Question 19749: Estimator Variance Remains Parameter-Indexed

Original question and givens

Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/33, 2/33, 3/33, 27/33], [2/37, 3/37, 4/37, 28/37]], and one parameter-independent total statistic T=[0, 1, 25, 26] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_variance_rows, certificate).

• G001 — Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/33, 2/33, 3/33, 27/33], [2/37, 3/37, 4/37, 28/37]], and one parameter-independent total statistic T=[0, 1, 25, 26] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_variance_rows, certificate).

Definition and scope

Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate.

• mko-estimator-variance

Step-by-step derivation

source operation=estvar-family-readback;original snapshot and mathematics retained

explicit ordered source output fields=[parameter_variance_rows, certificate]

variance parameter0 complete law mass := sum(1/33, 2/33, 1/11, 9/11)

variance parameter0 complete law mass = 1

original parameter SLOT=0;ALL original outcome masses=[1/33, 2/33, 1/11, 9/11];ONE fixed statistic=[0, 1, 25, 26]

complete same-law mass=1;all masses nonnegative;positive-mass support=[0, 1, 2, 3];null outcomes RETAINED

variance parameter1 complete law mass := sum(2/37, 3/37, 4/37, 28/37)

variance parameter1 complete law mass = 1

original parameter SLOT=1;ALL original outcome masses=[2/37, 3/37, 4/37, 28/37];ONE fixed statistic=[0, 1, 25, 26]

complete same-law mass=1;all masses nonnegative;positive-mass support=[0, 1, 2, 3];null outcomes RETAINED

variance parameter0 outcome0 weighted T := mul(1/33, 0)

variance parameter0 outcome0 weighted T = 0

variance parameter0 outcome1 weighted T := mul(2/33, 1)

variance parameter0 outcome1 weighted T = 2/33

variance parameter0 outcome2 weighted T := mul(1/11, 25)

variance parameter0 outcome2 weighted T = 25/11

variance parameter0 outcome3 weighted T := mul(9/11, 26)

variance parameter0 outcome3 weighted T = 234/11

variance parameter0 SAME-law mean := sum(0, 2/33, 25/11, 234/11)

variance parameter0 SAME-law mean = 779/33

variance parameter0 outcome0 T minus SAME mean := sub(0, 779/33)

variance parameter0 outcome0 T minus SAME mean = -779/33

variance parameter0 outcome0 centered square := mul(-779/33, -779/33)

variance parameter0 outcome0 centered square = 606841/1089

variance parameter0 outcome0 weighted centered square := mul(1/33, 606841/1089)

variance parameter0 outcome0 weighted centered square = 606841/35937

variance parameter0 outcome1 T minus SAME mean := sub(1, 779/33)

variance parameter0 outcome1 T minus SAME mean = -746/33

variance parameter0 outcome1 centered square := mul(-746/33, -746/33)

variance parameter0 outcome1 centered square = 556516/1089

variance parameter0 outcome1 weighted centered square := mul(2/33, 556516/1089)

variance parameter0 outcome1 weighted centered square = 1113032/35937

variance parameter0 outcome2 T minus SAME mean := sub(25, 779/33)

variance parameter0 outcome2 T minus SAME mean = 46/33

variance parameter0 outcome2 centered square := mul(46/33, 46/33)

variance parameter0 outcome2 centered square = 2116/1089

variance parameter0 outcome2 weighted centered square := mul(1/11, 2116/1089)

variance parameter0 outcome2 weighted centered square = 2116/11979

variance parameter0 outcome3 T minus SAME mean := sub(26, 779/33)

variance parameter0 outcome3 T minus SAME mean = 79/33

variance parameter0 outcome3 centered square := mul(79/33, 79/33)

variance parameter0 outcome3 centered square = 6241/1089

variance parameter0 outcome3 weighted centered square := mul(9/11, 6241/1089)

variance parameter0 outcome3 weighted centered square = 6241/1331

variance parameter0 full centered variance := sum(606841/35937, 1113032/35937, 2116/11979, 6241/1331)

variance parameter0 full centered variance = 57416/1089

FIXED parameter SLOT=0;original same law=[1/33, 2/33, 1/11, 9/11];same statistic=[0, 1, 25, 26];its own mean=779/33;ALL centered deviations=[-779/33, -746/33, 46/33, 79/33]

ALL nonnegative weighted centered squares=[606841/35937, 1113032/35937, 2116/11979, 6241/1331];variance=57416/1089;variance0 iff EVERY positive-mass outcome has T=THIS mean

variance parameter1 outcome0 weighted T := mul(2/37, 0)

variance parameter1 outcome0 weighted T = 0

variance parameter1 outcome1 weighted T := mul(3/37, 1)

variance parameter1 outcome1 weighted T = 3/37

variance parameter1 outcome2 weighted T := mul(4/37, 25)

variance parameter1 outcome2 weighted T = 100/37

variance parameter1 outcome3 weighted T := mul(28/37, 26)

variance parameter1 outcome3 weighted T = 728/37

variance parameter1 SAME-law mean := sum(0, 3/37, 100/37, 728/37)

variance parameter1 SAME-law mean = 831/37

variance parameter1 outcome0 T minus SAME mean := sub(0, 831/37)

variance parameter1 outcome0 T minus SAME mean = -831/37

variance parameter1 outcome0 centered square := mul(-831/37, -831/37)

variance parameter1 outcome0 centered square = 690561/1369

variance parameter1 outcome0 weighted centered square := mul(2/37, 690561/1369)

variance parameter1 outcome0 weighted centered square = 1381122/50653

variance parameter1 outcome1 T minus SAME mean := sub(1, 831/37)

variance parameter1 outcome1 T minus SAME mean = -794/37

variance parameter1 outcome1 centered square := mul(-794/37, -794/37)

variance parameter1 outcome1 centered square = 630436/1369

variance parameter1 outcome1 weighted centered square := mul(3/37, 630436/1369)

variance parameter1 outcome1 weighted centered square = 1891308/50653

variance parameter1 outcome2 T minus SAME mean := sub(25, 831/37)

variance parameter1 outcome2 T minus SAME mean = 94/37

variance parameter1 outcome2 centered square := mul(94/37, 94/37)

variance parameter1 outcome2 centered square = 8836/1369

variance parameter1 outcome2 weighted centered square := mul(4/37, 8836/1369)

variance parameter1 outcome2 weighted centered square = 35344/50653

variance parameter1 outcome3 T minus SAME mean := sub(26, 831/37)

variance parameter1 outcome3 T minus SAME mean = 131/37

variance parameter1 outcome3 centered square := mul(131/37, 131/37)

variance parameter1 outcome3 centered square = 17161/1369

variance parameter1 outcome3 weighted centered square := mul(28/37, 17161/1369)

variance parameter1 outcome3 weighted centered square = 480508/50653

variance parameter1 full centered variance := sum(1381122/50653, 1891308/50653, 35344/50653, 480508/50653)

variance parameter1 full centered variance = 102386/1369

FIXED parameter SLOT=1;original same law=[2/37, 3/37, 4/37, 28/37];same statistic=[0, 1, 25, 26];its own mean=831/37;ALL centered deviations=[-831/37, -794/37, 94/37, 131/37]

ALL nonnegative weighted centered squares=[1381122/50653, 1891308/50653, 35344/50653, 480508/50653];variance=102386/1369;variance0 iff EVERY positive-mass outcome has T=THIS mean

Reverse checks

• S043 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S042 · premise_and_result_audit — Recompute variance parameter1 full centered variance with exact rational arithmetic, not rounded decimals.
• S041 · premise_and_result_audit — Recompute variance parameter1 outcome3 weighted centered square with exact rational arithmetic, not rounded decimals.
• S040 · premise_and_result_audit — Recompute variance parameter1 outcome3 centered square with exact rational arithmetic, not rounded decimals.
• S039 · premise_and_result_audit — Recompute variance parameter1 outcome3 T minus SAME mean with exact rational arithmetic, not rounded decimals.
• S038 · premise_and_result_audit — Recompute variance parameter1 outcome2 weighted centered square with exact rational arithmetic, not rounded decimals.
• S037 · premise_and_result_audit — Recompute variance parameter1 outcome2 centered square with exact rational arithmetic, not rounded decimals.
• S036 · premise_and_result_audit — Recompute variance parameter1 outcome2 T minus SAME mean with exact rational arithmetic, not rounded decimals.
• S035 · premise_and_result_audit — Recompute variance parameter1 outcome1 weighted centered square with exact rational arithmetic, not rounded decimals.
• S034 · premise_and_result_audit — Recompute variance parameter1 outcome1 centered square with exact rational arithmetic, not rounded decimals.
• S033 · premise_and_result_audit — Recompute variance parameter1 outcome1 T minus SAME mean with exact rational arithmetic, not rounded decimals.
• S032 · premise_and_result_audit — Recompute variance parameter1 outcome0 weighted centered square with exact rational arithmetic, not rounded decimals.
• S031 · premise_and_result_audit — Recompute variance parameter1 outcome0 centered square with exact rational arithmetic, not rounded decimals.
• S030 · premise_and_result_audit — Recompute variance parameter1 outcome0 T minus SAME mean with exact rational arithmetic, not rounded decimals.
• S029 · premise_and_result_audit — Recompute variance parameter1 SAME-law mean with exact rational arithmetic, not rounded decimals.
• S028 · premise_and_result_audit — Recompute variance parameter1 outcome3 weighted T with exact rational arithmetic, not rounded decimals.
• S027 · premise_and_result_audit — Recompute variance parameter1 outcome2 weighted T with exact rational arithmetic, not rounded decimals.
• S026 · premise_and_result_audit — Recompute variance parameter1 outcome1 weighted T with exact rational arithmetic, not rounded decimals.
• S025 · premise_and_result_audit — Recompute variance parameter1 outcome0 weighted T with exact rational arithmetic, not rounded decimals.
• S024 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S023 · premise_and_result_audit — Recompute variance parameter0 full centered variance with exact rational arithmetic, not rounded decimals.
• S022 · premise_and_result_audit — Recompute variance parameter0 outcome3 weighted centered square with exact rational arithmetic, not rounded decimals.
• S021 · premise_and_result_audit — Recompute variance parameter0 outcome3 centered square with exact rational arithmetic, not rounded decimals.
• S020 · premise_and_result_audit — Recompute variance parameter0 outcome3 T minus SAME mean with exact rational arithmetic, not rounded decimals.
• S019 · premise_and_result_audit — Recompute variance parameter0 outcome2 weighted centered square with exact rational arithmetic, not rounded decimals.
• S018 · premise_and_result_audit — Recompute variance parameter0 outcome2 centered square with exact rational arithmetic, not rounded decimals.
• S017 · premise_and_result_audit — Recompute variance parameter0 outcome2 T minus SAME mean with exact rational arithmetic, not rounded decimals.
• S016 · premise_and_result_audit — Recompute variance parameter0 outcome1 weighted centered square with exact rational arithmetic, not rounded decimals.
• S015 · premise_and_result_audit — Recompute variance parameter0 outcome1 centered square with exact rational arithmetic, not rounded decimals.
• S014 · premise_and_result_audit — Recompute variance parameter0 outcome1 T minus SAME mean with exact rational arithmetic, not rounded decimals.
• S013 · premise_and_result_audit — Recompute variance parameter0 outcome0 weighted centered square with exact rational arithmetic, not rounded decimals.
• S012 · premise_and_result_audit — Recompute variance parameter0 outcome0 centered square with exact rational arithmetic, not rounded decimals.
• S011 · premise_and_result_audit — Recompute variance parameter0 outcome0 T minus SAME mean with exact rational arithmetic, not rounded decimals.
• S010 · premise_and_result_audit — Recompute variance parameter0 SAME-law mean with exact rational arithmetic, not rounded decimals.
• S009 · premise_and_result_audit — Recompute variance parameter0 outcome3 weighted T with exact rational arithmetic, not rounded decimals.
• S008 · premise_and_result_audit — Recompute variance parameter0 outcome2 weighted T with exact rational arithmetic, not rounded decimals.
• S007 · premise_and_result_audit — Recompute variance parameter0 outcome1 weighted T with exact rational arithmetic, not rounded decimals.
• S006 · premise_and_result_audit — Recompute variance parameter0 outcome0 weighted T with exact rational arithmetic, not rounded decimals.
• S005 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S004 · premise_and_result_audit — Recompute variance parameter1 complete law mass with exact rational arithmetic, not rounded decimals.
• S003 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S002 · premise_and_result_audit — Recompute variance parameter0 complete law mass with exact rational arithmetic, not rounded decimals.
• S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.

Result

([[0, 779/33, 57416/1089], [1, 831/37, 102386/1369]], variance_is_parameter_indexed_family)

The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.

This question review and source binding

Review date: 2026-10-26

Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.