Question 14927: The CDF Jump Is the Point Mass, Not the Whole CDF Value

Original question and givens

Use the complete finite discrete law with distinct integer support=[-11, -4, 2] and aligned nonnegative exact masses=[1/4, 1/4, 1/2], summing exactly to 1. Preserve support-to-mass alignment even when support is not sorted. F(t)=P(X≤t) includes equality. The CDF is not a point mass or density: its jump at t is P(X=t), it is nondecreasing and right-continuous, with endpoint limits 0 and 1. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Threshold t=-4. Give (t,left_limit,F_t,point_mass,jump_identity?).

• G001 — Use the complete finite discrete law with distinct integer support=[-11, -4, 2] and aligned nonnegative exact masses=[1/4, 1/4, 1/2], summing exactly to 1. Preserve support-to-mass alignment even when support is not sorted. F(t)=P(X≤t) includes equality. The CDF is not a point mass or density: its jump at t is P(X=t), it is nondecreasing and right-continuous, with endpoint limits 0 and 1. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Threshold t=-4. Give (t,left_limit,F_t,point_mass,jump_identity?).

Definition and scope

Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate.

• mko-cumulative-distribution-function

Step-by-step derivation

CDF law total mass := sum(1/4, 1/4, 1/2)

CDF law total mass = 1

original ordered support=[-11, -4, 2];raw aligned masses=[[1, 4], [1, 4], [1, 2]]

normalized aligned masses=[1/4, 1/4, 1/2];nonnegative=Yes;total=1

threshold CDF:original support index=0;value=-11;threshold=-4

test value<=threshold=Yes;aligned mass=1/4

threshold CDF:original support index=1;value=-4;threshold=-4

test value<=threshold=Yes;aligned mass=1/4

threshold CDF:original support index=2;value=2;threshold=-4

test value<=threshold=No;aligned mass=1/2

threshold CDF selected mass sum := sum(1/4, 1/4)

threshold CDF selected mass sum = 1/2

CDF left limit:original support index=0;value=-11;threshold=-4

test value<threshold=Yes;aligned mass=1/4

CDF left limit:original support index=1;value=-4;threshold=-4

test value<threshold=No;aligned mass=1/4

CDF left limit:original support index=2;value=2;threshold=-4

test value<threshold=No;aligned mass=1/2

CDF left limit selected mass sum := sum(1/4)

CDF left limit selected mass sum = 1/4

CDF threshold point mass := sum(1/4)

CDF threshold point mass = 1/4

CDF left limit plus point mass := sum(1/4, 1/4)

CDF left limit plus point mass = 1/2

threshold=-4;strict-below support mass=1/4;equal-point original indices=[1];point mass=1/4

left limit+point mass=1/2=F_t=1/2

Reverse checks

• S013 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S012 · premise_and_result_audit — Recompute CDF left limit plus point mass with exact rational arithmetic, not rounded decimals.
• S011 · premise_and_result_audit — Recompute CDF threshold point mass with exact rational arithmetic, not rounded decimals.
• S010 · premise_and_result_audit — Recompute CDF left limit selected mass sum with exact rational arithmetic, not rounded decimals.
• S009 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S008 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S007 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S006 · premise_and_result_audit — Recompute threshold CDF selected mass sum with exact rational arithmetic, not rounded decimals.
• S005 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S004 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S003 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S001 · premise_and_result_audit — Recompute CDF law total mass with exact rational arithmetic, not rounded decimals.

Result

(-4, 1/4, 1/2, 1/4, Yes)

The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.

This question review and source binding

Review date: 2026-10-26

Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.