Sampling distribution

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

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 — Each parameter supplies a sample law.
• Statistical estimator — One total statistic is independent of the unknown parameter.
• 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.