# The complete CDF of a three-point law

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This is an owner-adopted natural-language teaching guide. Finite/arithmetic checks are not universal proofs; no Lean was run, and review scope is recorded per revision. Original source status labels are retained as historical data. Formulas retain exact TeX; no new semantic AST is invented.

derivation-mko-cumulative-distribution-function-01 · ocme-new142-reader-adoption-20261021



## Goal and scope

Compute the CDF on the whole real line, including its right-continuous values and its jumps.

A complete finite discrete law, not a density calculation.





## Givens and symbols


- work


- G1 — X has support{-1,1,3} with masses 1/4,1/2,1/4.


- F — cumulative distribution function
- t — real threshold





## Rules and sources


- mko_definition · cited — 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. · mko-cumulative-distribution-function
- work · prerequisite — Event probabilities sum included masses; F(t)=P(X≤t), with jump F(t)-F(t-).





## Main derivation

S01 · impliesBefore

p=(-1:1/4,1:1/2,3:1/4)

After

1/4+1/2+1/4=1

Conditions: none

Rule: work · Uses: G1, work

Leaf basis: work

The three nonnegative rational masses form a complete law.



S02 · impliesBefore

F(t)=P(X\le t),\ t<-1

After

F(t)=0

Conditions: none

Rule: work · Uses: G1, S01, work

Leaf basis: work

Below the least support value the cumulative event is empty.



S03 · impliesBefore

-1\le t<1

After

\{X\le t\}=\{X=-1\},\ F(t)=1/4

Conditions: none

Rule: work · Uses: G1, S02, work

Leaf basis: work

This whole interval includes only the mass at -1, including its left endpoint.



S04 · impliesBefore

1\le t<3

After

F(t)=1/4+1/2=3/4

Conditions: none

Rule: work · Uses: G1, S03, work

Leaf basis: work

Crossing1 adds its full mass 1/2.



S05 · impliesBefore

t\ge3

After

F(t)=1

Conditions: none

Rule: work · Uses: G1, S04, work

Leaf basis: work

The final interval includes all three points.



S06 · impliesBefore

F(-1)-F(-1^-),F(1)-F(1^-),F(3)-F(3^-)

After

(1/4,1/2,1/4)

Conditions: none

Rule: work · Uses: G1, S05, work

Leaf basis: work

The three jumps recover the original PMF.



S07 · impliesBefore

0\le1/4\le3/4\le1

After

F\text{ is nondecreasing and right-continuous}

Conditions: none

Rule: work · Uses: G1, S06, work

Leaf basis: work

Constant intervals joined left-closed and right-open give right continuity at each support point.



S08 · impliesBefore

t<-1\text{ or }t\ge3

After

\lim_{t\to-\infty}F(t)=0,\quad\lim_{t\to\infty}F(t)=1

Conditions: none

Rule: work · Uses: G1, S07, work

Leaf basis: work

The tails are identically0 and 1, proving both endpoint limits.







## Substeps and lemmas

No substeps in this guide.





## Reverse audits


- S08 · premise_audit · G1, S07, work — All four real-line regions are covered, not merely three test points.
- S07 · premise_audit · G1, S06, work — Jumps are allowed; right continuity is not two-sided continuity.
- S06 · premise_audit · G1, S05, work — Use the preceding interval for left limits, not another right value.
- S05 · premise_audit · G1, S04, work — The point 3 belongs here, not in the preceding interval.
- S04 · premise_audit · G1, S03, work — The CDF is 3/4, not the point mass P(X=1)=1/2.
- S03 · premise_audit · G1, S02, work — At t=-1 the left limit0 is not the function value.
- S02 · premise_audit · G1, S01, work — The endpoint -1 is excluded because the CDF includes equality.
- S01 · premise_audit · G1, work — The support values are distinct; no duplicate mass is hidden.





## Error controls and negative controls

EndpointsUse left-closed/right-open pieces and include3 in the last piece.

CDF versus PMFRecover point masses by differences of adjacent cumulative values.


- F(1)=1/2. → reject — The mass 1/4 at -1 is also included, giving 3/4.
- F(-1)=0. → reject — The event uses≤ and includes-1.
- This CDF is continuous at 1. → reject — Its jump is 1/2; it is only right-continuous there.





## Result and limitations

The four CDF regions have values 0,1/4,3/4,1; its jumps are precisely the masses.







## Verification and source binding

2 declared checks for this guide; only the listed expressions are covered.


- source commit: b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f
- adopted source SHA-256: 1df2921a8fc4b01d41e743dfcb4bed7628ea9a2b4e9edb112f1cbe3af187c073
- original source SHA-256: 48d250722a2d877fdc939dece0ff74b714aaa7b3d447e8ea86c8f60e1ca87802
- Lean: NOT_RUN · independent AI: SCOPED_LEAD_AND_BILINGUAL_CONTEXT_REVIEW

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