# Cumulative distribution function

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

For a real-valued random variable, F(t)=P(X≤t) is nondecreasing and right-continuous; in a finite discrete law, its jump is the point mass.

mko-cumulative-distribution-function · definition · v0.10.0



## Mathematical statement

For a complete finite real-valued law, aggregate exactly every mass with x≤t. The CDF is not a PMF or density. Its value at a support point includes that point's mass, its jump is P(X=t), and its endpoint limits are 0 and 1.





## Representative formula (exact TeX)

F(t)





## Assumptions


- Integer support values are distinct and have exact nonnegative rational masses summing to 1.
- Comparisons retain ≤ and every output remains an exact rational.





## Symbol scope


- F — cumulative distribution function
- t — real threshold





## Intuition, definition, and misconceptions

The threshold includes equality.

A discrete jump is exactly the point mass.

Monotonicity and right continuity follow from the cumulative event structure.

A CDF value is not generally the single-point PMF, equality must not be omitted, and left continuity must not replace right continuity.





## Adopted derivation guide

[The complete CDF of a three-point law](https://ocme.evemisslab.com/en/derivations/derivation-mko-cumulative-distribution-function-01/)The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.





## Prerequisites


- [Finite discrete probability distribution](https://ocme.evemisslab.com/en/mko/mko-finite-discrete-probability-distribution/) — The CDF aggregates the complete PMF.
- [Usual order on the real numbers](https://ocme.evemisslab.com/en/mko/mko-order-relation/) — The event uses X≤t.
- [Probability on finite sample spaces](https://ocme.evemisslab.com/en/mko/mko-probability/) — A CDF value is an event probability.
- [Function mapping](https://ocme.evemisslab.com/en/mko/mko-function-mapping/) — F maps real thresholds to [0,1].





## Verification boundary

Only complete finite discrete laws are handled; no density or continuous-distribution claim is asserted.

not_formalized

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





## Next obligation

Add convergence in distribution.





## 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/cumulative-distribution-function.json) · [Plain Markdown](https://ocme.evemisslab.com/en/mko/mko-cumulative-distribution-function/index.md)