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Question 27181: A Uniform Coverage Target Is Checked against the Minimum, Not an Average

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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R0053-27181 · q-20261015-advance-20261118-r03-finite-confidence-interval-coverage-0056

Original question and givens

Use complete theta values=[0, 1], samples=[a, b], theta/sample-aligned probability rows=[[6/8, 2/8], [2/8, 6/8]], and sample-aligned closed intervals=[[0, 0], [1, 1]]. The interval procedure is total, precommitted and independent of the unknown theta. The declared target is=1/3. Give (pointwise_coverage_table, minimum_coverage, target, uniform_target_met, minimum_definition).

  • G001 — Use complete theta values=[0, 1], samples=[a, b], theta/sample-aligned probability rows=[[6/8, 2/8], [2/8, 6/8]], and sample-aligned closed intervals=[[0, 0], [1, 1]]. The interval procedure is total, precommitted and independent of the unknown theta. The declared target is=1/3. Give (pointwise_coverage_table, minimum_coverage, target, uniform_target_met, minimum_definition).

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 = [ci-worst-case-coverage]

After

derived = [[pointwise_coverage_table, minimum_coverage, target, uniform_target_met, minimum_definition]]

Conditions: none

List the complete source field order and mode-specific roles without JavaScript or answer-shape inference;No,undefined and infinity retain their distinct meanings.

Step identifiers and dependencies

Rule: source_ordered_fields_without_JS · Uses: G001

S002 · implies

Before

source roles = [[0, 1], [a, b], [[[6, 8], [2, 8]], [[2, 8], [6, 8]]], [[0, 0], [1, 1]]]

After

derived = [[0, 1], [a, b], [[0, 0], [1, 1]], [[3/4, 1/4], [1/4, 3/4]], [[a, [0, 0], 0], [b, [1, 1], 1]]]

Conditions: none

Bind intervals to ORIGINAL samples and probability rows/columns to ORIGINAL theta/samples BEFORE sorting those data together.Source13 theta order is numeric and sample order English canonical.Intervals are ordered and closed;each theta law is nonnegative and normalized.The same precommitted,total I(X) does not depend on unknown theta.

Step identifiers and dependencies

Rule: CI_ORIGINAL_theta_sample_interval_probability_alignment_Source13_numeric · Uses: G001, S001

S003 · calculation

Before

CI theta0 ORIGINAL probability normalization column0 := sum(0, 3/4)

After

CI theta0 ORIGINAL probability normalization column0 = 3/4

Conditions: none

Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S002

S004 · calculation

Before

CI theta0 ORIGINAL probability normalization column1 := sum(3/4, 1/4)

After

CI theta0 ORIGINAL probability normalization column1 = 1

Conditions: none

Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S003

S005 · calculation

Before

CI theta1 ORIGINAL probability normalization column0 := sum(0, 1/4)

After

CI theta1 ORIGINAL probability normalization column0 = 1/4

Conditions: none

Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S004

S006 · calculation

Before

CI theta1 ORIGINAL probability normalization column1 := sum(1/4, 3/4)

After

CI theta1 ORIGINAL probability normalization column1 = 1

Conditions: none

Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S005

S007 · calculation

Before

CI theta0 ORIGINAL included coverage column0 := sum(0, 3/4)

After

CI theta0 ORIGINAL included coverage column0 = 3/4

Conditions: none

Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S006

S008 · implies

Before

source roles = [0, [a, b], [[0, 0], [1, 1]], [3/4, 1/4]]

After

derived = [[[a, 0, 0, 0, Yes, 3/4], [b, 1, 1, 0, No, 1/4]], 3/4]

Conditions: none

Fix the true theta and test EVERY possible sample for L(x)<=theta<=U(x),including equality at either endpoint.Sum P_theta(x) only over hits.This is repeated-sampling coverage,not a posterior for theta after seeing data.

Step identifiers and dependencies

Rule: CI_EVERY_sample_closed_endpoint_membership_AND_weighted_coverage · Uses: G001, S007

S009 · calculation

Before

CI theta1 ORIGINAL included coverage column1 := sum(0, 3/4)

After

CI theta1 ORIGINAL included coverage column1 = 3/4

Conditions: none

Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S008

S010 · implies

Before

source roles = [1, [a, b], [[0, 0], [1, 1]], [1/4, 3/4]]

After

derived = [[[a, 0, 0, 1, No, 1/4], [b, 1, 1, 1, Yes, 3/4]], 3/4]

Conditions: none

Fix the true theta and test EVERY possible sample for L(x)<=theta<=U(x),including equality at either endpoint.Sum P_theta(x) only over hits.This is repeated-sampling coverage,not a posterior for theta after seeing data.

Step identifiers and dependencies

Rule: CI_EVERY_sample_closed_endpoint_membership_AND_weighted_coverage · Uses: G001, S009

S011 · implies

Before

source roles = [[[0, 3/4], [1, 3/4]], 1/3]

After

derived = [3/4, Yes]

Conditions: none

A uniform target requires EVERY theta;on this finite carrier this is equivalent to minimum coverage>=target.Keep the source rational target without inventing a[0,1] restriction;a target>1 can simply fail.

Step identifiers and dependencies

Rule: CI_ALL_theta_MINIMUM_uniform_target_NOT_average · Uses: G001, S010

Reverse checks

  • S011 · premise_and_result_audit — Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct.
  • S010 · premise_and_result_audit — Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct.
  • S009 · premise_and_result_audit — Recompute CI theta1 ORIGINAL included coverage column1 with exact rational arithmetic, not rounded decimals.
  • S008 · premise_and_result_audit — Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct.
  • S007 · premise_and_result_audit — Recompute CI theta0 ORIGINAL included coverage column0 with exact rational arithmetic, not rounded decimals.
  • S006 · premise_and_result_audit — Recompute CI theta1 ORIGINAL probability normalization column1 with exact rational arithmetic, not rounded decimals.
  • S005 · premise_and_result_audit — Recompute CI theta1 ORIGINAL probability normalization column0 with exact rational arithmetic, not rounded decimals.
  • S004 · premise_and_result_audit — Recompute CI theta0 ORIGINAL probability normalization column1 with exact rational arithmetic, not rounded decimals.
  • S003 · premise_and_result_audit — Recompute CI theta0 ORIGINAL probability normalization column0 with exact rational arithmetic, not rounded decimals.
  • S002 · premise_and_result_audit — Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct.
  • S001 · premise_and_result_audit — Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct.

Result

([[0, 3/4], [1, 3/4]], 3/4, 1/3, Yes, worst_case_is_minimum_over_finite_theta)

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: 5fd7f6f2db77530ef27a34a2bacc6064dd5ec586412ba6a19d4d6d6553c51d74
  • candidate record SHA-256: c1a0d53ae172598dfb09ddd9e3f0142dfbfc7637c583e11ca668b77d47b98d2f
  • semantic review receipt SHA-256: 1b7a62514347b82518305d5feb4369a52d7d44c442a0f2b131e9eb3cd7b45f62
  • question-first witness SHA-256: 79dc2a1077081f3ea2a03f75e32778163bf05e814cdf4c29ed86e567295f9ee1
  • Lean: NOT_RUN · human review: NOT_PERFORMED
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