A complete fibre-pushforward sampling distribution

Goal and scope

Aggregate equal-valued outcomes of U under both theta laws.

A sampling distribution is not a parameter distribution after one observation.

Givens and symbols

• work

• G1 — Theta={1/4,3/4}, Omega={00,01,10,11}; P_(1/4)=(9,3,3,1)/16 and P_(3/4)=(1,3,3,9)/16.
• G2 — U(x)=(x1+x2)/4 has total table(0,1/4,1/4,1/2) in Omega order, with target g(theta)=theta.

• θ — fixed unknown parameter
• U — one total statistic

Rules and sources

Some rule statements have reader clarifications in this revision: they are paraphrases, not verbatim quotations. Referenced original source bytes and hashes are unchanged.

• mko_definition · cited — On one common finite sample space, fix a total statistic T independent of the unknown parameter. For each parameter, sum probabilities over statistic fibres and recheck nonnegativity and total mass 1. A sampling distribution is not an observed estimate, likelihood, posterior or distribution of the parameter itself. · mko-sampling-distribution
• work · prerequisite — Finite laws have nonnegative probabilities summing to one; a statistic depends only on data, and expectation sums all weighted outcomes.

Main derivation

U=(0,1/4,1/4,1/2)

U(\Omega)=\{0,1/4,1/2\}

U^{-1}(0),U^{-1}(1/4),U^{-1}(1/2)

\{00\},\{01,10\},\{11\}

P_{1/4}(U=0)

9/16

P_{1/4}(U=1/4)

3/16+3/16=3/8

P_{1/4}(U=1/2)=1/16

P_{1/4}^U=(9/16,3/8,1/16)

P_{3/4}\text{ on the same fibres}

P_{3/4}^U=(1/16,3/8,9/16)

9/16+3/8+1/16,\quad1/16+3/8+9/16

1,\quad1

x_{\rm obs}=01

U(x_{\rm obs})=1/4\ne\text{the entire law }P_\theta^U

Substeps and lemmas

No substeps in this guide.

Reverse audits

• S08 · premise_audit · G1, G2, S07, work — This distinguishes objects, not an assertion that numerical values never coincide.
• S07 · premise_audit · G1, G2, S06, work — Normalization relies on the complete fibre partition.
• S06 · premise_audit · G1, G2, S05, work — Equal middle entries do not make the full laws equal.
• S05 · premise_audit · G1, G2, S04, work — Entries are ordered by 0,1/4,1/2.
• S04 · premise_audit · G1, G2, S03, work — Using only the01 mass 3/16 would omit10.
• S03 · premise_audit · G1, G2, S02, work — This is a statistic probability, not a parameter probability.
• S02 · premise_audit · G1, G2, S01, work — Each sample belongs to exactly one fibre.
• S01 · premise_audit · G1, G2, work — 01 and 10 belong to one fibre.

Error controls and negative controls

Complete fibres

List{00},{01,10},{11} before summing masses.

Parameterwise

Keep pushforwards separate rather than averaging laws.

• P_(1/4)(U=1/4)=3/16. → reject — Both01 and 10 contribute, giving 3/8.
• The two laws are identical. → reject — Their U=0 masses are 9/16 and 1/16.
• Only observed 01 is needed for the sampling distribution. → reject — All carrier masses are required for normalization.

Result and limitations

The sampling laws of U are(9/16,3/8,1/16) and(1/16,3/8,9/16).

Verification and source binding

3 declared checks for this guide; only the listed expressions are covered.

• source commit: b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f
• adopted source SHA-256: 34dc1cd85c3efadbcfd415dbe36f253fdad355a17c78ddeef279fa0cbb2bf808
• original source SHA-256: e67f4ec52577c1384b1c87cc1e0aa76a24df5697a005983130df8ae74bd5704c
• Lean: NOT_RUN · independent AI: SCOPED_LEAD_AND_BILINGUAL_CONTEXT_REVIEW

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