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Question 25521: Unbiasedness Gives MSE Equal to Variance, Not Zero Error or Consistency

On this page
  1. Original question and givens
  2. Definition and scope
  3. Step-by-step derivation
  4. Reverse checks
  5. Result
  6. This question review and source binding

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R0050-25521 · q-20261015-advance-20261115-r03-bias-variance-decomposition-0096

Original question and givens

Use scalar theta=8, complete estimator values=[5, 8, 11] and aligned exact probabilities=[1/4, 1/2, 1/4]. Under this same P_theta the nonnegative probabilities sum to1. Retain zero-mass listed rows and sort values with their own probabilities. Given listed observed estimate=5. Give (theta, observed, observed_squared_error, bias, unbiased, variance, MSE, variance_comparison_holds, unbiased_and_zero_MSE, consistency_claim, boundary).

  • G001 — Use scalar theta=8, complete estimator values=[5, 8, 11] and aligned exact probabilities=[1/4, 1/2, 1/4]. Under this same P_theta the nonnegative probabilities sum to1. Retain zero-mass listed rows and sort values with their own probabilities. Given listed observed estimate=5. Give (theta, observed, observed_squared_error, bias, unbiased, variance, MSE, variance_comparison_holds, unbiased_and_zero_MSE, consistency_claim, boundary).

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.

Step-by-step derivation

S001 · implies

Before

source roles = [biasvar-biased-unbiased-boundary]

After

derived = [[theta, observed, observed_squared_error, bias, unbiased, variance, MSE, variance_comparison_holds, unbiased_and_zero_MSE, consistency_claim, boundary]]

Conditions: none

List the full source field order without JavaScript.LP weak uses13 fields,zero/strong15;premises and roles follow the original operation/mode without padding.

Step identifiers and dependencies

Rule: source_ordered_fields_without_JS · Uses: G001

S002 · calculation

Before

Biasvar ORIGINAL single theta ORIGINAL mass sum := sum(1/4, 1/2, 1/4)

After

Biasvar ORIGINAL single theta ORIGINAL mass sum = 1

Conditions: none

Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S001

S003 · implies

Before

source roles = [Biasvar ORIGINAL single theta, 8, [5, 8, 11], [[1, 4], [1, 2], [1, 4]]]

After

derived = [[(5, 1/4), (8, 1/2), (11, 1/4)], 1]

Conditions: none

Qualify this same complete P_theta law,with nonnegative original-order mass summing to1,and sort values numerically with their masses.Keep null rows.Unlike the earlier MSE node,this node explicitly proves the decomposition.

Step identifiers and dependencies

Rule: Biasvar_complete_SAME_law_numeric_sorted_ALIGNED_null_rows · Uses: G001, S002

S004 · calculation

Before

Biasvar ORIGINAL single theta mean product0 := mul(5, 1/4)

After

Biasvar ORIGINAL single theta mean product0 = 5/4

Conditions: none

Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S003

S005 · calculation

Before

Biasvar ORIGINAL single theta mean product1 := mul(8, 1/2)

After

Biasvar ORIGINAL single theta mean product1 = 4

Conditions: none

Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S004

S006 · calculation

Before

Biasvar ORIGINAL single theta mean product2 := mul(11, 1/4)

After

Biasvar ORIGINAL single theta mean product2 = 11/4

Conditions: none

Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S005

S007 · calculation

Before

Biasvar ORIGINAL single theta mean sum := sum(5/4, 4, 11/4)

After

Biasvar ORIGINAL single theta mean sum = 8

Conditions: none

Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S006

S008 · calculation

Before

Biasvar ORIGINAL single theta bias mean minus theta := sub(8, 8)

After

Biasvar ORIGINAL single theta bias mean minus theta = 0

Conditions: none

Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S007

S009 · calculation

Before

Biasvar ORIGINAL single theta centered deviation0 := sub(5, 8)

After

Biasvar ORIGINAL single theta centered deviation0 = -3

Conditions: none

Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S008

S010 · calculation

Before

Biasvar ORIGINAL single theta centered square0 := mul(-3, -3)

After

Biasvar ORIGINAL single theta centered square0 = 9

Conditions: none

Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S009

S011 · calculation

Before

Biasvar ORIGINAL single theta variance weighted term0 := mul(1/4, 9)

After

Biasvar ORIGINAL single theta variance weighted term0 = 9/4

Conditions: none

Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S010

S012 · calculation

Before

Biasvar ORIGINAL single theta centered deviation1 := sub(8, 8)

After

Biasvar ORIGINAL single theta centered deviation1 = 0

Conditions: none

Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S011

S013 · calculation

Before

Biasvar ORIGINAL single theta centered square1 := mul(0, 0)

After

Biasvar ORIGINAL single theta centered square1 = 0

Conditions: none

Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S012

S014 · calculation

Before

Biasvar ORIGINAL single theta variance weighted term1 := mul(1/2, 0)

After

Biasvar ORIGINAL single theta variance weighted term1 = 0

Conditions: none

Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S013

S015 · calculation

Before

Biasvar ORIGINAL single theta centered deviation2 := sub(11, 8)

After

Biasvar ORIGINAL single theta centered deviation2 = 3

Conditions: none

Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S014

S016 · calculation

Before

Biasvar ORIGINAL single theta centered square2 := mul(3, 3)

After

Biasvar ORIGINAL single theta centered square2 = 9

Conditions: none

Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S015

S017 · calculation

Before

Biasvar ORIGINAL single theta variance weighted term2 := mul(1/4, 9)

After

Biasvar ORIGINAL single theta variance weighted term2 = 9/4

Conditions: none

Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S016

S018 · calculation

Before

Biasvar ORIGINAL single theta variance sum := sum(9/4, 0, 9/4)

After

Biasvar ORIGINAL single theta variance sum = 9/2

Conditions: none

Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S017

S019 · calculation

Before

Biasvar ORIGINAL single theta theta error0 := sub(5, 8)

After

Biasvar ORIGINAL single theta theta error0 = -3

Conditions: none

Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S018

S020 · calculation

Before

Biasvar ORIGINAL single theta theta square0 := mul(-3, -3)

After

Biasvar ORIGINAL single theta theta square0 = 9

Conditions: none

Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S019

S021 · calculation

Before

Biasvar ORIGINAL single theta direct MSE term0 := mul(1/4, 9)

After

Biasvar ORIGINAL single theta direct MSE term0 = 9/4

Conditions: none

Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S020

S022 · calculation

Before

Biasvar ORIGINAL single theta theta error1 := sub(8, 8)

After

Biasvar ORIGINAL single theta theta error1 = 0

Conditions: none

Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S021

S023 · calculation

Before

Biasvar ORIGINAL single theta theta square1 := mul(0, 0)

After

Biasvar ORIGINAL single theta theta square1 = 0

Conditions: none

Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S022

S024 · calculation

Before

Biasvar ORIGINAL single theta direct MSE term1 := mul(1/2, 0)

After

Biasvar ORIGINAL single theta direct MSE term1 = 0

Conditions: none

Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S023

S025 · calculation

Before

Biasvar ORIGINAL single theta theta error2 := sub(11, 8)

After

Biasvar ORIGINAL single theta theta error2 = 3

Conditions: none

Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S024

S026 · calculation

Before

Biasvar ORIGINAL single theta theta square2 := mul(3, 3)

After

Biasvar ORIGINAL single theta theta square2 = 9

Conditions: none

Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S025

S027 · calculation

Before

Biasvar ORIGINAL single theta direct MSE term2 := mul(1/4, 9)

After

Biasvar ORIGINAL single theta direct MSE term2 = 9/4

Conditions: none

Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S026

S028 · calculation

Before

Biasvar ORIGINAL single theta direct MSE sum := sum(9/4, 0, 9/4)

After

Biasvar ORIGINAL single theta direct MSE sum = 9/2

Conditions: none

Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S027

S029 · calculation

Before

Biasvar ORIGINAL single theta centered mean product0 := mul(1/4, -3)

After

Biasvar ORIGINAL single theta centered mean product0 = -3/4

Conditions: none

Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S028

S030 · calculation

Before

Biasvar ORIGINAL single theta centered mean product1 := mul(1/2, 0)

After

Biasvar ORIGINAL single theta centered mean product1 = 0

Conditions: none

Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S029

S031 · calculation

Before

Biasvar ORIGINAL single theta centered mean product2 := mul(1/4, 3)

After

Biasvar ORIGINAL single theta centered mean product2 = 3/4

Conditions: none

Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S030

S032 · calculation

Before

Biasvar ORIGINAL single theta centered mean sum := sum(-3/4, 0, 3/4)

After

Biasvar ORIGINAL single theta centered mean sum = 0

Conditions: none

Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S031

S033 · calculation

Before

Biasvar ORIGINAL single theta bias times centered mean := mul(0, 0)

After

Biasvar ORIGINAL single theta bias times centered mean = 0

Conditions: none

Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S032

S034 · calculation

Before

Biasvar ORIGINAL single theta cross twice inner := mul(2, 0)

After

Biasvar ORIGINAL single theta cross twice inner = 0

Conditions: none

Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S033

S035 · calculation

Before

Biasvar ORIGINAL single theta squared bias := mul(0, 0)

After

Biasvar ORIGINAL single theta squared bias = 0

Conditions: none

Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S034

S036 · calculation

Before

Biasvar ORIGINAL single theta variance plus squared bias := sum(9/2, 0)

After

Biasvar ORIGINAL single theta variance plus squared bias = 9/2

Conditions: none

Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S035

S037 · implies

Before

source roles = [Biasvar ORIGINAL single theta, 8, 8, 0, [(5, 1/4), (8, 1/2), (11, 1/4)]]

After

derived = [[[5, 1/4, -3, 0, -3, 9, 0, 0, 9], [8, 1/2, 0, 0, 0, 0, 0, 0, 0], [11, 1/4, 3, 0, 3, 9, 0, 0, 9]], 0, 0, 9/2, 9/2, 9/2, Yes]

Conditions: none

For every value,T−theta=(T−mean)+bias.Square to centered²+2*bias*centered+bias² and weight under the same law.E[T−mean]=E[T]−mean*1=0 cancels the cross term,without requiring bias0 or every deviation0.Total mass1 leaves bias².Pointwise cross terms are extra exact teaching values,not new source-profile restrictions.

Step identifiers and dependencies

Rule: Biasvar_EVERY_value_square_expansion_and_GENERAL_cross_cancellation · Uses: G001, S036

S038 · calculation

Before

Biasvar ORIGINAL observed minus theta := sub(5, 8)

After

Biasvar ORIGINAL observed minus theta = -3

Conditions: none

Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S037

S039 · calculation

Before

Biasvar ORIGINAL observed square := mul(-3, -3)

After

Biasvar ORIGINAL observed square = 9

Conditions: none

Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S038

S040 · implies

Before

source roles = [5, 9, 0, 9/2, 9/2]

After

derived = [Yes, Yes, No, No]

Conditions: none

Unbiasedness only removes bias²,giving MSE=variance,not variance0 or consistency.When biased,MSE>=variance.One observed loss is computed separately and may coincide with MSE without defining it.

Step identifiers and dependencies

Rule: Biasvar_observed_error_unbiased_variance_risk_consistency_separate · Uses: G001, S039

Reverse checks

  • S040 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S039 · premise_and_result_audit — Recompute Biasvar ORIGINAL observed square with exact rational arithmetic, not rounded decimals.
  • S038 · premise_and_result_audit — Recompute Biasvar ORIGINAL observed minus theta with exact rational arithmetic, not rounded decimals.
  • S037 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S036 · premise_and_result_audit — Recompute Biasvar ORIGINAL single theta variance plus squared bias with exact rational arithmetic, not rounded decimals.
  • S035 · premise_and_result_audit — Recompute Biasvar ORIGINAL single theta squared bias with exact rational arithmetic, not rounded decimals.
  • S034 · premise_and_result_audit — Recompute Biasvar ORIGINAL single theta cross twice inner with exact rational arithmetic, not rounded decimals.
  • S033 · premise_and_result_audit — Recompute Biasvar ORIGINAL single theta bias times centered mean with exact rational arithmetic, not rounded decimals.
  • S032 · premise_and_result_audit — Recompute Biasvar ORIGINAL single theta centered mean sum with exact rational arithmetic, not rounded decimals.
  • S031 · premise_and_result_audit — Recompute Biasvar ORIGINAL single theta centered mean product2 with exact rational arithmetic, not rounded decimals.
  • S030 · premise_and_result_audit — Recompute Biasvar ORIGINAL single theta centered mean product1 with exact rational arithmetic, not rounded decimals.
  • S029 · premise_and_result_audit — Recompute Biasvar ORIGINAL single theta centered mean product0 with exact rational arithmetic, not rounded decimals.
  • S028 · premise_and_result_audit — Recompute Biasvar ORIGINAL single theta direct MSE sum with exact rational arithmetic, not rounded decimals.
  • S027 · premise_and_result_audit — Recompute Biasvar ORIGINAL single theta direct MSE term2 with exact rational arithmetic, not rounded decimals.
  • S026 · premise_and_result_audit — Recompute Biasvar ORIGINAL single theta theta square2 with exact rational arithmetic, not rounded decimals.
  • S025 · premise_and_result_audit — Recompute Biasvar ORIGINAL single theta theta error2 with exact rational arithmetic, not rounded decimals.
  • S024 · premise_and_result_audit — Recompute Biasvar ORIGINAL single theta direct MSE term1 with exact rational arithmetic, not rounded decimals.
  • S023 · premise_and_result_audit — Recompute Biasvar ORIGINAL single theta theta square1 with exact rational arithmetic, not rounded decimals.
  • S022 · premise_and_result_audit — Recompute Biasvar ORIGINAL single theta theta error1 with exact rational arithmetic, not rounded decimals.
  • S021 · premise_and_result_audit — Recompute Biasvar ORIGINAL single theta direct MSE term0 with exact rational arithmetic, not rounded decimals.
  • S020 · premise_and_result_audit — Recompute Biasvar ORIGINAL single theta theta square0 with exact rational arithmetic, not rounded decimals.
  • S019 · premise_and_result_audit — Recompute Biasvar ORIGINAL single theta theta error0 with exact rational arithmetic, not rounded decimals.
  • S018 · premise_and_result_audit — Recompute Biasvar ORIGINAL single theta variance sum with exact rational arithmetic, not rounded decimals.
  • S017 · premise_and_result_audit — Recompute Biasvar ORIGINAL single theta variance weighted term2 with exact rational arithmetic, not rounded decimals.
  • S016 · premise_and_result_audit — Recompute Biasvar ORIGINAL single theta centered square2 with exact rational arithmetic, not rounded decimals.
  • S015 · premise_and_result_audit — Recompute Biasvar ORIGINAL single theta centered deviation2 with exact rational arithmetic, not rounded decimals.
  • S014 · premise_and_result_audit — Recompute Biasvar ORIGINAL single theta variance weighted term1 with exact rational arithmetic, not rounded decimals.
  • S013 · premise_and_result_audit — Recompute Biasvar ORIGINAL single theta centered square1 with exact rational arithmetic, not rounded decimals.
  • S012 · premise_and_result_audit — Recompute Biasvar ORIGINAL single theta centered deviation1 with exact rational arithmetic, not rounded decimals.
  • S011 · premise_and_result_audit — Recompute Biasvar ORIGINAL single theta variance weighted term0 with exact rational arithmetic, not rounded decimals.
  • S010 · premise_and_result_audit — Recompute Biasvar ORIGINAL single theta centered square0 with exact rational arithmetic, not rounded decimals.
  • S009 · premise_and_result_audit — Recompute Biasvar ORIGINAL single theta centered deviation0 with exact rational arithmetic, not rounded decimals.
  • S008 · premise_and_result_audit — Recompute Biasvar ORIGINAL single theta bias mean minus theta with exact rational arithmetic, not rounded decimals.
  • S007 · premise_and_result_audit — Recompute Biasvar ORIGINAL single theta mean sum with exact rational arithmetic, not rounded decimals.
  • S006 · premise_and_result_audit — Recompute Biasvar ORIGINAL single theta mean product2 with exact rational arithmetic, not rounded decimals.
  • S005 · premise_and_result_audit — Recompute Biasvar ORIGINAL single theta mean product1 with exact rational arithmetic, not rounded decimals.
  • S004 · premise_and_result_audit — Recompute Biasvar ORIGINAL single theta mean product0 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 Biasvar ORIGINAL single theta ORIGINAL mass sum 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

(8, 5, 9, 0, Yes, 9/2, 9/2, Yes, No, No, unbiased_only_gives_MSE_equals_variance_not_zero_MSE_or_consistency_and_observed_error_is_not_MSE)

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.

Source hashes and review receipts
  • input commit: 0f7be61e2c22317a4075561dc822c87d881a50db
  • original question SHA-256: 6f9881f97b8b35d0392e706d1dd7160c99bd4f2f58866cd4da1bde0dffd36d01
  • candidate record SHA-256: e43df9ec44fce76ea689c3ac25a80008aa053405878831e457dec096036694bd
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  • Lean: NOT_RUN · human review: NOT_PERFORMED
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