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R0050-25534 · q-20261015-advance-20261115-r03-bias-variance-decomposition-0034
Use scalar theta=-4, complete estimator values=[-3, -2, -1] 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. Give (mean, centered_mean, bias, cross_term, cross_term_zero, certificate).
G001 — Use scalar theta=-4, complete estimator values=[-3, -2, -1] 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. Give (mean, centered_mean, bias, cross_term, cross_term_zero, certificate).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.
Before
source roles = [biasvar-cross-term-cancellation]
After
derived = [[mean, centered_mean, bias, cross_term, cross_term_zero, certificate]]
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.
Rule: source_ordered_fields_without_JS · Uses: G001
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.
Rule: exact_rational_arithmetic · Uses: G001, S001
Before
source roles = [Biasvar ORIGINAL single theta, -4, [-3, -2, -1], [[1, 4], [1, 2], [1, 4]]]
After
derived = [[(-3, 1/4), (-2, 1/2), (-1, 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.
Rule: Biasvar_complete_SAME_law_numeric_sorted_ALIGNED_null_rows · Uses: G001, S002
Before
Biasvar ORIGINAL single theta mean product0 := mul(-3, 1/4)
After
Biasvar ORIGINAL single theta 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.
Rule: exact_rational_arithmetic · Uses: G001, S003
Before
Biasvar ORIGINAL single theta mean product1 := mul(-2, 1/2)
After
Biasvar ORIGINAL single theta mean product1 = -1
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Rule: exact_rational_arithmetic · Uses: G001, S004
Before
Biasvar ORIGINAL single theta mean product2 := mul(-1, 1/4)
After
Biasvar ORIGINAL single theta mean product2 = -1/4
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Rule: exact_rational_arithmetic · Uses: G001, S005
Before
Biasvar ORIGINAL single theta mean sum := sum(-3/4, -1, -1/4)
After
Biasvar ORIGINAL single theta mean sum = -2
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Rule: exact_rational_arithmetic · Uses: G001, S006
Before
Biasvar ORIGINAL single theta bias mean minus theta := sub(-2, -4)
After
Biasvar ORIGINAL single theta bias mean minus theta = 2
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Rule: exact_rational_arithmetic · Uses: G001, S007
Before
Biasvar ORIGINAL single theta centered deviation0 := sub(-3, -2)
After
Biasvar ORIGINAL single theta centered deviation0 = -1
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Rule: exact_rational_arithmetic · Uses: G001, S008
Before
Biasvar ORIGINAL single theta centered square0 := mul(-1, -1)
After
Biasvar ORIGINAL single theta centered square0 = 1
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Rule: exact_rational_arithmetic · Uses: G001, S009
Before
Biasvar ORIGINAL single theta variance weighted term0 := mul(1/4, 1)
After
Biasvar ORIGINAL single theta variance weighted term0 = 1/4
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Rule: exact_rational_arithmetic · Uses: G001, S010
Before
Biasvar ORIGINAL single theta centered deviation1 := sub(-2, -2)
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.
Rule: exact_rational_arithmetic · Uses: G001, S011
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.
Rule: exact_rational_arithmetic · Uses: G001, S012
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.
Rule: exact_rational_arithmetic · Uses: G001, S013
Before
Biasvar ORIGINAL single theta centered deviation2 := sub(-1, -2)
After
Biasvar ORIGINAL single theta centered deviation2 = 1
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Rule: exact_rational_arithmetic · Uses: G001, S014
Before
Biasvar ORIGINAL single theta centered square2 := mul(1, 1)
After
Biasvar ORIGINAL single theta centered square2 = 1
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Rule: exact_rational_arithmetic · Uses: G001, S015
Before
Biasvar ORIGINAL single theta variance weighted term2 := mul(1/4, 1)
After
Biasvar ORIGINAL single theta variance weighted term2 = 1/4
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Rule: exact_rational_arithmetic · Uses: G001, S016
Before
Biasvar ORIGINAL single theta variance sum := sum(1/4, 0, 1/4)
After
Biasvar ORIGINAL single theta variance sum = 1/2
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Rule: exact_rational_arithmetic · Uses: G001, S017
Before
Biasvar ORIGINAL single theta theta error0 := sub(-3, -4)
After
Biasvar ORIGINAL single theta theta error0 = 1
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Rule: exact_rational_arithmetic · Uses: G001, S018
Before
Biasvar ORIGINAL single theta theta square0 := mul(1, 1)
After
Biasvar ORIGINAL single theta theta square0 = 1
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Rule: exact_rational_arithmetic · Uses: G001, S019
Before
Biasvar ORIGINAL single theta direct MSE term0 := mul(1/4, 1)
After
Biasvar ORIGINAL single theta direct MSE term0 = 1/4
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Rule: exact_rational_arithmetic · Uses: G001, S020
Before
Biasvar ORIGINAL single theta theta error1 := sub(-2, -4)
After
Biasvar ORIGINAL single theta theta error1 = 2
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Rule: exact_rational_arithmetic · Uses: G001, S021
Before
Biasvar ORIGINAL single theta theta square1 := mul(2, 2)
After
Biasvar ORIGINAL single theta theta square1 = 4
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Rule: exact_rational_arithmetic · Uses: G001, S022
Before
Biasvar ORIGINAL single theta direct MSE term1 := mul(1/2, 4)
After
Biasvar ORIGINAL single theta direct MSE term1 = 2
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Rule: exact_rational_arithmetic · Uses: G001, S023
Before
Biasvar ORIGINAL single theta theta error2 := sub(-1, -4)
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.
Rule: exact_rational_arithmetic · Uses: G001, S024
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.
Rule: exact_rational_arithmetic · Uses: G001, S025
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.
Rule: exact_rational_arithmetic · Uses: G001, S026
Before
Biasvar ORIGINAL single theta direct MSE sum := sum(1/4, 2, 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.
Rule: exact_rational_arithmetic · Uses: G001, S027
Before
Biasvar ORIGINAL single theta centered mean product0 := mul(1/4, -1)
After
Biasvar ORIGINAL single theta centered mean product0 = -1/4
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Rule: exact_rational_arithmetic · Uses: G001, S028
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.
Rule: exact_rational_arithmetic · Uses: G001, S029
Before
Biasvar ORIGINAL single theta centered mean product2 := mul(1/4, 1)
After
Biasvar ORIGINAL single theta centered mean product2 = 1/4
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Rule: exact_rational_arithmetic · Uses: G001, S030
Before
Biasvar ORIGINAL single theta centered mean sum := sum(-1/4, 0, 1/4)
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.
Rule: exact_rational_arithmetic · Uses: G001, S031
Before
Biasvar ORIGINAL single theta bias times centered mean := mul(2, 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.
Rule: exact_rational_arithmetic · Uses: G001, S032
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.
Rule: exact_rational_arithmetic · Uses: G001, S033
Before
Biasvar ORIGINAL single theta squared bias := mul(2, 2)
After
Biasvar ORIGINAL single theta squared bias = 4
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Rule: exact_rational_arithmetic · Uses: G001, S034
Before
Biasvar ORIGINAL single theta variance plus squared bias := sum(1/2, 4)
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.
Rule: exact_rational_arithmetic · Uses: G001, S035
Before
source roles = [Biasvar ORIGINAL single theta, -4, -2, 2, [(-3, 1/4), (-2, 1/2), (-1, 1/4)]]
After
derived = [[[-3, 1/4, -1, 2, 1, 1, -4, 4, 1], [-2, 1/2, 0, 2, 2, 0, 0, 4, 4], [-1, 1/4, 1, 2, 3, 1, 4, 4, 9]], 0, 0, 1/2, 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.
Rule: Biasvar_EVERY_value_square_expansion_and_GENERAL_cross_cancellation · Uses: G001, S036
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.(-2, 0, 2, 0, Yes, E_of_T_minus_E_T_is_zero_so_the_cross_term_vanishes)
The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.
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.
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