# Question 15724: Compare Bernoulli CDF Values at an Explicit Threshold

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This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth.

R0030-15724 · q-20261014-advance-20261031-r03-convergence-in-distribution-0024



## Original question and givens

Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=4/20, positive shift=25, 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=24, threshold=2/1. 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=4/20, positive shift=25, 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=24, threshold=2/1. 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](https://ocme.evemisslab.com/en/mko/mko-convergence-in-distribution/)





## Step-by-step derivation

S001 · calculationBefore

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

After

interior Bernoulli limit mass at0 = 1/2

Conditions: none

Compute exactly from original roles and established intermediates,retaining indices,signs,denominators and the operation's carrier.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001





S002 · impliesBefore

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

After

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

Conditions: none

Honor the source English restriction0<p<1,so both jumps genuinely have positive mass.Although the raw oracle allows endpoint parameters,do not extend this continuity-point protocol to degenerate distributions;their jump sets require separate analysis.

Step identifiers and dependenciesRule: Bernoulli_interior_English_guard_not_degenerate_jump_set · Uses: G001, S001





S003 · calculationBefore

Bernoulli first legal tail denominator := sum(1, 25)

After

Bernoulli first legal tail denominator = 26

Conditions: none

Compute exactly from original roles and established intermediates,retaining indices,signs,denominators and the operation's carrier.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002





S004 · calculationBefore

Bernoulli first tail signed correction := div(1/5, 26)

After

Bernoulli first tail signed correction = 1/130

Conditions: none

Compute exactly from original roles and established intermediates,retaining indices,signs,denominators and the operation's carrier.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003





S005 · calculationBefore

Bernoulli first legal parameter := sum(1/2, 1/130)

After

Bernoulli first legal parameter = 33/65

Conditions: none

Compute exactly from original roles and established intermediates,retaining indices,signs,denominators and the operation's carrier.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004





S006 · impliesBefore

declared all-n family p_n=1/2+(1/5)/(n+25);n>=1;first parameter=33/65;limit=1/2

After

for every n>=1,lambda=(1+25)/(n+25) in(0,1];p_n=(1-lambda)*1/2+lambda*(33/65) in[0,1]

Conditions: none

The signed c may be positive,negative or zero.Every parameter is a convex combination of the first legal parameter and the limit,so their probability bounds cover the whole tail,not merely one displayed n.No chosen coupling is required for this family of distributions.

Step identifiers and dependenciesRule: Bernoulli_whole_signed_parameter_tail_convex_combination · Uses: G001, S005





S007 · calculationBefore

Bernoulli selected tail denominator := sum(24, 25)

After

Bernoulli selected tail denominator = 49

Conditions: none

Compute exactly from original roles and established intermediates,retaining indices,signs,denominators and the operation's carrier.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S006





S008 · calculationBefore

Bernoulli selected signed correction := div(1/5, 49)

After

Bernoulli selected signed correction = 1/245

Conditions: none

Compute exactly from original roles and established intermediates,retaining indices,signs,denominators and the operation's carrier.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S007





S009 · calculationBefore

Bernoulli selected exact parameter := sum(1/2, 1/245)

After

Bernoulli selected exact parameter = 247/490

Conditions: none

Compute exactly from original roles and established intermediates,retaining indices,signs,denominators and the operation's carrier.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S008





S010 · impliesBefore

selected distribution:Bernoulli parameter=247/490;same exact threshold=2;CDF branch=t>=1

After

selected distribution exact CDF value=1

Conditions: none

A CDF is P(X≤t):at0 include mass1−p,and at1 include all mass.Use the same threshold branch for both CDFs.The continuity flag is distinct from equality of two finite-n values.

Step identifiers and dependenciesRule: Bernoulli_CDF_same_threshold_including_endpoints · Uses: G001, S009





S011 · impliesBefore

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

After

limiting distribution exact CDF value=1

Conditions: none

A CDF is P(X≤t):at0 include mass1−p,and at1 include all mass.Use the same threshold branch for both CDFs.The continuity flag is distinct from equality of two finite-n values.

Step identifiers and dependenciesRule: Bernoulli_CDF_same_threshold_including_endpoints · Uses: G001, S010





S012 · impliesBefore

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

After

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

Conditions: none

The final flag belongs to the limit CDF;equal displayed values do not establish continuity.Convergence in distribution does not imply convergence in probability,almost surely or in Lp.

Step identifiers and dependenciesRule: convdist_continuity_is_limit_CDF_property · Uses: G001, S011









## Reverse checks


- S012 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- 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 — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- 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

(24, 247/490, 2, 1, 1, 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.

Source hashes and review receipts
- input commit: 0f7be61e2c22317a4075561dc822c87d881a50db
- original question SHA-256: bf53349c0722bec025df18ae5765cc663428ff1b165a89b9cb7ecdb78317536b
- candidate record SHA-256: 1d475b77e429fe8c30e11f804ece5a156c62452ac224643767908fbb07c132be
- semantic review receipt SHA-256: 73bc08066ffb1a81eacb2d4e288d03f522aeefd3949fa0d97da41b484ed737bc
- question-first witness SHA-256: ac120452a208b257b8270f38351a09dcd832cdbc8d9db6b69933f83e43cc9336
- Lean: NOT_RUN · human review: NOT_PERFORMED

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