14 probability factor2 := mul(1/4, 1/2)

CLT joint ORIGINAL word14 probability factor2 = 1/8

CLT joint ORIGINAL word14 probability factor3 := mul(1/8, 1/2)

CLT joint ORIGINAL word14 probability factor3 = 1/16

CLT joint ORIGINAL sum-1 bucket word14 := sum(3/16, 1/16)

CLT joint ORIGINAL sum-1 bucket word14 = 1/4

source roles = [4, 14, [0, 0, 0, -1], [1/2, 1/2, 1/2, 1/2]]

derived = [-1, 1/16, 3/16, 1/4]

CLT joint ORIGINAL word15 sum := sum(0, 0, 0, 0)

CLT joint ORIGINAL word15 sum = 0

CLT joint ORIGINAL word15 probability factor0 := mul(1, 1/2)

CLT joint ORIGINAL word15 probability factor0 = 1/2

CLT joint ORIGINAL word15 probability factor1 := mul(1/2, 1/2)

CLT joint ORIGINAL word15 probability factor1 = 1/4

CLT joint ORIGINAL word15 probability factor2 := mul(1/4, 1/2)

CLT joint ORIGINAL word15 probability factor2 = 1/8

CLT joint ORIGINAL word15 probability factor3 := mul(1/8, 1/2)

CLT joint ORIGINAL word15 probability factor3 = 1/16

CLT joint ORIGINAL sum0 bucket word15 := sum(0, 1/16)

CLT joint ORIGINAL sum0 bucket word15 = 1/16

source roles = [4, 15, [0, 0, 0, 0], [1/2, 1/2, 1/2, 1/2]]

derived = [0, 1/16, 0, 1/16]

CLT SUM ORIGINAL total distribution mass0 := sum(0, 1/16)

CLT SUM ORIGINAL total distribution mass0 = 1/16

CLT SUM ORIGINAL total distribution mass1 := sum(1/16, 1/4)

CLT SUM ORIGINAL total distribution mass1 = 5/16

CLT SUM ORIGINAL total distribution mass2 := sum(5/16, 3/8)

CLT SUM ORIGINAL total distribution mass2 = 11/16

CLT SUM ORIGINAL total distribution mass3 := sum(11/16, 1/4)

CLT SUM ORIGINAL total distribution mass3 = 15/16

CLT SUM ORIGINAL total distribution mass4 := sum(15/16, 1/16)

CLT SUM ORIGINAL total distribution mass4 = 1

CLT SUM ORIGINAL value times probability0 := mul(-4, 1/16)

CLT SUM ORIGINAL value times probability0 = -1/4

CLT SUM ORIGINAL ascending mean sum0 := sum(0, -1/4)

CLT SUM ORIGINAL ascending mean sum0 = -1/4

CLT SUM exact value square0 := mul(-4, -4)

CLT SUM exact value square0 = 16

CLT SUM ORIGINAL square times probability0 := mul(16, 1/16)

CLT SUM ORIGINAL square times probability0 = 1

CLT SUM ORIGINAL ascending second moment sum0 := sum(0, 1)

CLT SUM ORIGINAL ascending second moment sum0 = 1

CLT SUM ORIGINAL value times probability1 := mul(-3, 1/4)

CLT SUM ORIGINAL value times probability1 = -3/4

CLT SUM ORIGINAL ascending mean sum1 := sum(-1/4, -3/4)

CLT SUM ORIGINAL ascending mean sum1 = -1

CLT SUM exact value square1 := mul(-3, -3)

CLT SUM exact value square1 = 9

CLT SUM ORIGINAL square times probability1 := mul(9, 1/4)

CLT SUM ORIGINAL square times probability1 = 9/4

CLT SUM ORIGINAL ascending second moment sum1 := sum(1, 9/4)

CLT SUM ORIGINAL ascending second moment sum1 = 13/4

CLT SUM ORIGINAL value times probability2 := mul(-2, 3/8)

CLT SUM ORIGINAL value times probability2 = -3/4

CLT SUM ORIGINAL ascending mean sum2 := sum(-1, -3/4)

CLT SUM ORIGINAL ascending mean sum2 = -7/4

CLT SUM exact value square2 := mul(-2, -2)

CLT SUM exact value square2 = 4

CLT SUM ORIGINAL square times probability2 := mul(4, 3/8)

CLT SUM ORIGINAL square times probability2 = 3/2

CLT SUM ORIGINAL ascending second moment sum2 := sum(13/4, 3/2)

CLT SUM ORIGINAL ascending second moment sum2 = 19/4

CLT SUM ORIGINAL value times probability3 := mul(-1, 1/4)

CLT SUM ORIGINAL value times probability3 = -1/4

CLT SUM ORIGINAL ascending mean sum3 := sum(-7/4, -1/4)

CLT SUM ORIGINAL ascending mean sum3 = -2

CLT SUM exact value square3 := mul(-1, -1)

CLT SUM exact value square3 = 1

CLT SUM ORIGINAL square times probability3 := mul(1, 1/4)

CLT SUM ORIGINAL square times probability3 = 1/4

CLT SUM ORIGINAL ascending second moment sum3 := sum(19/4, 1/4)

CLT SUM ORIGINAL ascending second moment sum3 = 5

CLT SUM ORIGINAL value times probability4 := mul(0, 1/16)

CLT SUM ORIGINAL value times probability4 = 0

CLT SUM ORIGINAL ascending mean sum4 := sum(-2, 0)

CLT SUM ORIGINAL ascending mean sum4 = -2

CLT SUM exact value square4 := mul(0, 0)

CLT SUM exact value square4 = 0

CLT SUM ORIGINAL square times probability4 := mul(0, 1/16)

CLT SUM ORIGINAL square times probability4 = 0

CLT SUM ORIGINAL ascending second moment sum4 := sum(5, 0)

CLT SUM ORIGINAL ascending second moment sum4 = 5

CLT SUM ORIGINAL mean square := mul(-2, -2)

CLT SUM ORIGINAL mean square = 4

CLT SUM ORIGINAL second minus mean square := sub(5, 4)

CLT SUM ORIGINAL second minus mean square = 1

CLT ORIGINAL n times common mean := mul(4, -1/2)

CLT ORIGINAL n times common mean = -2

CLT ORIGINAL n times common variance := mul(4, 1/4)

CLT ORIGINAL n times common variance = 1

source roles = [4, -1/2, 1/4, [(-4, 1/16), (-3, 1/4), (-2, 3/8), (-1, 1/4), (0, 1/16)], -2, 1]

derived = [1, -2, 1, 0, 1, Z_n=(S_n-n*mu)/(sigma*sqrt(n))]

逐步反向核對

• S165 · premise_and_result_audit — 回查原始完整資料、步驟操作數與本步前提；有限核對、條件式定理引用與一般形式證明分開。
• S164 · premise_and_result_audit — 以精確有理數重算 CLT ORIGINAL n times common variance，不以小數近似替代。
• S163 · premise_and_result_audit — 以精確有理數重算 CLT ORIGINAL n times common mean，不以小數近似替代。
• S162 · premise_and_result_audit — 以精確有理數重算 CLT SUM ORIGINAL second minus mean square，不以小數近似替代。
• S161 · premise_and_result_audit — 以精確有理數重算 CLT SUM ORIGINAL mean square，不以小數近似替代。
• S160 · premise_and_result_audit — 以精確有理數重算 CLT SUM ORIGINAL ascending second moment sum4，不以小數近似替代。
• S159 · premise_and_result_audit — 以精確有理數重算 CLT SUM ORIGINAL square times probability4，不以小數近似替代。
• S158 · premise_and_result_audit — 以精確有理數重算 CLT SUM exact value square4，不以小數近似替代。
• S157 · premise_and_result_audit — 以精確有理數重算 CLT SUM ORIGINAL ascending mean sum4，不以小數近似替代。
• S156 · premise_and_result_audit — 以精確有理數重算 CLT SUM ORIGINAL value times probability4，不以小數近似替代。
• S155 · premise_and_result_audit — 以精確有理數重算 CLT SUM ORIGINAL ascending second moment sum3，不以小數近似替代。
• S154 · premise_and_result_audit — 以精確有理數重算 CLT SUM ORIGINAL square times probability3，不以小數近似替代。
• S153 · premise_and_result_audit — 以精確有理數重算 CLT SUM exact value square3，不以小數近似替代。
• S152 · premise_and_result_audit — 以精確有理數重算 CLT SUM ORIGINAL ascending mean sum3，不以小數近似替代。
• S151 · premise_and_result_audit — 以精確有理數重算 CLT SUM ORIGINAL value times probability3，不以小數近似替代。
• S150 · premise_and_result_audit — 以精確有理數重算 CLT SUM ORIGINAL ascending second moment sum2，不以小數近似替代。
• S149 · premise_and_result_audit — 以精確有理數重算 CLT SUM ORIGINAL square times probability2，不以小數近似替代。
• S148 · premise_and_result_audit — 以精確有理數重算 CLT SUM exact value square2，不以小數近似替代。
• S147 · premise_and_result_audit — 以精確有理數重算 CLT SUM ORIGINAL ascending mean sum2，不以小數近似替代。
• S146 · premise_and_result_audit — 以精確有理數重算 CLT SUM ORIGINAL value times probability2，不以小數近似替代。
• S145 · premise_and_result_audit — 以精確有理數重算 CLT SUM ORIGINAL ascending second moment sum1，不以小數近似替代。
• S144 · premise_and_result_audit — 以精確有理數重算 CLT SUM ORIGINAL square times probability1，不以小數近似替代。
• S143 · premise_and_result_audit — 以精確有理數重算 CLT SUM exact value square1，不以小數近似替代。
• S142 · premise_and_result_audit — 以精確有理數重算 CLT SUM ORIGINAL ascending mean sum1，不以小數近似替代。
• S141 · premise_and_result_audit — 以精確有理數重算 CLT SUM ORIGINAL value times probability1，不以小數近似替代。
• S140 · premise_and_result_audit — 以精確有理數重算 CLT SUM ORIGINAL ascending second moment sum0，不以小數近似替代。
• S139 · premise_and_result_audit — 以精確有理數重算 CLT SUM ORIGINAL square times probability0，不以小數近似替代。
• S138 · premise_and_result_audit — 以精確有理數重算 CLT SUM exact value square0，不以小數近似替代。
• S137 · premise_and_result_audit — 以精確有理數重算 CLT SUM ORIGINAL ascending mean sum0，不以小數近似替代。
• S136 · premise_and_result_audit — 以精確有理數重算 CLT SUM ORIGINAL value times probability0，不以小數近似替代。
• S135 · premise_and_result_audit — 以精確有理數重算 CLT SUM ORIGINAL total distribution mass4，不以小數近似替代。
• S134 · premise_and_result_audit — 以精確有理數重算 CLT SUM ORIGINAL total distribution mass3，不以小數近似替代。
• S133 · premise_and_result_audit — 以精確有理數重算 CLT SUM ORIGINAL total distribution mass2，不以小數近似替代。
• S132 · premise_and_result_audit — 以精確有理數重算 CLT SUM ORIGINAL total distribution mass1，不以小數近似替代。
• S131 · premise_and_result_audit — 以精確有理數重算 CLT SUM ORIGINAL total distribution mass0，不以小數近似替代。
• S130 · premise_and_result_audit — 回查原始完整資料、步驟操作數與本步前提；有限核對、條件式定理引用與一般形式證明分開。
• S129 · premise_and_result_audit — 以精確有理數重算 CLT joint ORIGINAL sum0 bucket word15，不以小數近似替代。
• S128 · premise_and_result_audit — 以精確有理數重算 CLT joint ORIGINAL word15 probability factor3，不以小數近似替代。
• S127 · premise_and_result_audit — 以精確有理數重算 CLT joint ORIGINAL word15 probability factor2，不以小數近似替代。
• S126 · premise_and_result_audit — 以精確有理數重算 CLT joint ORIGINAL word15 probability factor1，不以小數近似替代。
• S125 · premise_and_result_audit — 以精確有理數重算 CLT joint ORIGINAL word15 probability factor0，不以小數近似替代。
• S124 · premise_and_result_audit — 以精確有理數重算 CLT joint ORIGINAL word15 sum，不以小數近似替代。
• S123 · premise_and_result_audit — 回查原始完整資料、步驟操作數與本步前提；有限核對、條件式定理引用與一般形式證明分開。
• S122 · premise_and_result_audit — 以精確有理數重算 CLT joint ORIGINAL sum-1 bucket word14，不以小數近似替代。
• S121 · premise_and_result_audit — 以精確有理數重算 CLT joint ORIGINAL word14 probability factor3，不以小數近似替代。
• S120 · premise_and_result_audit — 以精確有理數重算 CLT joint ORIGINAL word14 probability factor2，不以小數近似替代。
• S119 · premise_and_result_audit — 以精確有理數重算 CLT joint ORIGINAL word14 probability factor1，不以小數近似替代。
• S118 · premise_and_result_audit — 以精確有理數重算 CLT joint ORIGINAL word14 probability factor0，不以小數近似替代。
• S117 · premise_and_result_audit — 以精確有理數重算 CLT joint ORIGINAL word14 sum，不以小數近似替代。
• S116 · premise_and_result_audit — 回查原始完整資料、步驟操作數與本步前提；有限核對、條件式定理引用與一般形式證明分開。
• S115 · premise_and_result_audit — 以精確有理數重算 CLT joint ORIGINAL sum-1 bucket word13，不以小數近似替代。
• S114 · premise_and_result_audit — 以精確有理數重算 CLT joint ORIGINAL word13 probability factor3，不以小數近似替代。
• S113 · premise_and_result_audit — 以精確有理數重算 CLT joint ORIGINAL word13 probability factor2，不以小數近似替代。
• S112 · premise_and_result_audit — 以精確有理數重算 CLT joint ORIGINAL word13 probability factor1，不以小數近似替代。
• S111 · premise_and_result_audit — 以精確有理數重算 CLT joint ORIGINAL word13 probability factor0，不以小數近似替代。
• S110 · premise_and_result_audit — 以精確有理數重算 CLT joint ORIGINAL word13 sum，不以小數近似替代。
• S109 · premise_and_result_audit — 回查原始完整資料、步驟操作數與本步前提；有限核對、條件式定理引用與一般形式證明分開。
• S108 · premise_and_result_audit — 以精確有理數重算 CLT joint ORIGINAL sum-2 bucket word12，不以小數近似替代。
• S107 · premise_and_result_audit — 以精確有理數重算 CLT joint ORIGINAL word12 probability factor3，不以小數近似替代。
• S106 · premise_and_result_audit — 以精確有理數重算 CLT joint ORIGINAL word12 probability factor2，不以小數近似替代。
• S105 · premise_and_result_audit — 以精確有理數重算 CLT joint ORIGINAL word12 probability factor1，不以小數近似替代。
• S104 · premise_and_result_audit — 以精確有理數重算 CLT joint ORIGINAL word12 probability factor0，不以小數近似替代。
• S103 · premise_and_result_audit — 以精確有理數重算 CLT joint ORIGINAL word12 sum，不以小數近似替代。
• S102 · premise_and_result_audit — 回查原始完整資料、步驟操作數與本步前提；有限核對、條件式定理引用與一般形式證明分開。
• S101 · premise_and_result_audit — 以精確有理數重算 CLT joint ORIGINAL sum-1 bucket word11，不以小數近似替代。
• S100 · premise_and_result_audit — 以精確有理數重算 CLT joint ORIGINAL word11 probability factor3，不以小數近似替代。
• S099 · premise_and_result_audit — 以精確有理數重算 CLT joint ORIGINAL word11 probability factor2，不以小數近似替代。
• S098 · premise_and_result_audit — 以精確有理數重算 CLT joint ORIGINAL word11 probability factor1，不以小數近似替代。
• S097 · premise_and_result_audit — 以精確有理數重算 CLT joint ORIGINAL word11 probability factor0，不以小數近似替代。
• S096 · premise_and_result_audit — 以精確有理數重算 CLT joint ORIGINAL word11 sum，不以小數近似替代。
• S095 · premise_and_result_audit — 回查原始完整資料、步驟操作數與本步前提；有限核對、條件式定理引用與一般形式證明分開。
• S094 · premise_and_result_audit — 以精確有理數重算 CLT joint ORIGINAL sum-2 bucket word10，不以小數近似替代。
• S093 · premise_and_result_audit — 以精確有理數重算 CLT joint ORIGINAL word10 probability factor3，不以小數近似替代。
• S092 · premise_and_result_audit — 以精確有理數重算 CLT joint ORIGINAL word10 probability factor2，不以小數近似替代。
• S091 · premise_and_result_audit — 以精確有理數重算 CLT joint ORIGINAL word10 probability factor1，不以小數近似替代。
• S090 · premise_and_result_audit — 以精確有理數重算 CLT joint ORIGINAL word10 probability factor0，不以小數近似替代。
• S089 · premise_and_result_audit — 以精確有理數重算 CLT joint ORIGINAL word10 sum，不以小數近似替代。
• S088 · premise_and_result_audit — 回查原始完整資料、步驟操作數與本步前提；有限核對、條件式定理引用與一般形式證明分開。
• S087 · premise_and_result_audit — 以精確有理數重算 CLT joint ORIGINAL sum-2 bucket word9，不以小數近似替代。
• S086 · premise_and_result_audit — 以精確有理數重算 CLT joint ORIGINAL word9 probability factor3，不以小數近似替代。
• S085 · premise_and_result_audit — 以精確有理數重算 CLT joint ORIGINAL word9 probability factor2，不以小數近似替代。
• S084 · premise_and_result_audit — 以精確有理數重算 CLT joint ORIGINAL word9 probability factor1，不以小數近似替代。
• S083 · premise_and_result_audit — 以精確有理數重算 CLT joint ORIGINAL word9 probability factor0，不以小數近似替代。
• S082 · premise_and_result_audit — 以精確有理數重算 CLT joint ORIGINAL word9 sum，不以小數近似替代。
• S081 · premise_and_result_audit — 回查原始完整資料、步驟操作數與本步前提；有限核對、條件式定理引用與一般形式證明分開。
• S080 · premise_and_result_audit — 以精確有理數重算 CLT joint ORIGINAL sum-3 bucket word8，不以小數近似替代。
• S079 · premise_and_result_audit — 以精確有理數重算 CLT joint ORIGINAL word8 probability factor3，不以小數近似替代。
• S078 · premise_and_result_audit — 以精確有理數重算 CLT joint ORIGINAL word8 probability factor2，不以小數近似替代。
• S077 · premise_and_result_audit — 以精確有理數重算 CLT joint ORIGINAL word8 probability factor1，不以小數近似替代。
• S076 · premise_and_result_audit — 以精確有理數重算 CLT joint ORIGINAL word8 probability factor0，不以小數近似替代。
• S075 · premise_and_result_audit — 以精確有理數重算 CLT joint ORIGINAL word8 sum，不以小數近似替代。
• S074 · premise_and_result_audit — 回查原始完整資料、步驟操作數與本步前提；有限核對、條件式定理引用與一般形式證明分開。
• S073 · premise_and_result_audit — 以精確有理數重算 CLT joint ORIGINAL sum-1 bucket word7，不以小數近似替代。
• S072 · premise_and_result_audit — 以精確有理數重算 CLT joint ORIGINAL word7 probability factor3，不以小數近似替代。
• S071 · premise_and_result_audit — 以精確有理數重算 CLT joint ORIGINAL word7 probability factor2，不以小數近似替代。
• S070 · premise_and_result_audit — 以精確有理數重算 CLT joint ORIGINAL word7 probability factor1，不以小數近似替代。
• S069 · premise_and_result_audit — 以精確有理數重算 CLT joint ORIGINAL word7 probability factor0，不以小數近似替代。
• S068 · premise_and_result_audit — 以精確有理數重算 CLT joint ORIGINAL word7 sum，不以小數近似替代。
• S067 · premise_and_result_audit — 回查原始完整資料、步驟操作數與本步前提；有限核對、條件式定理引用與一般形式證明分開。
• S066 · premise_and_result_audit — 以精確有理數重算 CLT joint ORIGINAL sum-2 bucket word6，不以小數近似替代。
• S065 · premise_and_result_audit — 以精確有理數重算 CLT joint ORIGINAL word6 probability factor3，不以小數近似替代。
• S064 · premise_and_result_audit — 以精確有理數重算 CLT joint ORIGINAL word6 probability factor2，不以小數近似替代。
• S063 · premise_and_result_audit — 以精確有理數重算 CLT joint ORIGINAL word6 probability factor1，不以小數近似替代。
• S062 · premise_and_result_audit — 以精確有理數重算 CLT joint ORIGINAL word6 probability factor0，不以小數近似替代。
• S061 · premise_and_result_audit — 以精確有理數重算 CLT joint ORIGINAL word6 sum，不以小數近似替代。
• S060 · premise_and_result_audit — 回查原始完整資料、步驟操作數與本步前提；有限核對、條件式定理引用與一般形式證明分開。
• S059 · premise_and_result_audit — 以精確有理數重算 CLT joint ORIGINAL sum-2 bucket word5，不以小數近似替代。
• S058 · premise_and_result_audit — 以精確有理數重算 CLT joint ORIGINAL word5 probability factor3，不以小數近似替代。
• S057 · premise_and_result_audit — 以精確有理數重算 CLT joint ORIGINAL word5 probability factor2，不以小數近似替代。
• S056 · premise_and_result_audit — 以精確有理數重算 CLT joint ORIGINAL word5 probability factor1，不以小數近似替代。
• S055 · premise_and_result_audit — 以精確有理數重算 CLT joint ORIGINAL word5 probability factor0，不以小數近似替代。
• S054 · premise_and_result_audit — 以精確有理數重算 CLT joint ORIGINAL word5 sum，不以小數近似替代。
• S053 · premise_and_result_audit — 回查原始完整資料、步驟操作數與本步前提；有限核對、條件式定理引用與一般形式證明分開。
• S052 · premise_and_result_audit — 以精確有理數重算 CLT joint ORIGINAL sum-3 bucket word4，不以小數近似替代。
• S051 · premise_and_result_audit — 以精確有理數重算 CLT joint ORIGINAL word4 probability factor3，不以小數近似替代。
• S050 · premise_and_result_audit — 以精確有理數重算 CLT joint ORIGINAL word4 probability factor2，不以小數近似替代。
• S049 · premise_and_result_audit — 以精確有理數重算 CLT joint ORIGINAL word4 probability factor1，不以小數近似替代。
• S048 · premise_and_result_audit — 以精確有理數重算 CLT joint ORIGINAL word4 probability factor0，不以小數近似替代。
• S047 · premise_and_result_audit — 以精確有理數重算 CLT joint ORIGINAL word4 sum，不以小數近似替代。
• S046 · premise_and_result_audit — 回查原始完整資料、步驟操作數與本步前提；有限核對、條件式定理引用與一般形式證明分開。
• S045 · premise_and_result_audit — 以精確有理數重算 CLT joint ORIGINAL sum-2 bucket word3，不以小數近似替代。
• S044 · premise_and_result_audit — 以精確有理數重算 CLT joint ORIGINAL word3 probability factor3，不以小數近似替代。
• S043 · premise_and_result_audit — 以精確有理數重算 CLT joint ORIGINAL word3 probability factor2，不以小數近似替代。
• S042 · premise_and_result_audit — 以精確有理數重算 CLT joint ORIGINAL word3 probability factor1，不以小數近似替代。
• S041 · premise_and_result_audit — 以精確有理數重算 CLT joint ORIGINAL word3 probability factor0，不以小數近似替代。
• S040 · premise_and_result_audit — 以精確有理數重算 CLT joint ORIGINAL word3 sum，不以小數近似替代。
• S039 · premise_and_result_audit — 回查原始完整資料、步驟操作數與本步前提；有限核對、條件式定理引用與一般形式證明分開。
• S038 · premise_and_result_audit — 以精確有理數重算 CLT joint ORIGINAL sum-3 bucket word2，不以小數近似替代。
• S037 · premise_and_result_audit — 以精確有理數重算 CLT joint ORIGINAL word2 probability factor3，不以小數近似替代。
• S036 · premise_and_result_audit — 以精確有理數重算 CLT joint ORIGINAL word2 probability factor2，不以小數近似替代。
• S035 · premise_and_result_audit — 以精確有理數重算 CLT joint ORIGINAL word2 probability factor1，不以小數近似替代。
• S034 · premise_and_result_audit — 以精確有理數重算 CLT joint ORIGINAL word2 probability factor0，不以小數近似替代。
• S033 · premise_and_result_audit — 以精確有理數重算 CLT joint ORIGINAL word2 sum，不以小數近似替代。
• S032 · premise_and_result_audit — 回查原始完整資料、步驟操作數與本步前提；有限核對、條件式定理引用與一般形式證明分開。
• S031 · premise_and_result_audit — 以精確有理數重算 CLT joint ORIGINAL sum-3 bucket word1，不以小數近似替代。
• S030 · premise_and_result_audit — 以精確有理數重算 CLT joint ORIGINAL word1 probability factor3，不以小數近似替代。
• S029 · premise_and_result_audit — 以精確有理數重算 CLT joint ORIGINAL word1 probability factor2，不以小數近似替代。
• S028 · premise_and_result_audit — 以精確有理數重算 CLT joint ORIGINAL word1 probability factor1，不以小數近似替代。
• S027 · premise_and_result_audit — 以精確有理數重算 CLT joint ORIGINAL word1 probability factor0，不以小數近似替代。
• S026 · premise_and_result_audit — 以精確有理數重算 CLT joint ORIGINAL word1 sum，不以小數近似替代。
• S025 · premise_and_result_audit — 回查原始完整資料、步驟操作數與本步前提；有限核對、條件式定理引用與一般形式證明分開。
• S024 · premise_and_result_audit — 以精確有理數重算 CLT joint ORIGINAL sum-4 bucket word0，不以小數近似替代。
• S023 · premise_and_result_audit — 以精確有理數重算 CLT joint ORIGINAL word0 probability factor3，不以小數近似替代。
• S022 · premise_and_result_audit — 以精確有理數重算 CLT joint ORIGINAL word0 probability factor2，不以小數近似替代。
• S021 · premise_and_result_audit — 以精確有理數重算 CLT joint ORIGINAL word0 probability factor1，不以小數近似替代。
• S020 · premise_and_result_audit — 以精確有理數重算 CLT joint ORIGINAL word0 probability factor0，不以小數近似替代。
• S019 · premise_and_result_audit — 以精確有理數重算 CLT joint ORIGINAL word0 sum，不以小數近似替代。
• S018 · premise_and_result_audit — 回查原始完整資料、步驟操作數與本步前提；有限核對、條件式定理引用與一般形式證明分開。
• S017 · premise_and_result_audit — 以精確有理數重算 CLT ORIGINAL common marginal ORIGINAL second minus mean square，不以小數近似替代。
• S016 · premise_and_result_audit — 以精確有理數重算 CLT ORIGINAL common marginal ORIGINAL mean square，不以小數近似替代。
• S015 · premise_and_result_audit — 以精確有理數重算 CLT ORIGINAL common marginal ORIGINAL ascending second moment sum1，不以小數近似替代。
• S014 · premise_and_result_audit — 以精確有理數重算 CLT ORIGINAL common marginal ORIGINAL square times probability1，不以小數近似替代。
• S013 · premise_and_result_audit — 以精確有理數重算 CLT ORIGINAL common marginal exact value square1，不以小數近似替代。
• S012 · premise_and_result_audit — 以精確有理數重算 CLT ORIGINAL common marginal ORIGINAL ascending mean sum1，不以小數近似替代。
• S011 · premise_and_result_audit — 以精確有理數重算 CLT ORIGINAL common marginal ORIGINAL value times probability1，不以小數近似替代。
• S010 · premise_and_result_audit — 以精確有理數重算 CLT ORIGINAL common marginal ORIGINAL ascending second moment sum0，不以小數近似替代。
• S009 · premise_and_result_audit — 以精確有理數重算 CLT ORIGINAL common marginal ORIGINAL square times probability0，不以小數近似替代。
• S008 · premise_and_result_audit — 以精確有理數重算 CLT ORIGINAL common marginal exact value square0，不以小數近似替代。
• S007 · premise_and_result_audit — 以精確有理數重算 CLT ORIGINAL common marginal ORIGINAL ascending mean sum0，不以小數近似替代。
• S006 · premise_and_result_audit — 以精確有理數重算 CLT ORIGINAL common marginal ORIGINAL value times probability0，不以小數近似替代。
• S005 · premise_and_result_audit — 回查原始完整資料、步驟操作數與本步前提；有限核對、條件式定理引用與一般形式證明分開。
• S004 · premise_and_result_audit — 以精確有理數重算 CLT ORIGINAL common marginal ORIGINAL law mass sum1，不以小數近似替代。
• S003 · premise_and_result_audit — 以精確有理數重算 CLT ORIGINAL common marginal ORIGINAL law mass sum0，不以小數近似替代。
• S002 · premise_and_result_audit — 回查原始完整資料、步驟操作數與本步前提；有限核對、條件式定理引用與一般形式證明分開。
• S001 · premise_and_result_audit — 回查原始完整資料、步驟操作數與本步前提；有限核對、條件式定理引用與一般形式證明分開。

結論

(4, [[-4, 1/16], [-3, 1/4], [-2, 3/8], [-1, 1/4], [0, 1/16]], 1, -2, 1, -2, 1, 0, 1, Z_n=(S_n-n*mu)/(sigma*sqrt(n)), complete_joint_law_moments_do_not_prove_distributional_convergence)

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。