Question 19782: The Second Moment and Squared Mean Are Different Quantities

Original question and givens

Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/16, 2/16, 3/16, 10/16], [2/20, 3/20, 4/20, 11/20]], and one parameter-independent total statistic T=[0, 1, 8, 9] 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. Selected parameter index=0. Give (parameter_index, second_moment, expectation, second_minus_squared_mean, variance, equal).

• G001 — Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/16, 2/16, 3/16, 10/16], [2/20, 3/20, 4/20, 11/20]], and one parameter-independent total statistic T=[0, 1, 8, 9] 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. Selected parameter index=0. Give (parameter_index, second_moment, expectation, second_minus_squared_mean, variance, equal).

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-second-moment-identity;original snapshot and mathematics retained

explicit ordered source output fields=[parameter_index, second_moment, expectation, second_minus_squared_mean, variance, equal]

variance parameter0 complete law mass := sum(1/16, 1/8, 3/16, 5/8)

variance parameter0 complete law mass = 1

original parameter SLOT=0;ALL original outcome masses=[1/16, 1/8, 3/16, 5/8];ONE fixed statistic=[0, 1, 8, 9]

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

variance parameter1 complete law mass := sum(1/10, 3/20, 1/5, 11/20)

variance parameter1 complete law mass = 1

original parameter SLOT=1;ALL original outcome masses=[1/10, 3/20, 1/5, 11/20];ONE fixed statistic=[0, 1, 8, 9]

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

variance parameter0 outcome0 weighted T := mul(1/16, 0)

variance parameter0 outcome0 weighted T = 0

variance parameter0 outcome1 weighted T := mul(1/8, 1)

variance parameter0 outcome1 weighted T = 1/8

variance parameter0 outcome2 weighted T := mul(3/16, 8)

variance parameter0 outcome2 weighted T = 3/2

variance parameter0 outcome3 weighted T := mul(5/8, 9)

variance parameter0 outcome3 weighted T = 45/8

variance parameter0 SAME-law mean := sum(0, 1/8, 3/2, 45/8)

variance parameter0 SAME-law mean = 29/4

variance parameter0 outcome0 T minus SAME mean := sub(0, 29/4)

variance parameter0 outcome0 T minus SAME mean = -29/4

variance parameter0 outcome0 centered square := mul(-29/4, -29/4)

variance parameter0 outcome0 centered square = 841/16

variance parameter0 outcome0 weighted centered square := mul(1/16, 841/16)

variance parameter0 outcome0 weighted centered square = 841/256

variance parameter0 outcome1 T minus SAME mean := sub(1, 29/4)

variance parameter0 outcome1 T minus SAME mean = -25/4

variance parameter0 outcome1 centered square := mul(-25/4, -25/4)

variance parameter0 outcome1 centered square = 625/16

variance parameter0 outcome1 weighted centered square := mul(1/8, 625/16)

variance parameter0 outcome1 weighted centered square = 625/128

variance parameter0 outcome2 T minus SAME mean := sub(8, 29/4)

variance parameter0 outcome2 T minus SAME mean = 3/4

variance parameter0 outcome2 centered square := mul(3/4, 3/4)

variance parameter0 outcome2 centered square = 9/16

variance parameter0 outcome2 weighted centered square := mul(3/16, 9/16)

variance parameter0 outcome2 weighted centered square = 27/256

variance parameter0 outcome3 T minus SAME mean := sub(9, 29/4)

variance parameter0 outcome3 T minus SAME mean = 7/4

variance parameter0 outcome3 centered square := mul(7/4, 7/4)

variance parameter0 outcome3 centered square = 49/16

variance parameter0 outcome3 weighted centered square := mul(5/8, 49/16)

variance parameter0 outcome3 weighted centered square = 245/128

variance parameter0 full centered variance := sum(841/256, 625/128, 27/256, 245/128)

variance parameter0 full centered variance = 163/16

FIXED parameter SLOT=0;original same law=[1/16, 1/8, 3/16, 5/8];same statistic=[0, 1, 8, 9];its own mean=29/4;ALL centered deviations=[-29/4, -25/4, 3/4, 7/4]

ALL nonnegative weighted centered squares=[841/256, 625/128, 27/256, 245/128];variance=163/16;variance0 iff EVERY positive-mass outcome has T=THIS mean

variance second moment outcome0 T squared := mul(0, 0)

variance second moment outcome0 T squared = 0

variance second moment outcome0 weighted T squared := mul(1/16, 0)

variance second moment outcome0 weighted T squared = 0

variance second moment outcome1 T squared := mul(1, 1)

variance second moment outcome1 T squared = 1

variance second moment outcome1 weighted T squared := mul(1/8, 1)

variance second moment outcome1 weighted T squared = 1/8

variance second moment outcome2 T squared := mul(8, 8)

variance second moment outcome2 T squared = 64

variance second moment outcome2 weighted T squared := mul(3/16, 64)

variance second moment outcome2 weighted T squared = 12

variance second moment outcome3 T squared := mul(9, 9)

variance second moment outcome3 T squared = 81

variance second moment outcome3 weighted T squared := mul(5/8, 81)

variance second moment outcome3 weighted T squared = 405/8

variance full SAME-law second moment := sum(0, 1/8, 12, 405/8)

variance full SAME-law second moment = 251/4

variance square of SAME-law mean := mul(29/4, 29/4)

variance square of SAME-law mean = 841/16

variance second moment minus mean squared := sub(251/4, 841/16)

variance second moment minus mean squared = 163/16

FIXED original law at SLOT=0;mass sum1;E[T]=29/4;E[T^2]=251/4;mean squared=841/16

expand sum mass*(T-mean)^2=E[T^2]-2*mean*E[T]+mean^2*1=163/16=centered variance=163/16

Reverse checks

• S036 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S035 · premise_and_result_audit — Recompute variance second moment minus mean squared with exact rational arithmetic, not rounded decimals.
• S034 · premise_and_result_audit — Recompute variance square of SAME-law mean with exact rational arithmetic, not rounded decimals.
• S033 · premise_and_result_audit — Recompute variance full SAME-law second moment with exact rational arithmetic, not rounded decimals.
• S032 · premise_and_result_audit — Recompute variance second moment outcome3 weighted T squared with exact rational arithmetic, not rounded decimals.
• S031 · premise_and_result_audit — Recompute variance second moment outcome3 T squared with exact rational arithmetic, not rounded decimals.
• S030 · premise_and_result_audit — Recompute variance second moment outcome2 weighted T squared with exact rational arithmetic, not rounded decimals.
• S029 · premise_and_result_audit — Recompute variance second moment outcome2 T squared with exact rational arithmetic, not rounded decimals.
• S028 · premise_and_result_audit — Recompute variance second moment outcome1 weighted T squared with exact rational arithmetic, not rounded decimals.
• S027 · premise_and_result_audit — Recompute variance second moment outcome1 T squared with exact rational arithmetic, not rounded decimals.
• S026 · premise_and_result_audit — Recompute variance second moment outcome0 weighted T squared with exact rational arithmetic, not rounded decimals.
• S025 · premise_and_result_audit — Recompute variance second moment outcome0 T squared 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, 251/4, 29/4, 163/16, 163/16, Yes)

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.