Formal Note Example: Unbiasedness of the Sample Mean Under Integrability
A compact Basis manuscript showing that the system can produce a tight proof note with clear assumptions, a clean theorem statement, and an explicit failure case.
This example shows that Basis can stay concise. It is a short formal note, not a padded essay, and it still preserves a real artifact trail and an inspectable manuscript output.
ObjectiveGive a rigorous proof that the sample mean is an unbiased estimator under finite expectation, make the key assumption explicit, and include a concrete failure case when the expectation is not defined.
OutputsPDF · TeX manuscript
ViewerFull public edition of the short note with a cleaned title page and no placeholder author text.
Run brief
What this run had to deliver.
Basis had to satisfy a concrete objective, keep the assumptions explicit, and leave behind artifacts
a human could inspect and continue.
Objective
Formal Note Example: Unbiasedness of the Sample Mean Under Integrability
Give a rigorous proof that the sample mean is an unbiased estimator under finite expectation, make the key assumption explicit, and include a concrete failure case when the expectation is not defined.
Scope
What had to be covered
State the minimal assumption that makes the expectation well-defined.
Prove the unbiasedness identity cleanly with linearity of expectation.
Show why the claim breaks down for heavy-tailed cases such as the standard Cauchy law.
Artifacts
What persisted after the run
Public manuscript PDF
LaTeX source manuscript
Abstract and main theorem note
Curated public viewer edition
Inside the output
What the document actually says.
Key assumptionIntegrability makes the expectation well-defined and keeps the proof honest.
Proof moveUses linearity of expectation on a finite sum rather than hand-wavy intuition.
Failure modeExplains why the standard Cauchy law makes unbiasedness ill-posed.
What the note covers
Representative summary
The note isolates integrability as the key assumption, proves the unbiasedness identity directly, and makes clear that independence is not required for the expectation calculation itself.
It then closes with a concrete failure case: for the standard Cauchy distribution, the mean is undefined, so the usual unbiasedness statement is not mathematically well-posed.
Main theorem page with the proof and the concrete failure case setup.Closing page showing the heavy-tail counterexample and short conclusion.
Actual PDF
Read the output directly.
Full public edition of the short note with a cleaned title page and no placeholder author text.
