Question 642: A Total Statistic Partitions the Complete Sample Carrier into Fibers

Original question and givens

Use complete theta labels=[a16, b16], samples=[160, 161, 162, 163], aligned exact probability rows=[[[2, 5], [1, 10], [2, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 1, 0, 1]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Give (samples, aligned_statistic, fibers, partition, certificate).

• G001 — Use complete theta labels=[a16, b16], samples=[160, 161, 162, 163], aligned exact probability rows=[[[2, 5], [1, 10], [2, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 1, 0, 1]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Give (samples, aligned_statistic, fibers, partition, certificate).

Definition and scope

The statistic is total on the sample carrier; the fibers by value are disjoint and cover every sample.

• mko-finite-sufficient-statistic

Step-by-step derivation

Θ=[a16, b16]; Ω=[160, 161, 162, 163]

[[2/5, 1/10, 2/5, 1/10], [1/10, 2/5, 1/10, 2/5]]

P_a16(Ω) := sum(2/5, 1/10, 2/5, 1/10)

P_a16(Ω) = 1

P_b16(Ω) := sum(1/10, 2/5, 1/10, 2/5)

P_b16(Ω) = 1

sample=160

T(160)=0

sample=161

T(161)=1

sample=162

T(162)=0

sample=163

T(163)=1

[(160, 0), (161, 1), (162, 0), (163, 1)]

fibers=[[0, [160, 162]], [1, [161, 163]]]

Reverse checks

• S008 · premise_and_result_audit — Retain every parameter and sample label; distinguish null, empty tables, zero distributions and tied optima.
• S007 · premise_and_result_audit — Retain every parameter and sample label; distinguish null, empty tables, zero distributions and tied optima.
• S006 · premise_and_result_audit — Retain every parameter and sample label; distinguish null, empty tables, zero distributions and tied optima.
• S005 · premise_and_result_audit — Retain every parameter and sample label; distinguish null, empty tables, zero distributions and tied optima.
• S004 · premise_and_result_audit — Retain every parameter and sample label; distinguish null, empty tables, zero distributions and tied optima.
• S003 · premise_and_result_audit — Recompute P_b16(Ω) with exact rational arithmetic, not rounded decimals.
• S002 · premise_and_result_audit — Recompute P_a16(Ω) with exact rational arithmetic, not rounded decimals.
• S001 · premise_and_result_audit — Retain every parameter and sample label; distinguish null, empty tables, zero distributions and tied optima.

Result

([160, 161, 162, 163], [0, 1, 0, 1], [[0, [160, 162]], [1, [161, 163]]], Yes, total_statistic_fibers_partition_sample_space)

The full substitutions and premise audits recover the original answer, retaining its field order and verdict meanings.

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.