Question 648: A Total Statistic Partitions the Complete Sample Carrier into Fibers

Original question and givens

Use complete theta labels=[a22, b22], samples=[220, 221, 222, 223], 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=[a22, b22], samples=[220, 221, 222, 223], 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=[a22, b22]; Ω=[220, 221, 222, 223]

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

P_a22(Ω) := sum(2/5, 1/10, 2/5, 1/10)

P_a22(Ω) = 1

P_b22(Ω) := sum(1/10, 2/5, 1/10, 2/5)

P_b22(Ω) = 1

sample=220

T(220)=0

sample=221

T(221)=1

sample=222

T(222)=0

sample=223

T(223)=1

[(220, 0), (221, 1), (222, 0), (223, 1)]

fibers=[[0, [220, 222]], [1, [221, 223]]]

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_b22(Ω) with exact rational arithmetic, not rounded decimals.
• S002 · premise_and_result_audit — Recompute P_a22(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

([220, 221, 222, 223], [0, 1, 0, 1], [[0, [220, 222]], [1, [221, 223]]], 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.