# Exact Cumulative Masses, Jumps, Monotonicity and Right Continuity

This batch has passed Schema, deduplication, source-snapshot, and independent-answer recomputation checks; it is still not marked as human-reviewed.

This question bank has corresponding bilingual step-by-step derivations. Source candidate and English ai_candidate states are retained. AI review, finite computation and Lean formal proof are labelled separately.

batch-20261014-advance-20261030-r06-cumulative-distribution-function-foundations · 2026-10-14 · 100 questions

This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.



## 1. The CDF Includes Mass Exactly at the Threshold

Use the complete finite discrete law with distinct integer support=[-12, -5, 1] 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=-12. Give (t,F_t,uses_X_less_than_or_equal_t).

View answer and explanation(-12, 1/4, uses_X_less_than_or_equal_t)

Sum every mass whose aligned support value is at most t, including equality. Below all support the sum is zero; at or above the largest support it is one. The exact result, retaining the requested order, is (-12, 1/4, uses_X_less_than_or_equal_t).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r06-cumulative-distribution-function-0001/)

Cumulative distribution function · qf-mod30-cdf-evaluate · candidate





## 2. The CDF Includes Mass Exactly at the Threshold

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,F_t,uses_X_less_than_or_equal_t).

View answer and explanation(-4, 1/2, uses_X_less_than_or_equal_t)

Sum every mass whose aligned support value is at most t, including equality. Below all support the sum is zero; at or above the largest support it is one. The exact result, retaining the requested order, is (-4, 1/2, uses_X_less_than_or_equal_t).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r06-cumulative-distribution-function-0002/)

Cumulative distribution function · qf-mod30-cdf-evaluate · candidate





## 3. The CDF Includes Mass Exactly at the Threshold

Use the complete finite discrete law with distinct integer support=[-10, -3, 3] 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=3. Give (t,F_t,uses_X_less_than_or_equal_t).

View answer and explanation(3, 1, uses_X_less_than_or_equal_t)

Sum every mass whose aligned support value is at most t, including equality. Below all support the sum is zero; at or above the largest support it is one. The exact result, retaining the requested order, is (3, 1, uses_X_less_than_or_equal_t).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r06-cumulative-distribution-function-0003/)

Cumulative distribution function · qf-mod30-cdf-evaluate · candidate





## 4. The CDF Includes Mass Exactly at the Threshold

Use the complete finite discrete law with distinct integer support=[-9, -2, 4] 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=-9. Give (t,F_t,uses_X_less_than_or_equal_t).

View answer and explanation(-9, 1/4, uses_X_less_than_or_equal_t)

Sum every mass whose aligned support value is at most t, including equality. Below all support the sum is zero; at or above the largest support it is one. The exact result, retaining the requested order, is (-9, 1/4, uses_X_less_than_or_equal_t).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r06-cumulative-distribution-function-0004/)

Cumulative distribution function · qf-mod30-cdf-evaluate · candidate





## 5. The CDF Includes Mass Exactly at the Threshold

Use the complete finite discrete law with distinct integer support=[-8, -1, 5] 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=-1. Give (t,F_t,uses_X_less_than_or_equal_t).

View answer and explanation(-1, 1/2, uses_X_less_than_or_equal_t)

Sum every mass whose aligned support value is at most t, including equality. Below all support the sum is zero; at or above the largest support it is one. The exact result, retaining the requested order, is (-1, 1/2, uses_X_less_than_or_equal_t).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r06-cumulative-distribution-function-0005/)

Cumulative distribution function · qf-mod30-cdf-evaluate · candidate





## 6. The CDF Includes Mass Exactly at the Threshold

Use the complete finite discrete law with distinct integer support=[-7, 0, 6] 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=6. Give (t,F_t,uses_X_less_than_or_equal_t).

View answer and explanation(6, 1, uses_X_less_than_or_equal_t)

Sum every mass whose aligned support value is at most t, including equality. Below all support the sum is zero; at or above the largest support it is one. The exact result, retaining the requested order, is (6, 1, uses_X_less_than_or_equal_t).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r06-cumulative-distribution-function-0006/)

Cumulative distribution function · qf-mod30-cdf-evaluate · candidate





## 7. The CDF Includes Mass Exactly at the Threshold

Use the complete finite discrete law with distinct integer support=[-6, 1, 7] 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=-6. Give (t,F_t,uses_X_less_than_or_equal_t).

View answer and explanation(-6, 1/4, uses_X_less_than_or_equal_t)

Sum every mass whose aligned support value is at most t, including equality. Below all support the sum is zero; at or above the largest support it is one. The exact result, retaining the requested order, is (-6, 1/4, uses_X_less_than_or_equal_t).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r06-cumulative-distribution-function-0007/)

Cumulative distribution function · qf-mod30-cdf-evaluate · candidate





## 8. The CDF Includes Mass Exactly at the Threshold

Use the complete finite discrete law with distinct integer support=[-5, 2, 8] 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=2. Give (t,F_t,uses_X_less_than_or_equal_t).

View answer and explanation(2, 1/2, uses_X_less_than_or_equal_t)

Sum every mass whose aligned support value is at most t, including equality. Below all support the sum is zero; at or above the largest support it is one. The exact result, retaining the requested order, is (2, 1/2, uses_X_less_than_or_equal_t).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r06-cumulative-distribution-function-0008/)

Cumulative distribution function · qf-mod30-cdf-evaluate · candidate





## 9. The CDF Includes Mass Exactly at the Threshold

Use the complete finite discrete law with distinct integer support=[-4, 3, 9] 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=9. Give (t,F_t,uses_X_less_than_or_equal_t).

View answer and explanation(9, 1, uses_X_less_than_or_equal_t)

Sum every mass whose aligned support value is at most t, including equality. Below all support the sum is zero; at or above the largest support it is one. The exact result, retaining the requested order, is (9, 1, uses_X_less_than_or_equal_t).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r06-cumulative-distribution-function-0009/)

Cumulative distribution function · qf-mod30-cdf-evaluate · candidate





## 10. The CDF Includes Mass Exactly at the Threshold

Use the complete finite discrete law with distinct integer support=[-3, 4, 10] 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=-3. Give (t,F_t,uses_X_less_than_or_equal_t).

View answer and explanation(-3, 1/4, uses_X_less_than_or_equal_t)

Sum every mass whose aligned support value is at most t, including equality. Below all support the sum is zero; at or above the largest support it is one. The exact result, retaining the requested order, is (-3, 1/4, uses_X_less_than_or_equal_t).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r06-cumulative-distribution-function-0010/)

Cumulative distribution function · qf-mod30-cdf-evaluate · candidate





## 11. The CDF Includes Mass Exactly at the Threshold

Use the complete finite discrete law with distinct integer support=[-2, 5, 11] 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=5. Give (t,F_t,uses_X_less_than_or_equal_t).

View answer and explanation(5, 1/2, uses_X_less_than_or_equal_t)

Sum every mass whose aligned support value is at most t, including equality. Below all support the sum is zero; at or above the largest support it is one. The exact result, retaining the requested order, is (5, 1/2, uses_X_less_than_or_equal_t).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r06-cumulative-distribution-function-0011/)

Cumulative distribution function · qf-mod30-cdf-evaluate · candidate





## 12. The CDF Includes Mass Exactly at the Threshold

Use the complete finite discrete law with distinct integer support=[-1, 6, 12] 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=12. Give (t,F_t,uses_X_less_than_or_equal_t).

View answer and explanation(12, 1, uses_X_less_than_or_equal_t)

Sum every mass whose aligned support value is at most t, including equality. Below all support the sum is zero; at or above the largest support it is one. The exact result, retaining the requested order, is (12, 1, uses_X_less_than_or_equal_t).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r06-cumulative-distribution-function-0012/)

Cumulative distribution function · qf-mod30-cdf-evaluate · candidate





## 13. The CDF Includes Mass Exactly at the Threshold

Use the complete finite discrete law with distinct integer support=[0, 7, 13] 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=0. Give (t,F_t,uses_X_less_than_or_equal_t).

View answer and explanation(0, 1/4, uses_X_less_than_or_equal_t)

Sum every mass whose aligned support value is at most t, including equality. Below all support the sum is zero; at or above the largest support it is one. The exact result, retaining the requested order, is (0, 1/4, uses_X_less_than_or_equal_t).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r06-cumulative-distribution-function-0013/)

Cumulative distribution function · qf-mod30-cdf-evaluate · candidate





## 14. The CDF Includes Mass Exactly at the Threshold

Use the complete finite discrete law with distinct integer support=[1, 8, 14] 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=8. Give (t,F_t,uses_X_less_than_or_equal_t).

View answer and explanation(8, 1/2, uses_X_less_than_or_equal_t)

Sum every mass whose aligned support value is at most t, including equality. Below all support the sum is zero; at or above the largest support it is one. The exact result, retaining the requested order, is (8, 1/2, uses_X_less_than_or_equal_t).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r06-cumulative-distribution-function-0014/)

Cumulative distribution function · qf-mod30-cdf-evaluate · candidate





## 15. The CDF Includes Mass Exactly at the Threshold

Use the complete finite discrete law with distinct integer support=[2, 9, 15] 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=15. Give (t,F_t,uses_X_less_than_or_equal_t).

View answer and explanation(15, 1, uses_X_less_than_or_equal_t)

Sum every mass whose aligned support value is at most t, including equality. Below all support the sum is zero; at or above the largest support it is one. The exact result, retaining the requested order, is (15, 1, uses_X_less_than_or_equal_t).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r06-cumulative-distribution-function-0015/)

Cumulative distribution function · qf-mod30-cdf-evaluate · candidate





## 16. The CDF Includes Mass Exactly at the Threshold

Use the complete finite discrete law with distinct integer support=[3, 10, 16] 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=3. Give (t,F_t,uses_X_less_than_or_equal_t).

View answer and explanation(3, 1/4, uses_X_less_than_or_equal_t)

Sum every mass whose aligned support value is at most t, including equality. Below all support the sum is zero; at or above the largest support it is one. The exact result, retaining the requested order, is (3, 1/4, uses_X_less_than_or_equal_t).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r06-cumulative-distribution-function-0016/)

Cumulative distribution function · qf-mod30-cdf-evaluate · candidate





## 17. The CDF Includes Mass Exactly at the Threshold

Use the complete finite discrete law with distinct integer support=[4, 11, 17] 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=11. Give (t,F_t,uses_X_less_than_or_equal_t).

View answer and explanation(11, 1/2, uses_X_less_than_or_equal_t)

Sum every mass whose aligned support value is at most t, including equality. Below all support the sum is zero; at or above the largest support it is one. The exact result, retaining the requested order, is (11, 1/2, uses_X_less_than_or_equal_t).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r06-cumulative-distribution-function-0017/)

Cumulative distribution function · qf-mod30-cdf-evaluate · candidate





## 18. The CDF Includes Mass Exactly at the Threshold

Use the complete finite discrete law with distinct integer support=[5, 12, 18] 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=18. Give (t,F_t,uses_X_less_than_or_equal_t).

View answer and explanation(18, 1, uses_X_less_than_or_equal_t)

Sum every mass whose aligned support value is at most t, including equality. Below all support the sum is zero; at or above the largest support it is one. The exact result, retaining the requested order, is (18, 1, uses_X_less_than_or_equal_t).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r06-cumulative-distribution-function-0018/)

Cumulative distribution function · qf-mod30-cdf-evaluate · candidate





## 19. The CDF Includes Mass Exactly at the Threshold

Use the complete finite discrete law with distinct integer support=[6, 13, 19] 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=6. Give (t,F_t,uses_X_less_than_or_equal_t).

View answer and explanation(6, 1/4, uses_X_less_than_or_equal_t)

Sum every mass whose aligned support value is at most t, including equality. Below all support the sum is zero; at or above the largest support it is one. The exact result, retaining the requested order, is (6, 1/4, uses_X_less_than_or_equal_t).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r06-cumulative-distribution-function-0019/)

Cumulative distribution function · qf-mod30-cdf-evaluate · candidate





## 20. The CDF Includes Mass Exactly at the Threshold

Use the complete finite discrete law with distinct integer support=[7, 14, 20] 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=14. Give (t,F_t,uses_X_less_than_or_equal_t).

View answer and explanation(14, 1/2, uses_X_less_than_or_equal_t)

Sum every mass whose aligned support value is at most t, including equality. Below all support the sum is zero; at or above the largest support it is one. The exact result, retaining the requested order, is (14, 1/2, uses_X_less_than_or_equal_t).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r06-cumulative-distribution-function-0020/)

Cumulative distribution function · qf-mod30-cdf-evaluate · candidate





## 21. The CDF Includes Mass Exactly at the Threshold

Use the complete finite discrete law with distinct integer support=[8, 15, 21] 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=21. Give (t,F_t,uses_X_less_than_or_equal_t).

View answer and explanation(21, 1, uses_X_less_than_or_equal_t)

Sum every mass whose aligned support value is at most t, including equality. Below all support the sum is zero; at or above the largest support it is one. The exact result, retaining the requested order, is (21, 1, uses_X_less_than_or_equal_t).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r06-cumulative-distribution-function-0021/)

Cumulative distribution function · qf-mod30-cdf-evaluate · candidate





## 22. The CDF Includes Mass Exactly at the Threshold

Use the complete finite discrete law with distinct integer support=[9, 16, 22] 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=9. Give (t,F_t,uses_X_less_than_or_equal_t).

View answer and explanation(9, 1/4, uses_X_less_than_or_equal_t)

Sum every mass whose aligned support value is at most t, including equality. Below all support the sum is zero; at or above the largest support it is one. The exact result, retaining the requested order, is (9, 1/4, uses_X_less_than_or_equal_t).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r06-cumulative-distribution-function-0022/)

Cumulative distribution function · qf-mod30-cdf-evaluate · candidate





## 23. The CDF Includes Mass Exactly at the Threshold

Use the complete finite discrete law with distinct integer support=[10, 17, 23] 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=17. Give (t,F_t,uses_X_less_than_or_equal_t).

View answer and explanation(17, 1/2, uses_X_less_than_or_equal_t)

Sum every mass whose aligned support value is at most t, including equality. Below all support the sum is zero; at or above the largest support it is one. The exact result, retaining the requested order, is (17, 1/2, uses_X_less_than_or_equal_t).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r06-cumulative-distribution-function-0023/)

Cumulative distribution function · qf-mod30-cdf-evaluate · candidate





## 24. The CDF Includes Mass Exactly at the Threshold

Use the complete finite discrete law with distinct integer support=[11, 18, 24] 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=24. Give (t,F_t,uses_X_less_than_or_equal_t).

View answer and explanation(24, 1, uses_X_less_than_or_equal_t)

Sum every mass whose aligned support value is at most t, including equality. Below all support the sum is zero; at or above the largest support it is one. The exact result, retaining the requested order, is (24, 1, uses_X_less_than_or_equal_t).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r06-cumulative-distribution-function-0024/)

Cumulative distribution function · qf-mod30-cdf-evaluate · candidate





## 25. The CDF Includes Mass Exactly at the Threshold

Use the complete finite discrete law with distinct integer support=[12, 19, 25] 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=12. Give (t,F_t,uses_X_less_than_or_equal_t).

View answer and explanation(12, 1/4, uses_X_less_than_or_equal_t)

Sum every mass whose aligned support value is at most t, including equality. Below all support the sum is zero; at or above the largest support it is one. The exact result, retaining the requested order, is (12, 1/4, uses_X_less_than_or_equal_t).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r06-cumulative-distribution-function-0025/)

Cumulative distribution function · qf-mod30-cdf-evaluate · candidate





## 26. The CDF Jump Is the Point Mass, Not the Whole CDF Value

Use the complete finite discrete law with distinct integer support=[-12, -5, 1] 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=-12. Give (t,left_limit,F_t,point_mass,jump_identity?).

View answer and explanation(-12, 0, 1/4, 1/4, Yes)

Sum support strictly below t for the left limit, include equality for F(t), and use zero point mass when t is absent. Compare left_limit+point_mass with F(t) exactly. The exact result, retaining the requested order, is (-12, 0, 1/4, 1/4, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r06-cumulative-distribution-function-0026/)

Cumulative distribution function · qf-mod30-cdf-jump · candidate





## 27. The CDF Jump Is the Point Mass, Not the Whole CDF Value

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?).

View answer and explanation(-4, 1/4, 1/2, 1/4, Yes)

Sum support strictly below t for the left limit, include equality for F(t), and use zero point mass when t is absent. Compare left_limit+point_mass with F(t) exactly. The exact result, retaining the requested order, is (-4, 1/4, 1/2, 1/4, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r06-cumulative-distribution-function-0027/)

Cumulative distribution function · qf-mod30-cdf-jump · candidate





## 28. The CDF Jump Is the Point Mass, Not the Whole CDF Value

Use the complete finite discrete law with distinct integer support=[-10, -3, 3] 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=3. Give (t,left_limit,F_t,point_mass,jump_identity?).

View answer and explanation(3, 1/2, 1, 1/2, Yes)

Sum support strictly below t for the left limit, include equality for F(t), and use zero point mass when t is absent. Compare left_limit+point_mass with F(t) exactly. The exact result, retaining the requested order, is (3, 1/2, 1, 1/2, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r06-cumulative-distribution-function-0028/)

Cumulative distribution function · qf-mod30-cdf-jump · candidate





## 29. The CDF Jump Is the Point Mass, Not the Whole CDF Value

Use the complete finite discrete law with distinct integer support=[-9, -2, 4] 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=-9. Give (t,left_limit,F_t,point_mass,jump_identity?).

View answer and explanation(-9, 0, 1/4, 1/4, Yes)

Sum support strictly below t for the left limit, include equality for F(t), and use zero point mass when t is absent. Compare left_limit+point_mass with F(t) exactly. The exact result, retaining the requested order, is (-9, 0, 1/4, 1/4, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r06-cumulative-distribution-function-0029/)

Cumulative distribution function · qf-mod30-cdf-jump · candidate





## 30. The CDF Jump Is the Point Mass, Not the Whole CDF Value

Use the complete finite discrete law with distinct integer support=[-8, -1, 5] 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=-1. Give (t,left_limit,F_t,point_mass,jump_identity?).

View answer and explanation(-1, 1/4, 1/2, 1/4, Yes)

Sum support strictly below t for the left limit, include equality for F(t), and use zero point mass when t is absent. Compare left_limit+point_mass with F(t) exactly. The exact result, retaining the requested order, is (-1, 1/4, 1/2, 1/4, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r06-cumulative-distribution-function-0030/)

Cumulative distribution function · qf-mod30-cdf-jump · candidate





## 31. The CDF Jump Is the Point Mass, Not the Whole CDF Value

Use the complete finite discrete law with distinct integer support=[-7, 0, 6] 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=6. Give (t,left_limit,F_t,point_mass,jump_identity?).

View answer and explanation(6, 1/2, 1, 1/2, Yes)

Sum support strictly below t for the left limit, include equality for F(t), and use zero point mass when t is absent. Compare left_limit+point_mass with F(t) exactly. The exact result, retaining the requested order, is (6, 1/2, 1, 1/2, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r06-cumulative-distribution-function-0031/)

Cumulative distribution function · qf-mod30-cdf-jump · candidate





## 32. The CDF Jump Is the Point Mass, Not the Whole CDF Value

Use the complete finite discrete law with distinct integer support=[-6, 1, 7] 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=-6. Give (t,left_limit,F_t,point_mass,jump_identity?).

View answer and explanation(-6, 0, 1/4, 1/4, Yes)

Sum support strictly below t for the left limit, include equality for F(t), and use zero point mass when t is absent. Compare left_limit+point_mass with F(t) exactly. The exact result, retaining the requested order, is (-6, 0, 1/4, 1/4, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r06-cumulative-distribution-function-0032/)

Cumulative distribution function · qf-mod30-cdf-jump · candidate





## 33. The CDF Jump Is the Point Mass, Not the Whole CDF Value

Use the complete finite discrete law with distinct integer support=[-5, 2, 8] 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=2. Give (t,left_limit,F_t,point_mass,jump_identity?).

View answer and explanation(2, 1/4, 1/2, 1/4, Yes)

Sum support strictly below t for the left limit, include equality for F(t), and use zero point mass when t is absent. Compare left_limit+point_mass with F(t) exactly. The exact result, retaining the requested order, is (2, 1/4, 1/2, 1/4, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r06-cumulative-distribution-function-0033/)

Cumulative distribution function · qf-mod30-cdf-jump · candidate





## 34. The CDF Jump Is the Point Mass, Not the Whole CDF Value

Use the complete finite discrete law with distinct integer support=[-4, 3, 9] 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=9. Give (t,left_limit,F_t,point_mass,jump_identity?).

View answer and explanation(9, 1/2, 1, 1/2, Yes)

Sum support strictly below t for the left limit, include equality for F(t), and use zero point mass when t is absent. Compare left_limit+point_mass with F(t) exactly. The exact result, retaining the requested order, is (9, 1/2, 1, 1/2, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r06-cumulative-distribution-function-0034/)

Cumulative distribution function · qf-mod30-cdf-jump · candidate





## 35. The CDF Jump Is the Point Mass, Not the Whole CDF Value

Use the complete finite discrete law with distinct integer support=[-3, 4, 10] 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=-3. Give (t,left_limit,F_t,point_mass,jump_identity?).

View answer and explanation(-3, 0, 1/4, 1/4, Yes)

Sum support strictly below t for the left limit, include equality for F(t), and use zero point mass when t is absent. Compare left_limit+point_mass with F(t) exactly. The exact result, retaining the requested order, is (-3, 0, 1/4, 1/4, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r06-cumulative-distribution-function-0035/)

Cumulative distribution function · qf-mod30-cdf-jump · candidate





## 36. The CDF Jump Is the Point Mass, Not the Whole CDF Value

Use the complete finite discrete law with distinct integer support=[-2, 5, 11] 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=5. Give (t,left_limit,F_t,point_mass,jump_identity?).

View answer and explanation(5, 1/4, 1/2, 1/4, Yes)

Sum support strictly below t for the left limit, include equality for F(t), and use zero point mass when t is absent. Compare left_limit+point_mass with F(t) exactly. The exact result, retaining the requested order, is (5, 1/4, 1/2, 1/4, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r06-cumulative-distribution-function-0036/)

Cumulative distribution function · qf-mod30-cdf-jump · candidate





## 37. The CDF Jump Is the Point Mass, Not the Whole CDF Value

Use the complete finite discrete law with distinct integer support=[-1, 6, 12] 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=12. Give (t,left_limit,F_t,point_mass,jump_identity?).

View answer and explanation(12, 1/2, 1, 1/2, Yes)

Sum support strictly below t for the left limit, include equality for F(t), and use zero point mass when t is absent. Compare left_limit+point_mass with F(t) exactly. The exact result, retaining the requested order, is (12, 1/2, 1, 1/2, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r06-cumulative-distribution-function-0037/)

Cumulative distribution function · qf-mod30-cdf-jump · candidate





## 38. The CDF Jump Is the Point Mass, Not the Whole CDF Value

Use the complete finite discrete law with distinct integer support=[0, 7, 13] 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=0. Give (t,left_limit,F_t,point_mass,jump_identity?).

View answer and explanation(0, 0, 1/4, 1/4, Yes)

Sum support strictly below t for the left limit, include equality for F(t), and use zero point mass when t is absent. Compare left_limit+point_mass with F(t) exactly. The exact result, retaining the requested order, is (0, 0, 1/4, 1/4, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r06-cumulative-distribution-function-0038/)

Cumulative distribution function · qf-mod30-cdf-jump · candidate





## 39. The CDF Jump Is the Point Mass, Not the Whole CDF Value

Use the complete finite discrete law with distinct integer support=[1, 8, 14] 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=8. Give (t,left_limit,F_t,point_mass,jump_identity?).

View answer and explanation(8, 1/4, 1/2, 1/4, Yes)

Sum support strictly below t for the left limit, include equality for F(t), and use zero point mass when t is absent. Compare left_limit+point_mass with F(t) exactly. The exact result, retaining the requested order, is (8, 1/4, 1/2, 1/4, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r06-cumulative-distribution-function-0039/)

Cumulative distribution function · qf-mod30-cdf-jump · candidate





## 40. The CDF Jump Is the Point Mass, Not the Whole CDF Value

Use the complete finite discrete law with distinct integer support=[2, 9, 15] 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=15. Give (t,left_limit,F_t,point_mass,jump_identity?).

View answer and explanation(15, 1/2, 1, 1/2, Yes)

Sum support strictly below t for the left limit, include equality for F(t), and use zero point mass when t is absent. Compare left_limit+point_mass with F(t) exactly. The exact result, retaining the requested order, is (15, 1/2, 1, 1/2, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r06-cumulative-distribution-function-0040/)

Cumulative distribution function · qf-mod30-cdf-jump · candidate





## 41. The CDF Jump Is the Point Mass, Not the Whole CDF Value

Use the complete finite discrete law with distinct integer support=[3, 10, 16] 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=3. Give (t,left_limit,F_t,point_mass,jump_identity?).

View answer and explanation(3, 0, 1/4, 1/4, Yes)

Sum support strictly below t for the left limit, include equality for F(t), and use zero point mass when t is absent. Compare left_limit+point_mass with F(t) exactly. The exact result, retaining the requested order, is (3, 0, 1/4, 1/4, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r06-cumulative-distribution-function-0041/)

Cumulative distribution function · qf-mod30-cdf-jump · candidate





## 42. The CDF Jump Is the Point Mass, Not the Whole CDF Value

Use the complete finite discrete law with distinct integer support=[4, 11, 17] 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=11. Give (t,left_limit,F_t,point_mass,jump_identity?).

View answer and explanation(11, 1/4, 1/2, 1/4, Yes)

Sum support strictly below t for the left limit, include equality for F(t), and use zero point mass when t is absent. Compare left_limit+point_mass with F(t) exactly. The exact result, retaining the requested order, is (11, 1/4, 1/2, 1/4, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r06-cumulative-distribution-function-0042/)

Cumulative distribution function · qf-mod30-cdf-jump · candidate





## 43. The CDF Jump Is the Point Mass, Not the Whole CDF Value

Use the complete finite discrete law with distinct integer support=[5, 12, 18] 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=18. Give (t,left_limit,F_t,point_mass,jump_identity?).

View answer and explanation(18, 1/2, 1, 1/2, Yes)

Sum support strictly below t for the left limit, include equality for F(t), and use zero point mass when t is absent. Compare left_limit+point_mass with F(t) exactly. The exact result, retaining the requested order, is (18, 1/2, 1, 1/2, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r06-cumulative-distribution-function-0043/)

Cumulative distribution function · qf-mod30-cdf-jump · candidate





## 44. The CDF Jump Is the Point Mass, Not the Whole CDF Value

Use the complete finite discrete law with distinct integer support=[6, 13, 19] 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=6. Give (t,left_limit,F_t,point_mass,jump_identity?).

View answer and explanation(6, 0, 1/4, 1/4, Yes)

Sum support strictly below t for the left limit, include equality for F(t), and use zero point mass when t is absent. Compare left_limit+point_mass with F(t) exactly. The exact result, retaining the requested order, is (6, 0, 1/4, 1/4, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r06-cumulative-distribution-function-0044/)

Cumulative distribution function · qf-mod30-cdf-jump · candidate





## 45. The CDF Jump Is the Point Mass, Not the Whole CDF Value

Use the complete finite discrete law with distinct integer support=[7, 14, 20] 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=14. Give (t,left_limit,F_t,point_mass,jump_identity?).

View answer and explanation(14, 1/4, 1/2, 1/4, Yes)

Sum support strictly below t for the left limit, include equality for F(t), and use zero point mass when t is absent. Compare left_limit+point_mass with F(t) exactly. The exact result, retaining the requested order, is (14, 1/4, 1/2, 1/4, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r06-cumulative-distribution-function-0045/)

Cumulative distribution function · qf-mod30-cdf-jump · candidate





## 46. The CDF Jump Is the Point Mass, Not the Whole CDF Value

Use the complete finite discrete law with distinct integer support=[8, 15, 21] 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=21. Give (t,left_limit,F_t,point_mass,jump_identity?).

View answer and explanation(21, 1/2, 1, 1/2, Yes)

Sum support strictly below t for the left limit, include equality for F(t), and use zero point mass when t is absent. Compare left_limit+point_mass with F(t) exactly. The exact result, retaining the requested order, is (21, 1/2, 1, 1/2, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r06-cumulative-distribution-function-0046/)

Cumulative distribution function · qf-mod30-cdf-jump · candidate





## 47. The CDF Jump Is the Point Mass, Not the Whole CDF Value

Use the complete finite discrete law with distinct integer support=[9, 16, 22] 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=9. Give (t,left_limit,F_t,point_mass,jump_identity?).

View answer and explanation(9, 0, 1/4, 1/4, Yes)

Sum support strictly below t for the left limit, include equality for F(t), and use zero point mass when t is absent. Compare left_limit+point_mass with F(t) exactly. The exact result, retaining the requested order, is (9, 0, 1/4, 1/4, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r06-cumulative-distribution-function-0047/)

Cumulative distribution function · qf-mod30-cdf-jump · candidate





## 48. The CDF Jump Is the Point Mass, Not the Whole CDF Value

Use the complete finite discrete law with distinct integer support=[10, 17, 23] 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=17. Give (t,left_limit,F_t,point_mass,jump_identity?).

View answer and explanation(17, 1/4, 1/2, 1/4, Yes)

Sum support strictly below t for the left limit, include equality for F(t), and use zero point mass when t is absent. Compare left_limit+point_mass with F(t) exactly. The exact result, retaining the requested order, is (17, 1/4, 1/2, 1/4, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r06-cumulative-distribution-function-0048/)

Cumulative distribution function · qf-mod30-cdf-jump · candidate





## 49. The CDF Jump Is the Point Mass, Not the Whole CDF Value

Use the complete finite discrete law with distinct integer support=[11, 18, 24] 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=24. Give (t,left_limit,F_t,point_mass,jump_identity?).

View answer and explanation(24, 1/2, 1, 1/2, Yes)

Sum support strictly below t for the left limit, include equality for F(t), and use zero point mass when t is absent. Compare left_limit+point_mass with F(t) exactly. The exact result, retaining the requested order, is (24, 1/2, 1, 1/2, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r06-cumulative-distribution-function-0049/)

Cumulative distribution function · qf-mod30-cdf-jump · candidate





## 50. The CDF Jump Is the Point Mass, Not the Whole CDF Value

Use the complete finite discrete law with distinct integer support=[12, 19, 25] 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=12. Give (t,left_limit,F_t,point_mass,jump_identity?).

View answer and explanation(12, 0, 1/4, 1/4, Yes)

Sum support strictly below t for the left limit, include equality for F(t), and use zero point mass when t is absent. Compare left_limit+point_mass with F(t) exactly. The exact result, retaining the requested order, is (12, 0, 1/4, 1/4, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r06-cumulative-distribution-function-0050/)

Cumulative distribution function · qf-mod30-cdf-jump · candidate





## 51. Nested Threshold Events Give a Nondecreasing CDF

Use the complete finite discrete law with distinct integer support=[-12, -5, 1] 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. Ordered thresholds left=-12≤right=1. Give (left,right,F_left,F_right,monotone?).

View answer and explanation(-12, 1, 1/4, 1, Yes)

Aggregate both cumulative masses from the same complete law and compare exact rationals. Reversing the threshold premise is rejected; sampled values alone are not a general CDF construction. The exact result, retaining the requested order, is (-12, 1, 1/4, 1, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r06-cumulative-distribution-function-0051/)

Cumulative distribution function · qf-mod30-cdf-monotonicity · candidate





## 52. Nested Threshold Events Give a Nondecreasing CDF

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. Ordered thresholds left=-11≤right=2. Give (left,right,F_left,F_right,monotone?).

View answer and explanation(-11, 2, 1/4, 1, Yes)

Aggregate both cumulative masses from the same complete law and compare exact rationals. Reversing the threshold premise is rejected; sampled values alone are not a general CDF construction. The exact result, retaining the requested order, is (-11, 2, 1/4, 1, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r06-cumulative-distribution-function-0052/)

Cumulative distribution function · qf-mod30-cdf-monotonicity · candidate





## 53. Nested Threshold Events Give a Nondecreasing CDF

Use the complete finite discrete law with distinct integer support=[-10, -3, 3] 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. Ordered thresholds left=-10≤right=3. Give (left,right,F_left,F_right,monotone?).

View answer and explanation(-10, 3, 1/4, 1, Yes)

Aggregate both cumulative masses from the same complete law and compare exact rationals. Reversing the threshold premise is rejected; sampled values alone are not a general CDF construction. The exact result, retaining the requested order, is (-10, 3, 1/4, 1, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r06-cumulative-distribution-function-0053/)

Cumulative distribution function · qf-mod30-cdf-monotonicity · candidate





## 54. Nested Threshold Events Give a Nondecreasing CDF

Use the complete finite discrete law with distinct integer support=[-9, -2, 4] 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. Ordered thresholds left=-9≤right=4. Give (left,right,F_left,F_right,monotone?).

View answer and explanation(-9, 4, 1/4, 1, Yes)

Aggregate both cumulative masses from the same complete law and compare exact rationals. Reversing the threshold premise is rejected; sampled values alone are not a general CDF construction. The exact result, retaining the requested order, is (-9, 4, 1/4, 1, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r06-cumulative-distribution-function-0054/)

Cumulative distribution function · qf-mod30-cdf-monotonicity · candidate





## 55. Nested Threshold Events Give a Nondecreasing CDF

Use the complete finite discrete law with distinct integer support=[-8, -1, 5] 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. Ordered thresholds left=-8≤right=5. Give (left,right,F_left,F_right,monotone?).

View answer and explanation(-8, 5, 1/4, 1, Yes)

Aggregate both cumulative masses from the same complete law and compare exact rationals. Reversing the threshold premise is rejected; sampled values alone are not a general CDF construction. The exact result, retaining the requested order, is (-8, 5, 1/4, 1, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r06-cumulative-distribution-function-0055/)

Cumulative distribution function · qf-mod30-cdf-monotonicity · candidate





## 56. Nested Threshold Events Give a Nondecreasing CDF

Use the complete finite discrete law with distinct integer support=[-7, 0, 6] 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. Ordered thresholds left=-7≤right=6. Give (left,right,F_left,F_right,monotone?).

View answer and explanation(-7, 6, 1/4, 1, Yes)

Aggregate both cumulative masses from the same complete law and compare exact rationals. Reversing the threshold premise is rejected; sampled values alone are not a general CDF construction. The exact result, retaining the requested order, is (-7, 6, 1/4, 1, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r06-cumulative-distribution-function-0056/)

Cumulative distribution function · qf-mod30-cdf-monotonicity · candidate





## 57. Nested Threshold Events Give a Nondecreasing CDF

Use the complete finite discrete law with distinct integer support=[-6, 1, 7] 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. Ordered thresholds left=-6≤right=7. Give (left,right,F_left,F_right,monotone?).

View answer and explanation(-6, 7, 1/4, 1, Yes)

Aggregate both cumulative masses from the same complete law and compare exact rationals. Reversing the threshold premise is rejected; sampled values alone are not a general CDF construction. The exact result, retaining the requested order, is (-6, 7, 1/4, 1, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r06-cumulative-distribution-function-0057/)

Cumulative distribution function · qf-mod30-cdf-monotonicity · candidate





## 58. Nested Threshold Events Give a Nondecreasing CDF

Use the complete finite discrete law with distinct integer support=[-5, 2, 8] 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. Ordered thresholds left=-5≤right=8. Give (left,right,F_left,F_right,monotone?).

View answer and explanation(-5, 8, 1/4, 1, Yes)

Aggregate both cumulative masses from the same complete law and compare exact rationals. Reversing the threshold premise is rejected; sampled values alone are not a general CDF construction. The exact result, retaining the requested order, is (-5, 8, 1/4, 1, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r06-cumulative-distribution-function-0058/)

Cumulative distribution function · qf-mod30-cdf-monotonicity · candidate





## 59. Nested Threshold Events Give a Nondecreasing CDF

Use the complete finite discrete law with distinct integer support=[-4, 3, 9] 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. Ordered thresholds left=-4≤right=9. Give (left,right,F_left,F_right,monotone?).

View answer and explanation(-4, 9, 1/4, 1, Yes)

Aggregate both cumulative masses from the same complete law and compare exact rationals. Reversing the threshold premise is rejected; sampled values alone are not a general CDF construction. The exact result, retaining the requested order, is (-4, 9, 1/4, 1, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r06-cumulative-distribution-function-0059/)

Cumulative distribution function · qf-mod30-cdf-monotonicity · candidate





## 60. Nested Threshold Events Give a Nondecreasing CDF

Use the complete finite discrete law with distinct integer support=[-3, 4, 10] 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. Ordered thresholds left=-3≤right=10. Give (left,right,F_left,F_right,monotone?).

View answer and explanation(-3, 10, 1/4, 1, Yes)

Aggregate both cumulative masses from the same complete law and compare exact rationals. Reversing the threshold premise is rejected; sampled values alone are not a general CDF construction. The exact result, retaining the requested order, is (-3, 10, 1/4, 1, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r06-cumulative-distribution-function-0060/)

Cumulative distribution function · qf-mod30-cdf-monotonicity · candidate





## 61. Nested Threshold Events Give a Nondecreasing CDF

Use the complete finite discrete law with distinct integer support=[-2, 5, 11] 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. Ordered thresholds left=-2≤right=11. Give (left,right,F_left,F_right,monotone?).

View answer and explanation(-2, 11, 1/4, 1, Yes)

Aggregate both cumulative masses from the same complete law and compare exact rationals. Reversing the threshold premise is rejected; sampled values alone are not a general CDF construction. The exact result, retaining the requested order, is (-2, 11, 1/4, 1, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r06-cumulative-distribution-function-0061/)

Cumulative distribution function · qf-mod30-cdf-monotonicity · candidate





## 62. Nested Threshold Events Give a Nondecreasing CDF

Use the complete finite discrete law with distinct integer support=[-1, 6, 12] 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. Ordered thresholds left=-1≤right=12. Give (left,right,F_left,F_right,monotone?).

View answer and explanation(-1, 12, 1/4, 1, Yes)

Aggregate both cumulative masses from the same complete law and compare exact rationals. Reversing the threshold premise is rejected; sampled values alone are not a general CDF construction. The exact result, retaining the requested order, is (-1, 12, 1/4, 1, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r06-cumulative-distribution-function-0062/)

Cumulative distribution function · qf-mod30-cdf-monotonicity · candidate





## 63. Nested Threshold Events Give a Nondecreasing CDF

Use the complete finite discrete law with distinct integer support=[0, 7, 13] 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. Ordered thresholds left=0≤right=13. Give (left,right,F_left,F_right,monotone?).

View answer and explanation(0, 13, 1/4, 1, Yes)

Aggregate both cumulative masses from the same complete law and compare exact rationals. Reversing the threshold premise is rejected; sampled values alone are not a general CDF construction. The exact result, retaining the requested order, is (0, 13, 1/4, 1, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r06-cumulative-distribution-function-0063/)

Cumulative distribution function · qf-mod30-cdf-monotonicity · candidate





## 64. Nested Threshold Events Give a Nondecreasing CDF

Use the complete finite discrete law with distinct integer support=[1, 8, 14] 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. Ordered thresholds left=1≤right=14. Give (left,right,F_left,F_right,monotone?).

View answer and explanation(1, 14, 1/4, 1, Yes)

Aggregate both cumulative masses from the same complete law and compare exact rationals. Reversing the threshold premise is rejected; sampled values alone are not a general CDF construction. The exact result, retaining the requested order, is (1, 14, 1/4, 1, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r06-cumulative-distribution-function-0064/)

Cumulative distribution function · qf-mod30-cdf-monotonicity · candidate





## 65. Nested Threshold Events Give a Nondecreasing CDF

Use the complete finite discrete law with distinct integer support=[2, 9, 15] 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. Ordered thresholds left=2≤right=15. Give (left,right,F_left,F_right,monotone?).

View answer and explanation(2, 15, 1/4, 1, Yes)

Aggregate both cumulative masses from the same complete law and compare exact rationals. Reversing the threshold premise is rejected; sampled values alone are not a general CDF construction. The exact result, retaining the requested order, is (2, 15, 1/4, 1, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r06-cumulative-distribution-function-0065/)

Cumulative distribution function · qf-mod30-cdf-monotonicity · candidate





## 66. Nested Threshold Events Give a Nondecreasing CDF

Use the complete finite discrete law with distinct integer support=[3, 10, 16] 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. Ordered thresholds left=3≤right=16. Give (left,right,F_left,F_right,monotone?).

View answer and explanation(3, 16, 1/4, 1, Yes)

Aggregate both cumulative masses from the same complete law and compare exact rationals. Reversing the threshold premise is rejected; sampled values alone are not a general CDF construction. The exact result, retaining the requested order, is (3, 16, 1/4, 1, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r06-cumulative-distribution-function-0066/)

Cumulative distribution function · qf-mod30-cdf-monotonicity · candidate





## 67. Nested Threshold Events Give a Nondecreasing CDF

Use the complete finite discrete law with distinct integer support=[4, 11, 17] 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. Ordered thresholds left=4≤right=17. Give (left,right,F_left,F_right,monotone?).

View answer and explanation(4, 17, 1/4, 1, Yes)

Aggregate both cumulative masses from the same complete law and compare exact rationals. Reversing the threshold premise is rejected; sampled values alone are not a general CDF construction. The exact result, retaining the requested order, is (4, 17, 1/4, 1, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r06-cumulative-distribution-function-0067/)

Cumulative distribution function · qf-mod30-cdf-monotonicity · candidate





## 68. Nested Threshold Events Give a Nondecreasing CDF

Use the complete finite discrete law with distinct integer support=[5, 12, 18] 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. Ordered thresholds left=5≤right=18. Give (left,right,F_left,F_right,monotone?).

View answer and explanation(5, 18, 1/4, 1, Yes)

Aggregate both cumulative masses from the same complete law and compare exact rationals. Reversing the threshold premise is rejected; sampled values alone are not a general CDF construction. The exact result, retaining the requested order, is (5, 18, 1/4, 1, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r06-cumulative-distribution-function-0068/)

Cumulative distribution function · qf-mod30-cdf-monotonicity · candidate





## 69. Nested Threshold Events Give a Nondecreasing CDF

Use the complete finite discrete law with distinct integer support=[6, 13, 19] 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. Ordered thresholds left=6≤right=19. Give (left,right,F_left,F_right,monotone?).

View answer and explanation(6, 19, 1/4, 1, Yes)

Aggregate both cumulative masses from the same complete law and compare exact rationals. Reversing the threshold premise is rejected; sampled values alone are not a general CDF construction. The exact result, retaining the requested order, is (6, 19, 1/4, 1, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r06-cumulative-distribution-function-0069/)

Cumulative distribution function · qf-mod30-cdf-monotonicity · candidate





## 70. Nested Threshold Events Give a Nondecreasing CDF

Use the complete finite discrete law with distinct integer support=[7, 14, 20] 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. Ordered thresholds left=7≤right=20. Give (left,right,F_left,F_right,monotone?).

View answer and explanation(7, 20, 1/4, 1, Yes)

Aggregate both cumulative masses from the same complete law and compare exact rationals. Reversing the threshold premise is rejected; sampled values alone are not a general CDF construction. The exact result, retaining the requested order, is (7, 20, 1/4, 1, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r06-cumulative-distribution-function-0070/)

Cumulative distribution function · qf-mod30-cdf-monotonicity · candidate





## 71. Nested Threshold Events Give a Nondecreasing CDF

Use the complete finite discrete law with distinct integer support=[8, 15, 21] 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. Ordered thresholds left=8≤right=21. Give (left,right,F_left,F_right,monotone?).

View answer and explanation(8, 21, 1/4, 1, Yes)

Aggregate both cumulative masses from the same complete law and compare exact rationals. Reversing the threshold premise is rejected; sampled values alone are not a general CDF construction. The exact result, retaining the requested order, is (8, 21, 1/4, 1, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r06-cumulative-distribution-function-0071/)

Cumulative distribution function · qf-mod30-cdf-monotonicity · candidate





## 72. Nested Threshold Events Give a Nondecreasing CDF

Use the complete finite discrete law with distinct integer support=[9, 16, 22] 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. Ordered thresholds left=9≤right=22. Give (left,right,F_left,F_right,monotone?).

View answer and explanation(9, 22, 1/4, 1, Yes)

Aggregate both cumulative masses from the same complete law and compare exact rationals. Reversing the threshold premise is rejected; sampled values alone are not a general CDF construction. The exact result, retaining the requested order, is (9, 22, 1/4, 1, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r06-cumulative-distribution-function-0072/)

Cumulative distribution function · qf-mod30-cdf-monotonicity · candidate





## 73. Nested Threshold Events Give a Nondecreasing CDF

Use the complete finite discrete law with distinct integer support=[10, 17, 23] 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. Ordered thresholds left=10≤right=23. Give (left,right,F_left,F_right,monotone?).

View answer and explanation(10, 23, 1/4, 1, Yes)

Aggregate both cumulative masses from the same complete law and compare exact rationals. Reversing the threshold premise is rejected; sampled values alone are not a general CDF construction. The exact result, retaining the requested order, is (10, 23, 1/4, 1, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r06-cumulative-distribution-function-0073/)

Cumulative distribution function · qf-mod30-cdf-monotonicity · candidate





## 74. Nested Threshold Events Give a Nondecreasing CDF

Use the complete finite discrete law with distinct integer support=[11, 18, 24] 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. Ordered thresholds left=11≤right=24. Give (left,right,F_left,F_right,monotone?).

View answer and explanation(11, 24, 1/4, 1, Yes)

Aggregate both cumulative masses from the same complete law and compare exact rationals. Reversing the threshold premise is rejected; sampled values alone are not a general CDF construction. The exact result, retaining the requested order, is (11, 24, 1/4, 1, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r06-cumulative-distribution-function-0074/)

Cumulative distribution function · qf-mod30-cdf-monotonicity · candidate





## 75. Nested Threshold Events Give a Nondecreasing CDF

Use the complete finite discrete law with distinct integer support=[12, 19, 25] 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. Ordered thresholds left=12≤right=25. Give (left,right,F_left,F_right,monotone?).

View answer and explanation(12, 25, 1/4, 1, Yes)

Aggregate both cumulative masses from the same complete law and compare exact rationals. Reversing the threshold premise is rejected; sampled values alone are not a general CDF construction. The exact result, retaining the requested order, is (12, 25, 1/4, 1, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r06-cumulative-distribution-function-0075/)

Cumulative distribution function · qf-mod30-cdf-monotonicity · candidate





## 76. A Finite-Law CDF Stays Constant Just to the Right

Use the complete finite discrete law with distinct integer support=[-12, -5, 1] 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=-12. Give (t,F_t,next_support,constant_on_right_neighborhood_before_next_support,right_continuous?). next_support is the least support value strictly above t, or null when none exists.

View answer and explanation(-12, 1/4, -5, constant_on_right_neighborhood_before_next_support, Yes)

The right value includes any mass at t and remains unchanged before the next support point. With no next support it remains constant for all larger thresholds. This is right continuity, not left continuity or a density claim. The exact result, retaining the requested order, is (-12, 1/4, -5, constant_on_right_neighborhood_before_next_support, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r06-cumulative-distribution-function-0076/)

Cumulative distribution function · qf-mod30-cdf-right-continuity · candidate





## 77. A Finite-Law CDF Stays Constant Just to the Right

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,F_t,next_support,constant_on_right_neighborhood_before_next_support,right_continuous?). next_support is the least support value strictly above t, or null when none exists.

View answer and explanation(-4, 1/2, 2, constant_on_right_neighborhood_before_next_support, Yes)

The right value includes any mass at t and remains unchanged before the next support point. With no next support it remains constant for all larger thresholds. This is right continuity, not left continuity or a density claim. The exact result, retaining the requested order, is (-4, 1/2, 2, constant_on_right_neighborhood_before_next_support, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r06-cumulative-distribution-function-0077/)

Cumulative distribution function · qf-mod30-cdf-right-continuity · candidate





## 78. A Finite-Law CDF Stays Constant Just to the Right

Use the complete finite discrete law with distinct integer support=[-10, -3, 3] 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=3. Give (t,F_t,next_support,constant_on_right_neighborhood_before_next_support,right_continuous?). next_support is the least support value strictly above t, or null when none exists.

View answer and explanation(3, 1, null, constant_on_right_neighborhood_before_next_support, Yes)

The right value includes any mass at t and remains unchanged before the next support point. With no next support it remains constant for all larger thresholds. This is right continuity, not left continuity or a density claim. The exact result, retaining the requested order, is (3, 1, null, constant_on_right_neighborhood_before_next_support, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r06-cumulative-distribution-function-0078/)

Cumulative distribution function · qf-mod30-cdf-right-continuity · candidate





## 79. A Finite-Law CDF Stays Constant Just to the Right

Use the complete finite discrete law with distinct integer support=[-9, -2, 4] 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=-9. Give (t,F_t,next_support,constant_on_right_neighborhood_before_next_support,right_continuous?). next_support is the least support value strictly above t, or null when none exists.

View answer and explanation(-9, 1/4, -2, constant_on_right_neighborhood_before_next_support, Yes)

The right value includes any mass at t and remains unchanged before the next support point. With no next support it remains constant for all larger thresholds. This is right continuity, not left continuity or a density claim. The exact result, retaining the requested order, is (-9, 1/4, -2, constant_on_right_neighborhood_before_next_support, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r06-cumulative-distribution-function-0079/)

Cumulative distribution function · qf-mod30-cdf-right-continuity · candidate





## 80. A Finite-Law CDF Stays Constant Just to the Right

Use the complete finite discrete law with distinct integer support=[-8, -1, 5] 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=-1. Give (t,F_t,next_support,constant_on_right_neighborhood_before_next_support,right_continuous?). next_support is the least support value strictly above t, or null when none exists.

View answer and explanation(-1, 1/2, 5, constant_on_right_neighborhood_before_next_support, Yes)

The right value includes any mass at t and remains unchanged before the next support point. With no next support it remains constant for all larger thresholds. This is right continuity, not left continuity or a density claim. The exact result, retaining the requested order, is (-1, 1/2, 5, constant_on_right_neighborhood_before_next_support, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r06-cumulative-distribution-function-0080/)

Cumulative distribution function · qf-mod30-cdf-right-continuity · candidate





## 81. A Finite-Law CDF Stays Constant Just to the Right

Use the complete finite discrete law with distinct integer support=[-7, 0, 6] 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=6. Give (t,F_t,next_support,constant_on_right_neighborhood_before_next_support,right_continuous?). next_support is the least support value strictly above t, or null when none exists.

View answer and explanation(6, 1, null, constant_on_right_neighborhood_before_next_support, Yes)

The right value includes any mass at t and remains unchanged before the next support point. With no next support it remains constant for all larger thresholds. This is right continuity, not left continuity or a density claim. The exact result, retaining the requested order, is (6, 1, null, constant_on_right_neighborhood_before_next_support, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r06-cumulative-distribution-function-0081/)

Cumulative distribution function · qf-mod30-cdf-right-continuity · candidate





## 82. A Finite-Law CDF Stays Constant Just to the Right

Use the complete finite discrete law with distinct integer support=[-6, 1, 7] 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=-6. Give (t,F_t,next_support,constant_on_right_neighborhood_before_next_support,right_continuous?). next_support is the least support value strictly above t, or null when none exists.

View answer and explanation(-6, 1/4, 1, constant_on_right_neighborhood_before_next_support, Yes)

The right value includes any mass at t and remains unchanged before the next support point. With no next support it remains constant for all larger thresholds. This is right continuity, not left continuity or a density claim. The exact result, retaining the requested order, is (-6, 1/4, 1, constant_on_right_neighborhood_before_next_support, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r06-cumulative-distribution-function-0082/)

Cumulative distribution function · qf-mod30-cdf-right-continuity · candidate





## 83. A Finite-Law CDF Stays Constant Just to the Right

Use the complete finite discrete law with distinct integer support=[-5, 2, 8] 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=2. Give (t,F_t,next_support,constant_on_right_neighborhood_before_next_support,right_continuous?). next_support is the least support value strictly above t, or null when none exists.

View answer and explanation(2, 1/2, 8, constant_on_right_neighborhood_before_next_support, Yes)

The right value includes any mass at t and remains unchanged before the next support point. With no next support it remains constant for all larger thresholds. This is right continuity, not left continuity or a density claim. The exact result, retaining the requested order, is (2, 1/2, 8, constant_on_right_neighborhood_before_next_support, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r06-cumulative-distribution-function-0083/)

Cumulative distribution function · qf-mod30-cdf-right-continuity · candidate





## 84. A Finite-Law CDF Stays Constant Just to the Right

Use the complete finite discrete law with distinct integer support=[-4, 3, 9] 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=9. Give (t,F_t,next_support,constant_on_right_neighborhood_before_next_support,right_continuous?). next_support is the least support value strictly above t, or null when none exists.

View answer and explanation(9, 1, null, constant_on_right_neighborhood_before_next_support, Yes)

The right value includes any mass at t and remains unchanged before the next support point. With no next support it remains constant for all larger thresholds. This is right continuity, not left continuity or a density claim. The exact result, retaining the requested order, is (9, 1, null, constant_on_right_neighborhood_before_next_support, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r06-cumulative-distribution-function-0084/)

Cumulative distribution function · qf-mod30-cdf-right-continuity · candidate





## 85. A Finite-Law CDF Stays Constant Just to the Right

Use the complete finite discrete law with distinct integer support=[-3, 4, 10] 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=-3. Give (t,F_t,next_support,constant_on_right_neighborhood_before_next_support,right_continuous?). next_support is the least support value strictly above t, or null when none exists.

View answer and explanation(-3, 1/4, 4, constant_on_right_neighborhood_before_next_support, Yes)

The right value includes any mass at t and remains unchanged before the next support point. With no next support it remains constant for all larger thresholds. This is right continuity, not left continuity or a density claim. The exact result, retaining the requested order, is (-3, 1/4, 4, constant_on_right_neighborhood_before_next_support, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r06-cumulative-distribution-function-0085/)

Cumulative distribution function · qf-mod30-cdf-right-continuity · candidate





## 86. A Finite-Law CDF Stays Constant Just to the Right

Use the complete finite discrete law with distinct integer support=[-2, 5, 11] 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=5. Give (t,F_t,next_support,constant_on_right_neighborhood_before_next_support,right_continuous?). next_support is the least support value strictly above t, or null when none exists.

View answer and explanation(5, 1/2, 11, constant_on_right_neighborhood_before_next_support, Yes)

The right value includes any mass at t and remains unchanged before the next support point. With no next support it remains constant for all larger thresholds. This is right continuity, not left continuity or a density claim. The exact result, retaining the requested order, is (5, 1/2, 11, constant_on_right_neighborhood_before_next_support, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r06-cumulative-distribution-function-0086/)

Cumulative distribution function · qf-mod30-cdf-right-continuity · candidate





## 87. A Finite-Law CDF Stays Constant Just to the Right

Use the complete finite discrete law with distinct integer support=[-1, 6, 12] 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=12. Give (t,F_t,next_support,constant_on_right_neighborhood_before_next_support,right_continuous?). next_support is the least support value strictly above t, or null when none exists.

View answer and explanation(12, 1, null, constant_on_right_neighborhood_before_next_support, Yes)

The right value includes any mass at t and remains unchanged before the next support point. With no next support it remains constant for all larger thresholds. This is right continuity, not left continuity or a density claim. The exact result, retaining the requested order, is (12, 1, null, constant_on_right_neighborhood_before_next_support, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r06-cumulative-distribution-function-0087/)

Cumulative distribution function · qf-mod30-cdf-right-continuity · candidate





## 88. A Finite-Law CDF Stays Constant Just to the Right

Use the complete finite discrete law with distinct integer support=[0, 7, 13] 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=0. Give (t,F_t,next_support,constant_on_right_neighborhood_before_next_support,right_continuous?). next_support is the least support value strictly above t, or null when none exists.

View answer and explanation(0, 1/4, 7, constant_on_right_neighborhood_before_next_support, Yes)

The right value includes any mass at t and remains unchanged before the next support point. With no next support it remains constant for all larger thresholds. This is right continuity, not left continuity or a density claim. The exact result, retaining the requested order, is (0, 1/4, 7, constant_on_right_neighborhood_before_next_support, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r06-cumulative-distribution-function-0088/)

Cumulative distribution function · qf-mod30-cdf-right-continuity · candidate





## 89. A Finite-Law CDF Stays Constant Just to the Right

Use the complete finite discrete law with distinct integer support=[1, 8, 14] 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=8. Give (t,F_t,next_support,constant_on_right_neighborhood_before_next_support,right_continuous?). next_support is the least support value strictly above t, or null when none exists.

View answer and explanation(8, 1/2, 14, constant_on_right_neighborhood_before_next_support, Yes)

The right value includes any mass at t and remains unchanged before the next support point. With no next support it remains constant for all larger thresholds. This is right continuity, not left continuity or a density claim. The exact result, retaining the requested order, is (8, 1/2, 14, constant_on_right_neighborhood_before_next_support, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r06-cumulative-distribution-function-0089/)

Cumulative distribution function · qf-mod30-cdf-right-continuity · candidate





## 90. A Finite-Law CDF Stays Constant Just to the Right

Use the complete finite discrete law with distinct integer support=[2, 9, 15] 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=15. Give (t,F_t,next_support,constant_on_right_neighborhood_before_next_support,right_continuous?). next_support is the least support value strictly above t, or null when none exists.

View answer and explanation(15, 1, null, constant_on_right_neighborhood_before_next_support, Yes)

The right value includes any mass at t and remains unchanged before the next support point. With no next support it remains constant for all larger thresholds. This is right continuity, not left continuity or a density claim. The exact result, retaining the requested order, is (15, 1, null, constant_on_right_neighborhood_before_next_support, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r06-cumulative-distribution-function-0090/)

Cumulative distribution function · qf-mod30-cdf-right-continuity · candidate





## 91. A Finite-Law CDF Stays Constant Just to the Right

Use the complete finite discrete law with distinct integer support=[3, 10, 16] 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=3. Give (t,F_t,next_support,constant_on_right_neighborhood_before_next_support,right_continuous?). next_support is the least support value strictly above t, or null when none exists.

View answer and explanation(3, 1/4, 10, constant_on_right_neighborhood_before_next_support, Yes)

The right value includes any mass at t and remains unchanged before the next support point. With no next support it remains constant for all larger thresholds. This is right continuity, not left continuity or a density claim. The exact result, retaining the requested order, is (3, 1/4, 10, constant_on_right_neighborhood_before_next_support, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r06-cumulative-distribution-function-0091/)

Cumulative distribution function · qf-mod30-cdf-right-continuity · candidate





## 92. A Finite-Law CDF Stays Constant Just to the Right

Use the complete finite discrete law with distinct integer support=[4, 11, 17] 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=11. Give (t,F_t,next_support,constant_on_right_neighborhood_before_next_support,right_continuous?). next_support is the least support value strictly above t, or null when none exists.

View answer and explanation(11, 1/2, 17, constant_on_right_neighborhood_before_next_support, Yes)

The right value includes any mass at t and remains unchanged before the next support point. With no next support it remains constant for all larger thresholds. This is right continuity, not left continuity or a density claim. The exact result, retaining the requested order, is (11, 1/2, 17, constant_on_right_neighborhood_before_next_support, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r06-cumulative-distribution-function-0092/)

Cumulative distribution function · qf-mod30-cdf-right-continuity · candidate





## 93. A Finite-Law CDF Stays Constant Just to the Right

Use the complete finite discrete law with distinct integer support=[5, 12, 18] 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=18. Give (t,F_t,next_support,constant_on_right_neighborhood_before_next_support,right_continuous?). next_support is the least support value strictly above t, or null when none exists.

View answer and explanation(18, 1, null, constant_on_right_neighborhood_before_next_support, Yes)

The right value includes any mass at t and remains unchanged before the next support point. With no next support it remains constant for all larger thresholds. This is right continuity, not left continuity or a density claim. The exact result, retaining the requested order, is (18, 1, null, constant_on_right_neighborhood_before_next_support, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r06-cumulative-distribution-function-0093/)

Cumulative distribution function · qf-mod30-cdf-right-continuity · candidate





## 94. A Finite-Law CDF Stays Constant Just to the Right

Use the complete finite discrete law with distinct integer support=[6, 13, 19] 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=6. Give (t,F_t,next_support,constant_on_right_neighborhood_before_next_support,right_continuous?). next_support is the least support value strictly above t, or null when none exists.

View answer and explanation(6, 1/4, 13, constant_on_right_neighborhood_before_next_support, Yes)

The right value includes any mass at t and remains unchanged before the next support point. With no next support it remains constant for all larger thresholds. This is right continuity, not left continuity or a density claim. The exact result, retaining the requested order, is (6, 1/4, 13, constant_on_right_neighborhood_before_next_support, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r06-cumulative-distribution-function-0094/)

Cumulative distribution function · qf-mod30-cdf-right-continuity · candidate





## 95. A Finite-Law CDF Stays Constant Just to the Right

Use the complete finite discrete law with distinct integer support=[7, 14, 20] 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=14. Give (t,F_t,next_support,constant_on_right_neighborhood_before_next_support,right_continuous?). next_support is the least support value strictly above t, or null when none exists.

View answer and explanation(14, 1/2, 20, constant_on_right_neighborhood_before_next_support, Yes)

The right value includes any mass at t and remains unchanged before the next support point. With no next support it remains constant for all larger thresholds. This is right continuity, not left continuity or a density claim. The exact result, retaining the requested order, is (14, 1/2, 20, constant_on_right_neighborhood_before_next_support, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r06-cumulative-distribution-function-0095/)

Cumulative distribution function · qf-mod30-cdf-right-continuity · candidate





## 96. A Finite-Law CDF Stays Constant Just to the Right

Use the complete finite discrete law with distinct integer support=[8, 15, 21] 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=21. Give (t,F_t,next_support,constant_on_right_neighborhood_before_next_support,right_continuous?). next_support is the least support value strictly above t, or null when none exists.

View answer and explanation(21, 1, null, constant_on_right_neighborhood_before_next_support, Yes)

The right value includes any mass at t and remains unchanged before the next support point. With no next support it remains constant for all larger thresholds. This is right continuity, not left continuity or a density claim. The exact result, retaining the requested order, is (21, 1, null, constant_on_right_neighborhood_before_next_support, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r06-cumulative-distribution-function-0096/)

Cumulative distribution function · qf-mod30-cdf-right-continuity · candidate





## 97. A Finite-Law CDF Stays Constant Just to the Right

Use the complete finite discrete law with distinct integer support=[9, 16, 22] 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=9. Give (t,F_t,next_support,constant_on_right_neighborhood_before_next_support,right_continuous?). next_support is the least support value strictly above t, or null when none exists.

View answer and explanation(9, 1/4, 16, constant_on_right_neighborhood_before_next_support, Yes)

The right value includes any mass at t and remains unchanged before the next support point. With no next support it remains constant for all larger thresholds. This is right continuity, not left continuity or a density claim. The exact result, retaining the requested order, is (9, 1/4, 16, constant_on_right_neighborhood_before_next_support, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r06-cumulative-distribution-function-0097/)

Cumulative distribution function · qf-mod30-cdf-right-continuity · candidate





## 98. A Finite-Law CDF Stays Constant Just to the Right

Use the complete finite discrete law with distinct integer support=[10, 17, 23] 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=17. Give (t,F_t,next_support,constant_on_right_neighborhood_before_next_support,right_continuous?). next_support is the least support value strictly above t, or null when none exists.

View answer and explanation(17, 1/2, 23, constant_on_right_neighborhood_before_next_support, Yes)

The right value includes any mass at t and remains unchanged before the next support point. With no next support it remains constant for all larger thresholds. This is right continuity, not left continuity or a density claim. The exact result, retaining the requested order, is (17, 1/2, 23, constant_on_right_neighborhood_before_next_support, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r06-cumulative-distribution-function-0098/)

Cumulative distribution function · qf-mod30-cdf-right-continuity · candidate





## 99. A Finite-Law CDF Stays Constant Just to the Right

Use the complete finite discrete law with distinct integer support=[11, 18, 24] 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=24. Give (t,F_t,next_support,constant_on_right_neighborhood_before_next_support,right_continuous?). next_support is the least support value strictly above t, or null when none exists.

View answer and explanation(24, 1, null, constant_on_right_neighborhood_before_next_support, Yes)

The right value includes any mass at t and remains unchanged before the next support point. With no next support it remains constant for all larger thresholds. This is right continuity, not left continuity or a density claim. The exact result, retaining the requested order, is (24, 1, null, constant_on_right_neighborhood_before_next_support, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r06-cumulative-distribution-function-0099/)

Cumulative distribution function · qf-mod30-cdf-right-continuity · candidate





## 100. A Finite-Law CDF Stays Constant Just to the Right

Use the complete finite discrete law with distinct integer support=[12, 19, 25] 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=12. Give (t,F_t,next_support,constant_on_right_neighborhood_before_next_support,right_continuous?). next_support is the least support value strictly above t, or null when none exists.

View answer and explanation(12, 1/4, 19, constant_on_right_neighborhood_before_next_support, Yes)

The right value includes any mass at t and remains unchanged before the next support point. With no next support it remains constant for all larger thresholds. This is right continuity, not left continuity or a density claim. The exact result, retaining the requested order, is (12, 1/4, 19, constant_on_right_neighborhood_before_next_support, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r06-cumulative-distribution-function-0100/)

Cumulative distribution function · qf-mod30-cdf-right-continuity · candidate



[Canonical question JSONL](https://ocme.evemisslab.com/data/questions/batches/batch-20261014-advance-20261030-r06-cumulative-distribution-function-foundations/questions.jsonl) · [English JSONL](https://ocme.evemisslab.com/data/i18n/questions/batch-20261014-advance-20261030-r06-cumulative-distribution-function-foundations.en.jsonl) · [Validation report](https://ocme.evemisslab.com/data/questions/batches/batch-20261014-advance-20261030-r06-cumulative-distribution-function-foundations/validation-report.json)