# Sampling distribution

This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.

At a fixed parameter, an estimator pushes the sample law forward to its sampling distribution; all parameters give a family of such laws.

mko-sampling-distribution · definition · v0.10.0



## Mathematical statement

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 mass1. A sampling distribution is not an observed estimate, likelihood, posterior or distribution of the parameter itself.





## Representative formula (exact TeX)

P_\theta(T)





## Assumptions


- All parameters share the same fixed finite sample space and total statistic.
- Every parameter law is a complete exact rational distribution.





## Symbol scope


- T — estimator
- P_\theta^T — sampling distribution





## Intuition, definition, and misconceptions

Each parameter has its own pushforward law.

One observed estimate is only the value of T on one sample.

Unbiasedness, low variance, consistency and posterior claims are not supplied here.

An observed numerical estimate, likelihood or posterior is not the sampling distribution.





## Adopted derivation guide

[A complete fibre-pushforward sampling distribution](https://ocme.evemisslab.com/en/derivations/derivation-mko-sampling-distribution-01/)The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.





## Prerequisites


- [Parametric statistical model](https://ocme.evemisslab.com/en/mko/mko-parametric-statistical-model/) — Each parameter supplies a sample law.
- [Statistical estimator](https://ocme.evemisslab.com/en/mko/mko-statistical-estimator/) — One total statistic is independent of the unknown parameter.
- [Finite discrete probability distribution](https://ocme.evemisslab.com/en/mko/mko-finite-discrete-probability-distribution/) — The pushforward is another probability law.





## Verification boundary

Only complete finite models and exact pushforwards are handled; statistical properties remain later obligations.

not_formalized

This does not mean the content is false; it means no replayable computational or formal evidence is registered yet.





## Next obligation

Add estimator bias and estimator variance.





## Same-source structured data

This page and the interactive view use the same object and existing language edition; this adds no proof or human review.

[Canonical MKO JSON](https://ocme.evemisslab.com/data/mko/sampling-distribution.json) · [Plain Markdown](https://ocme.evemisslab.com/en/mko/mko-sampling-distribution/index.md)