Exact Cumulative Masses, Jumps, Monotonicity and Right Continuity
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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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
Cumulative distribution function · qf-mod30-cdf-right-continuity · candidate
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