Question 15708: Compare Bernoulli CDF Values at an Explicit Threshold

Original question and givens

Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=3/20, positive shift=9, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. n=8, threshold=1/2. Give (n,p_n,threshold,F_n,F,limit_CDF_continuity_point?).

• G001 — Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=3/20, positive shift=9, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. n=8, threshold=1/2. Give (n,p_n,threshold,F_n,F,limit_CDF_continuity_point?).

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-convergence-in-distribution

Step-by-step derivation

interior Bernoulli limit mass at0 := sub(1, 1/2)

interior Bernoulli limit mass at0 = 1/2

original limit parameter=[1, 2];reduced p=1/2;mass at0=1/2;mass at1=1/2

both endpoint masses positive;only limit-CDF jumps at0 and1;registered interior projection

Bernoulli first legal tail denominator := sum(1, 9)

Bernoulli first legal tail denominator = 10

Bernoulli first tail signed correction := div(3/20, 10)

Bernoulli first tail signed correction = 3/200

Bernoulli first legal parameter := sum(1/2, 3/200)

Bernoulli first legal parameter = 103/200

declared all-n family p_n=1/2+(3/20)/(n+9);n>=1;first parameter=103/200;limit=1/2

for every n>=1,lambda=(1+9)/(n+9) in(0,1];p_n=(1-lambda)*1/2+lambda*(103/200) in[0,1]

Bernoulli selected tail denominator := sum(8, 9)

Bernoulli selected tail denominator = 17

Bernoulli selected signed correction := div(3/20, 17)

Bernoulli selected signed correction = 3/340

Bernoulli selected exact parameter := sum(1/2, 3/340)

Bernoulli selected exact parameter = 173/340

selected distribution one minus Bernoulli parameter := sub(1, 173/340)

selected distribution one minus Bernoulli parameter = 167/340

selected distribution:Bernoulli parameter=173/340;same exact threshold=1/2;CDF branch=0<=t<1

selected distribution exact CDF value=167/340

limiting distribution one minus Bernoulli parameter := sub(1, 1/2)

limiting distribution one minus Bernoulli parameter = 1/2

limiting distribution:Bernoulli parameter=1/2;same exact threshold=1/2;CDF branch=0<=t<1

limiting distribution exact CDF value=1/2

limit parameter=1/2;threshold=1/2;jump set=[0, 1];selected CDF=167/340;limit CDF=1/2

continuity-point flag=Yes;finite-n equality does not decide this flag

Reverse checks

• S014 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• 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 limiting distribution one minus Bernoulli parameter with exact rational arithmetic, not rounded decimals.
• S011 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S010 · premise_and_result_audit — Recompute selected distribution one minus Bernoulli parameter with exact rational arithmetic, not rounded decimals.
• S009 · premise_and_result_audit — Recompute Bernoulli selected exact parameter with exact rational arithmetic, not rounded decimals.
• S008 · premise_and_result_audit — Recompute Bernoulli selected signed correction with exact rational arithmetic, not rounded decimals.
• S007 · premise_and_result_audit — Recompute Bernoulli selected tail denominator with exact rational arithmetic, not rounded decimals.
• S006 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S005 · premise_and_result_audit — Recompute Bernoulli first legal parameter with exact rational arithmetic, not rounded decimals.
• S004 · premise_and_result_audit — Recompute Bernoulli first tail signed correction with exact rational arithmetic, not rounded decimals.
• S003 · premise_and_result_audit — Recompute Bernoulli first legal tail denominator with exact rational arithmetic, not rounded decimals.
• 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 interior Bernoulli limit mass at0 with exact rational arithmetic, not rounded decimals.

Result

(8, 173/340, 1/2, 167/340, 1/2, 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.