The complete CDF of a three-point law

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

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

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

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

F(t)=0

-1\le t<1

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

1\le t<3

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

t\ge3

F(t)=1

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

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

0\le1/4\le3/4\le1

F\text{ is nondecreasing and right-continuous}

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

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

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

Endpoints

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

CDF versus PMF

Recover 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

Same-source full JSON (formulas, ASTs, controls and revision log)