Question 19729: Estimator Variance Remains Parameter-Indexed

Original question and givens

Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/13, 2/13, 3/13, 7/13], [2/17, 3/17, 4/17, 8/17]], and one parameter-independent total statistic T=[0, 1, 5, 6] 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=[[1/13, 2/13, 3/13, 7/13], [2/17, 3/17, 4/17, 8/17]], and one parameter-independent total statistic T=[0, 1, 5, 6] 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(1/13, 2/13, 3/13, 7/13)

variance parameter0 complete law mass = 1

original parameter SLOT=0;ALL original outcome masses=[1/13, 2/13, 3/13, 7/13];ONE fixed statistic=[0, 1, 5, 6]

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

variance parameter1 complete law mass := sum(2/17, 3/17, 4/17, 8/17)

variance parameter1 complete law mass = 1

original parameter SLOT=1;ALL original outcome masses=[2/17, 3/17, 4/17, 8/17];ONE fixed statistic=[0, 1, 5, 6]

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

variance parameter0 outcome0 weighted T := mul(1/13, 0)

variance parameter0 outcome0 weighted T = 0

variance parameter0 outcome1 weighted T := mul(2/13, 1)

variance parameter0 outcome1 weighted T = 2/13

variance parameter0 outcome2 weighted T := mul(3/13, 5)

variance parameter0 outcome2 weighted T = 15/13

variance parameter0 outcome3 weighted T := mul(7/13, 6)

variance parameter0 outcome3 weighted T = 42/13

variance parameter0 SAME-law mean := sum(0, 2/13, 15/13, 42/13)

variance parameter0 SAME-law mean = 59/13

variance parameter0 outcome0 T minus SAME mean := sub(0, 59/13)

variance parameter0 outcome0 T minus SAME mean = -59/13

variance parameter0 outcome0 centered square := mul(-59/13, -59/13)

variance parameter0 outcome0 centered square = 3481/169

variance parameter0 outcome0 weighted centered square := mul(1/13, 3481/169)

variance parameter0 outcome0 weighted centered square = 3481/2197

variance parameter0 outcome1 T minus SAME mean := sub(1, 59/13)

variance parameter0 outcome1 T minus SAME mean = -46/13

variance parameter0 outcome1 centered square := mul(-46/13, -46/13)

variance parameter0 outcome1 centered square = 2116/169

variance parameter0 outcome1 weighted centered square := mul(2/13, 2116/169)

variance parameter0 outcome1 weighted centered square = 4232/2197

variance parameter0 outcome2 T minus SAME mean := sub(5, 59/13)

variance parameter0 outcome2 T minus SAME mean = 6/13

variance parameter0 outcome2 centered square := mul(6/13, 6/13)

variance parameter0 outcome2 centered square = 36/169

variance parameter0 outcome2 weighted centered square := mul(3/13, 36/169)

variance parameter0 outcome2 weighted centered square = 108/2197

variance parameter0 outcome3 T minus SAME mean := sub(6, 59/13)

variance parameter0 outcome3 T minus SAME mean = 19/13

variance parameter0 outcome3 centered square := mul(19/13, 19/13)

variance parameter0 outcome3 centered square = 361/169

variance parameter0 outcome3 weighted centered square := mul(7/13, 361/169)

variance parameter0 outcome3 weighted centered square = 2527/2197

variance parameter0 full centered variance := sum(3481/2197, 4232/2197, 108/2197, 2527/2197)

variance parameter0 full centered variance = 796/169

FIXED parameter SLOT=0;original same law=[1/13, 2/13, 3/13, 7/13];same statistic=[0, 1, 5, 6];its own mean=59/13;ALL centered deviations=[-59/13, -46/13, 6/13, 19/13]

ALL nonnegative weighted centered squares=[3481/2197, 4232/2197, 108/2197, 2527/2197];variance=796/169;variance0 iff EVERY positive-mass outcome has T=THIS mean

variance parameter1 outcome0 weighted T := mul(2/17, 0)

variance parameter1 outcome0 weighted T = 0

variance parameter1 outcome1 weighted T := mul(3/17, 1)

variance parameter1 outcome1 weighted T = 3/17

variance parameter1 outcome2 weighted T := mul(4/17, 5)

variance parameter1 outcome2 weighted T = 20/17

variance parameter1 outcome3 weighted T := mul(8/17, 6)

variance parameter1 outcome3 weighted T = 48/17

variance parameter1 SAME-law mean := sum(0, 3/17, 20/17, 48/17)

variance parameter1 SAME-law mean = 71/17

variance parameter1 outcome0 T minus SAME mean := sub(0, 71/17)

variance parameter1 outcome0 T minus SAME mean = -71/17

variance parameter1 outcome0 centered square := mul(-71/17, -71/17)

variance parameter1 outcome0 centered square = 5041/289

variance parameter1 outcome0 weighted centered square := mul(2/17, 5041/289)

variance parameter1 outcome0 weighted centered square = 10082/4913

variance parameter1 outcome1 T minus SAME mean := sub(1, 71/17)

variance parameter1 outcome1 T minus SAME mean = -54/17

variance parameter1 outcome1 centered square := mul(-54/17, -54/17)

variance parameter1 outcome1 centered square = 2916/289

variance parameter1 outcome1 weighted centered square := mul(3/17, 2916/289)

variance parameter1 outcome1 weighted centered square = 8748/4913

variance parameter1 outcome2 T minus SAME mean := sub(5, 71/17)

variance parameter1 outcome2 T minus SAME mean = 14/17

variance parameter1 outcome2 centered square := mul(14/17, 14/17)

variance parameter1 outcome2 centered square = 196/289

variance parameter1 outcome2 weighted centered square := mul(4/17, 196/289)

variance parameter1 outcome2 weighted centered square = 784/4913

variance parameter1 outcome3 T minus SAME mean := sub(6, 71/17)

variance parameter1 outcome3 T minus SAME mean = 31/17

variance parameter1 outcome3 centered square := mul(31/17, 31/17)

variance parameter1 outcome3 centered square = 961/289

variance parameter1 outcome3 weighted centered square := mul(8/17, 961/289)

variance parameter1 outcome3 weighted centered square = 7688/4913

variance parameter1 full centered variance := sum(10082/4913, 8748/4913, 784/4913, 7688/4913)

variance parameter1 full centered variance = 1606/289

FIXED parameter SLOT=1;original same law=[2/17, 3/17, 4/17, 8/17];same statistic=[0, 1, 5, 6];its own mean=71/17;ALL centered deviations=[-71/17, -54/17, 14/17, 31/17]

ALL nonnegative weighted centered squares=[10082/4913, 8748/4913, 784/4913, 7688/4913];variance=1606/289;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, 59/13, 796/169], [1, 71/17, 1606/289]], 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.