Cumulative distribution function

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 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 — The CDF aggregates the complete PMF.
• Usual order on the real numbers — The event uses X≤t.
• Probability on finite sample spaces — A CDF value is an event probability.
• 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.