# Finite Random Variable Foundations

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

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

batch-20261004-advance-20261005-r05-random-variable-foundations · 2026-10-04 · 100 questions

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



## 1. Random-Variable Value at an Elementary Outcome

The nonempty finite sample space is S={0}, with event family F=𝒫(S). The total function X:S→R has table {(0, 0)}. Find X(0), reading only the function value without assigning unspecified probabilities.

View answer and explanation0

The elementary outcome belongs to S and its unique value is X(0)=0. No unspecified probability law or equiprobability is inferred.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r05-random-variable-0001/)

Real random variables on finite sample spaces · qf-random-variable-evaluation · candidate





## 2. Random-Variable Value at an Elementary Outcome

The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -1), (1, -1), (2, 3), (3, 1)}. Find X(1), reading only the function value without assigning unspecified probabilities.

View answer and explanation-1

The elementary outcome belongs to S and its unique value is X(1)=-1. No unspecified probability law or equiprobability is inferred.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r05-random-variable-0002/)

Real random variables on finite sample spaces · qf-random-variable-evaluation · candidate





## 3. Random-Variable Value at an Elementary Outcome

The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -2), (1, -2), (2, 4), (3, 0)}. Find X(2), reading only the function value without assigning unspecified probabilities.

View answer and explanation4

The elementary outcome belongs to S and its unique value is X(2)=4. No unspecified probability law or equiprobability is inferred.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r05-random-variable-0003/)

Real random variables on finite sample spaces · qf-random-variable-evaluation · candidate





## 4. Random-Variable Value at an Elementary Outcome

The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -3), (1, -3), (2, 5), (3, 5)}. Find X(3), reading only the function value without assigning unspecified probabilities.

View answer and explanation5

The elementary outcome belongs to S and its unique value is X(3)=5. No unspecified probability law or equiprobability is inferred.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r05-random-variable-0004/)

Real random variables on finite sample spaces · qf-random-variable-evaluation · candidate





## 5. Random-Variable Value at an Elementary Outcome

The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -4), (1, -4), (2, 6), (3, -2)}. Find X(0), reading only the function value without assigning unspecified probabilities.

View answer and explanation-4

The elementary outcome belongs to S and its unique value is X(0)=-4. No unspecified probability law or equiprobability is inferred.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r05-random-variable-0005/)

Real random variables on finite sample spaces · qf-random-variable-evaluation · candidate





## 6. Random-Variable Value at an Elementary Outcome

The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -5), (1, -5), (2, 7), (3, -3)}. Find X(1), reading only the function value without assigning unspecified probabilities.

View answer and explanation-5

The elementary outcome belongs to S and its unique value is X(1)=-5. No unspecified probability law or equiprobability is inferred.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r05-random-variable-0006/)

Real random variables on finite sample spaces · qf-random-variable-evaluation · candidate





## 7. Random-Variable Value at an Elementary Outcome

The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -6), (1, -6), (2, 8), (3, 8)}. Find X(2), reading only the function value without assigning unspecified probabilities.

View answer and explanation8

The elementary outcome belongs to S and its unique value is X(2)=8. No unspecified probability law or equiprobability is inferred.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r05-random-variable-0007/)

Real random variables on finite sample spaces · qf-random-variable-evaluation · candidate





## 8. Random-Variable Value at an Elementary Outcome

The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -7), (1, -7), (2, 9), (3, -5)}. Find X(3), reading only the function value without assigning unspecified probabilities.

View answer and explanation-5

The elementary outcome belongs to S and its unique value is X(3)=-5. No unspecified probability law or equiprobability is inferred.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r05-random-variable-0008/)

Real random variables on finite sample spaces · qf-random-variable-evaluation · candidate





## 9. Random-Variable Value at an Elementary Outcome

The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -8), (1, -8), (2, 10), (3, -6)}. Find X(0), reading only the function value without assigning unspecified probabilities.

View answer and explanation-8

The elementary outcome belongs to S and its unique value is X(0)=-8. No unspecified probability law or equiprobability is inferred.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r05-random-variable-0009/)

Real random variables on finite sample spaces · qf-random-variable-evaluation · candidate





## 10. Random-Variable Value at an Elementary Outcome

The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -9), (1, -9), (2, 11), (3, 11)}. Find X(1), reading only the function value without assigning unspecified probabilities.

View answer and explanation-9

The elementary outcome belongs to S and its unique value is X(1)=-9. No unspecified probability law or equiprobability is inferred.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r05-random-variable-0010/)

Real random variables on finite sample spaces · qf-random-variable-evaluation · candidate





## 11. Random-Variable Value at an Elementary Outcome

The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -10), (1, -10), (2, 12), (3, -8)}. Find X(2), reading only the function value without assigning unspecified probabilities.

View answer and explanation12

The elementary outcome belongs to S and its unique value is X(2)=12. No unspecified probability law or equiprobability is inferred.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r05-random-variable-0011/)

Real random variables on finite sample spaces · qf-random-variable-evaluation · candidate





## 12. Random-Variable Value at an Elementary Outcome

The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -11), (1, -11), (2, 13), (3, -9)}. Find X(3), reading only the function value without assigning unspecified probabilities.

View answer and explanation-9

The elementary outcome belongs to S and its unique value is X(3)=-9. No unspecified probability law or equiprobability is inferred.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r05-random-variable-0012/)

Real random variables on finite sample spaces · qf-random-variable-evaluation · candidate





## 13. Random-Variable Value at an Elementary Outcome

The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -12), (1, -12), (2, 14), (3, 14)}. Find X(0), reading only the function value without assigning unspecified probabilities.

View answer and explanation-12

The elementary outcome belongs to S and its unique value is X(0)=-12. No unspecified probability law or equiprobability is inferred.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r05-random-variable-0013/)

Real random variables on finite sample spaces · qf-random-variable-evaluation · candidate





## 14. Random-Variable Value at an Elementary Outcome

The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -13), (1, -13), (2, 15), (3, -11)}. Find X(1), reading only the function value without assigning unspecified probabilities.

View answer and explanation-13

The elementary outcome belongs to S and its unique value is X(1)=-13. No unspecified probability law or equiprobability is inferred.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r05-random-variable-0014/)

Real random variables on finite sample spaces · qf-random-variable-evaluation · candidate





## 15. Random-Variable Value at an Elementary Outcome

The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -14), (1, -14), (2, 16), (3, -12)}. Find X(2), reading only the function value without assigning unspecified probabilities.

View answer and explanation16

The elementary outcome belongs to S and its unique value is X(2)=16. No unspecified probability law or equiprobability is inferred.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r05-random-variable-0015/)

Real random variables on finite sample spaces · qf-random-variable-evaluation · candidate





## 16. Random-Variable Value at an Elementary Outcome

The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -15), (1, -15), (2, 17), (3, 17)}. Find X(3), reading only the function value without assigning unspecified probabilities.

View answer and explanation17

The elementary outcome belongs to S and its unique value is X(3)=17. No unspecified probability law or equiprobability is inferred.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r05-random-variable-0016/)

Real random variables on finite sample spaces · qf-random-variable-evaluation · candidate





## 17. Random-Variable Value at an Elementary Outcome

The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -16), (1, -16), (2, 18), (3, -14)}. Find X(0), reading only the function value without assigning unspecified probabilities.

View answer and explanation-16

The elementary outcome belongs to S and its unique value is X(0)=-16. No unspecified probability law or equiprobability is inferred.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r05-random-variable-0017/)

Real random variables on finite sample spaces · qf-random-variable-evaluation · candidate





## 18. Random-Variable Value at an Elementary Outcome

The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -17), (1, -17), (2, 19), (3, -15)}. Find X(1), reading only the function value without assigning unspecified probabilities.

View answer and explanation-17

The elementary outcome belongs to S and its unique value is X(1)=-17. No unspecified probability law or equiprobability is inferred.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r05-random-variable-0018/)

Real random variables on finite sample spaces · qf-random-variable-evaluation · candidate





## 19. Random-Variable Value at an Elementary Outcome

The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -18), (1, -18), (2, 20), (3, 20)}. Find X(2), reading only the function value without assigning unspecified probabilities.

View answer and explanation20

The elementary outcome belongs to S and its unique value is X(2)=20. No unspecified probability law or equiprobability is inferred.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r05-random-variable-0019/)

Real random variables on finite sample spaces · qf-random-variable-evaluation · candidate





## 20. Random-Variable Value at an Elementary Outcome

The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -19), (1, -19), (2, 21), (3, -17)}. Find X(3), reading only the function value without assigning unspecified probabilities.

View answer and explanation-17

The elementary outcome belongs to S and its unique value is X(3)=-17. No unspecified probability law or equiprobability is inferred.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r05-random-variable-0020/)

Real random variables on finite sample spaces · qf-random-variable-evaluation · candidate





## 21. Random-Variable Value at an Elementary Outcome

The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -20), (1, -20), (2, 22), (3, -18)}. Find X(0), reading only the function value without assigning unspecified probabilities.

View answer and explanation-20

The elementary outcome belongs to S and its unique value is X(0)=-20. No unspecified probability law or equiprobability is inferred.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r05-random-variable-0021/)

Real random variables on finite sample spaces · qf-random-variable-evaluation · candidate





## 22. Random-Variable Value at an Elementary Outcome

The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -21), (1, -21), (2, 23), (3, 23)}. Find X(1), reading only the function value without assigning unspecified probabilities.

View answer and explanation-21

The elementary outcome belongs to S and its unique value is X(1)=-21. No unspecified probability law or equiprobability is inferred.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r05-random-variable-0022/)

Real random variables on finite sample spaces · qf-random-variable-evaluation · candidate





## 23. Random-Variable Value at an Elementary Outcome

The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -22), (1, -22), (2, 24), (3, -20)}. Find X(2), reading only the function value without assigning unspecified probabilities.

View answer and explanation24

The elementary outcome belongs to S and its unique value is X(2)=24. No unspecified probability law or equiprobability is inferred.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r05-random-variable-0023/)

Real random variables on finite sample spaces · qf-random-variable-evaluation · candidate





## 24. Random-Variable Value at an Elementary Outcome

The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -23), (1, -23), (2, 25), (3, -21)}. Find X(3), reading only the function value without assigning unspecified probabilities.

View answer and explanation-21

The elementary outcome belongs to S and its unique value is X(3)=-21. No unspecified probability law or equiprobability is inferred.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r05-random-variable-0024/)

Real random variables on finite sample spaces · qf-random-variable-evaluation · candidate





## 25. Random-Variable Value at an Elementary Outcome

The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -24), (1, -24), (2, 26), (3, 26)}. Find X(0), reading only the function value without assigning unspecified probabilities.

View answer and explanation-24

The elementary outcome belongs to S and its unique value is X(0)=-24. No unspecified probability law or equiprobability is inferred.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r05-random-variable-0025/)

Real random variables on finite sample spaces · qf-random-variable-evaluation · candidate





## 26. Preimage Event of a Threshold Condition

The nonempty finite sample space is S={0}, with event family F=𝒫(S). The total function X:S→R has table {(0, 0)}. List the event {s∈S:X(s)≥-1} in increasing order. Return elementary outcomes, not a numerical probability.

View answer and explanation{0}

Check the inclusive threshold at each outcome to obtain {0}. F is the full powerset, so this subset is an event; without weights, no event probability is asserted.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r05-random-variable-0026/)

Real random variables on finite sample spaces · qf-random-variable-threshold-event · candidate





## 27. Preimage Event of a Threshold Condition

The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -1), (1, -1), (2, 3), (3, 1)}. List the event {s∈S:X(s)≥4} in increasing order. Return elementary outcomes, not a numerical probability.

View answer and explanation∅

Check the inclusive threshold at each outcome to obtain ∅. F is the full powerset, so this subset is an event; without weights, no event probability is asserted.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r05-random-variable-0027/)

Real random variables on finite sample spaces · qf-random-variable-threshold-event · candidate





## 28. Preimage Event of a Threshold Condition

The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -2), (1, -2), (2, 4), (3, 0)}. List the event {s∈S:X(s)≥0} in increasing order. Return elementary outcomes, not a numerical probability.

View answer and explanation{2, 3}

Check the inclusive threshold at each outcome to obtain {2, 3}. F is the full powerset, so this subset is an event; without weights, no event probability is asserted.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r05-random-variable-0028/)

Real random variables on finite sample spaces · qf-random-variable-threshold-event · candidate





## 29. Preimage Event of a Threshold Condition

The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -3), (1, -3), (2, 5), (3, 5)}. List the event {s∈S:X(s)≥-3} in increasing order. Return elementary outcomes, not a numerical probability.

View answer and explanation{0, 1, 2, 3}

Check the inclusive threshold at each outcome to obtain {0, 1, 2, 3}. F is the full powerset, so this subset is an event; without weights, no event probability is asserted.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r05-random-variable-0029/)

Real random variables on finite sample spaces · qf-random-variable-threshold-event · candidate





## 30. Preimage Event of a Threshold Condition

The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -4), (1, -4), (2, 6), (3, -2)}. List the event {s∈S:X(s)≥-5} in increasing order. Return elementary outcomes, not a numerical probability.

View answer and explanation{0, 1, 2, 3}

Check the inclusive threshold at each outcome to obtain {0, 1, 2, 3}. F is the full powerset, so this subset is an event; without weights, no event probability is asserted.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r05-random-variable-0030/)

Real random variables on finite sample spaces · qf-random-variable-threshold-event · candidate





## 31. Preimage Event of a Threshold Condition

The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -5), (1, -5), (2, 7), (3, -3)}. List the event {s∈S:X(s)≥8} in increasing order. Return elementary outcomes, not a numerical probability.

View answer and explanation∅

Check the inclusive threshold at each outcome to obtain ∅. F is the full powerset, so this subset is an event; without weights, no event probability is asserted.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r05-random-variable-0031/)

Real random variables on finite sample spaces · qf-random-variable-threshold-event · candidate





## 32. Preimage Event of a Threshold Condition

The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -6), (1, -6), (2, 8), (3, 8)}. List the event {s∈S:X(s)≥0} in increasing order. Return elementary outcomes, not a numerical probability.

View answer and explanation{2, 3}

Check the inclusive threshold at each outcome to obtain {2, 3}. F is the full powerset, so this subset is an event; without weights, no event probability is asserted.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r05-random-variable-0032/)

Real random variables on finite sample spaces · qf-random-variable-threshold-event · candidate





## 33. Preimage Event of a Threshold Condition

The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -7), (1, -7), (2, 9), (3, -5)}. List the event {s∈S:X(s)≥-7} in increasing order. Return elementary outcomes, not a numerical probability.

View answer and explanation{0, 1, 2, 3}

Check the inclusive threshold at each outcome to obtain {0, 1, 2, 3}. F is the full powerset, so this subset is an event; without weights, no event probability is asserted.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r05-random-variable-0033/)

Real random variables on finite sample spaces · qf-random-variable-threshold-event · candidate





## 34. Preimage Event of a Threshold Condition

The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -8), (1, -8), (2, 10), (3, -6)}. List the event {s∈S:X(s)≥-9} in increasing order. Return elementary outcomes, not a numerical probability.

View answer and explanation{0, 1, 2, 3}

Check the inclusive threshold at each outcome to obtain {0, 1, 2, 3}. F is the full powerset, so this subset is an event; without weights, no event probability is asserted.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r05-random-variable-0034/)

Real random variables on finite sample spaces · qf-random-variable-threshold-event · candidate





## 35. Preimage Event of a Threshold Condition

The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -9), (1, -9), (2, 11), (3, 11)}. List the event {s∈S:X(s)≥12} in increasing order. Return elementary outcomes, not a numerical probability.

View answer and explanation∅

Check the inclusive threshold at each outcome to obtain ∅. F is the full powerset, so this subset is an event; without weights, no event probability is asserted.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r05-random-variable-0035/)

Real random variables on finite sample spaces · qf-random-variable-threshold-event · candidate





## 36. Preimage Event of a Threshold Condition

The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -10), (1, -10), (2, 12), (3, -8)}. List the event {s∈S:X(s)≥0} in increasing order. Return elementary outcomes, not a numerical probability.

View answer and explanation{2}

Check the inclusive threshold at each outcome to obtain {2}. F is the full powerset, so this subset is an event; without weights, no event probability is asserted.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r05-random-variable-0036/)

Real random variables on finite sample spaces · qf-random-variable-threshold-event · candidate





## 37. Preimage Event of a Threshold Condition

The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -11), (1, -11), (2, 13), (3, -9)}. List the event {s∈S:X(s)≥-11} in increasing order. Return elementary outcomes, not a numerical probability.

View answer and explanation{0, 1, 2, 3}

Check the inclusive threshold at each outcome to obtain {0, 1, 2, 3}. F is the full powerset, so this subset is an event; without weights, no event probability is asserted.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r05-random-variable-0037/)

Real random variables on finite sample spaces · qf-random-variable-threshold-event · candidate





## 38. Preimage Event of a Threshold Condition

The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -12), (1, -12), (2, 14), (3, 14)}. List the event {s∈S:X(s)≥-13} in increasing order. Return elementary outcomes, not a numerical probability.

View answer and explanation{0, 1, 2, 3}

Check the inclusive threshold at each outcome to obtain {0, 1, 2, 3}. F is the full powerset, so this subset is an event; without weights, no event probability is asserted.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r05-random-variable-0038/)

Real random variables on finite sample spaces · qf-random-variable-threshold-event · candidate





## 39. Preimage Event of a Threshold Condition

The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -13), (1, -13), (2, 15), (3, -11)}. List the event {s∈S:X(s)≥16} in increasing order. Return elementary outcomes, not a numerical probability.

View answer and explanation∅

Check the inclusive threshold at each outcome to obtain ∅. F is the full powerset, so this subset is an event; without weights, no event probability is asserted.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r05-random-variable-0039/)

Real random variables on finite sample spaces · qf-random-variable-threshold-event · candidate





## 40. Preimage Event of a Threshold Condition

The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -14), (1, -14), (2, 16), (3, -12)}. List the event {s∈S:X(s)≥0} in increasing order. Return elementary outcomes, not a numerical probability.

View answer and explanation{2}

Check the inclusive threshold at each outcome to obtain {2}. F is the full powerset, so this subset is an event; without weights, no event probability is asserted.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r05-random-variable-0040/)

Real random variables on finite sample spaces · qf-random-variable-threshold-event · candidate





## 41. Preimage Event of a Threshold Condition

The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -15), (1, -15), (2, 17), (3, 17)}. List the event {s∈S:X(s)≥-15} in increasing order. Return elementary outcomes, not a numerical probability.

View answer and explanation{0, 1, 2, 3}

Check the inclusive threshold at each outcome to obtain {0, 1, 2, 3}. F is the full powerset, so this subset is an event; without weights, no event probability is asserted.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r05-random-variable-0041/)

Real random variables on finite sample spaces · qf-random-variable-threshold-event · candidate





## 42. Preimage Event of a Threshold Condition

The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -16), (1, -16), (2, 18), (3, -14)}. List the event {s∈S:X(s)≥-17} in increasing order. Return elementary outcomes, not a numerical probability.

View answer and explanation{0, 1, 2, 3}

Check the inclusive threshold at each outcome to obtain {0, 1, 2, 3}. F is the full powerset, so this subset is an event; without weights, no event probability is asserted.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r05-random-variable-0042/)

Real random variables on finite sample spaces · qf-random-variable-threshold-event · candidate





## 43. Preimage Event of a Threshold Condition

The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -17), (1, -17), (2, 19), (3, -15)}. List the event {s∈S:X(s)≥20} in increasing order. Return elementary outcomes, not a numerical probability.

View answer and explanation∅

Check the inclusive threshold at each outcome to obtain ∅. F is the full powerset, so this subset is an event; without weights, no event probability is asserted.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r05-random-variable-0043/)

Real random variables on finite sample spaces · qf-random-variable-threshold-event · candidate





## 44. Preimage Event of a Threshold Condition

The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -18), (1, -18), (2, 20), (3, 20)}. List the event {s∈S:X(s)≥0} in increasing order. Return elementary outcomes, not a numerical probability.

View answer and explanation{2, 3}

Check the inclusive threshold at each outcome to obtain {2, 3}. F is the full powerset, so this subset is an event; without weights, no event probability is asserted.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r05-random-variable-0044/)

Real random variables on finite sample spaces · qf-random-variable-threshold-event · candidate





## 45. Preimage Event of a Threshold Condition

The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -19), (1, -19), (2, 21), (3, -17)}. List the event {s∈S:X(s)≥-19} in increasing order. Return elementary outcomes, not a numerical probability.

View answer and explanation{0, 1, 2, 3}

Check the inclusive threshold at each outcome to obtain {0, 1, 2, 3}. F is the full powerset, so this subset is an event; without weights, no event probability is asserted.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r05-random-variable-0045/)

Real random variables on finite sample spaces · qf-random-variable-threshold-event · candidate





## 46. Preimage Event of a Threshold Condition

The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -20), (1, -20), (2, 22), (3, -18)}. List the event {s∈S:X(s)≥-21} in increasing order. Return elementary outcomes, not a numerical probability.

View answer and explanation{0, 1, 2, 3}

Check the inclusive threshold at each outcome to obtain {0, 1, 2, 3}. F is the full powerset, so this subset is an event; without weights, no event probability is asserted.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r05-random-variable-0046/)

Real random variables on finite sample spaces · qf-random-variable-threshold-event · candidate





## 47. Preimage Event of a Threshold Condition

The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -21), (1, -21), (2, 23), (3, 23)}. List the event {s∈S:X(s)≥24} in increasing order. Return elementary outcomes, not a numerical probability.

View answer and explanation∅

Check the inclusive threshold at each outcome to obtain ∅. F is the full powerset, so this subset is an event; without weights, no event probability is asserted.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r05-random-variable-0047/)

Real random variables on finite sample spaces · qf-random-variable-threshold-event · candidate





## 48. Preimage Event of a Threshold Condition

The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -22), (1, -22), (2, 24), (3, -20)}. List the event {s∈S:X(s)≥0} in increasing order. Return elementary outcomes, not a numerical probability.

View answer and explanation{2}

Check the inclusive threshold at each outcome to obtain {2}. F is the full powerset, so this subset is an event; without weights, no event probability is asserted.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r05-random-variable-0048/)

Real random variables on finite sample spaces · qf-random-variable-threshold-event · candidate





## 49. Preimage Event of a Threshold Condition

The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -23), (1, -23), (2, 25), (3, -21)}. List the event {s∈S:X(s)≥-23} in increasing order. Return elementary outcomes, not a numerical probability.

View answer and explanation{0, 1, 2, 3}

Check the inclusive threshold at each outcome to obtain {0, 1, 2, 3}. F is the full powerset, so this subset is an event; without weights, no event probability is asserted.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r05-random-variable-0049/)

Real random variables on finite sample spaces · qf-random-variable-threshold-event · candidate





## 50. Preimage Event of a Threshold Condition

The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -24), (1, -24), (2, 26), (3, 26)}. List the event {s∈S:X(s)≥-25} in increasing order. Return elementary outcomes, not a numerical probability.

View answer and explanation{0, 1, 2, 3}

Check the inclusive threshold at each outcome to obtain {0, 1, 2, 3}. F is the full powerset, so this subset is an event; without weights, no event probability is asserted.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r05-random-variable-0050/)

Real random variables on finite sample spaces · qf-random-variable-threshold-event · candidate





## 51. Attained Range of a Random Variable

The nonempty finite sample space is S={0}, with event family F=𝒫(S). The total function X:S→R has table {(0, 0)}. List the attained range X[S] in increasing order with each output value once.

View answer and explanation{0}

The range is {0}, a set of attained output values rather than elementary outcomes or probability weights. Removing duplicate output values does not remove their preimage weights from a probability calculation.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r05-random-variable-0051/)

Real random variables on finite sample spaces · qf-random-variable-range · candidate





## 52. Attained Range of a Random Variable

The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -1), (1, -1), (2, 3), (3, 1)}. List the attained range X[S] in increasing order with each output value once.

View answer and explanation{-1, 1, 3}

The range is {-1, 1, 3}, a set of attained output values rather than elementary outcomes or probability weights. Removing duplicate output values does not remove their preimage weights from a probability calculation.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r05-random-variable-0052/)

Real random variables on finite sample spaces · qf-random-variable-range · candidate





## 53. Attained Range of a Random Variable

The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -2), (1, -2), (2, 4), (3, 0)}. List the attained range X[S] in increasing order with each output value once.

View answer and explanation{-2, 0, 4}

The range is {-2, 0, 4}, a set of attained output values rather than elementary outcomes or probability weights. Removing duplicate output values does not remove their preimage weights from a probability calculation.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r05-random-variable-0053/)

Real random variables on finite sample spaces · qf-random-variable-range · candidate





## 54. Attained Range of a Random Variable

The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -3), (1, -3), (2, 5), (3, 5)}. List the attained range X[S] in increasing order with each output value once.

View answer and explanation{-3, 5}

The range is {-3, 5}, a set of attained output values rather than elementary outcomes or probability weights. Removing duplicate output values does not remove their preimage weights from a probability calculation.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r05-random-variable-0054/)

Real random variables on finite sample spaces · qf-random-variable-range · candidate





## 55. Attained Range of a Random Variable

The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -4), (1, -4), (2, 6), (3, -2)}. List the attained range X[S] in increasing order with each output value once.

View answer and explanation{-4, -2, 6}

The range is {-4, -2, 6}, a set of attained output values rather than elementary outcomes or probability weights. Removing duplicate output values does not remove their preimage weights from a probability calculation.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r05-random-variable-0055/)

Real random variables on finite sample spaces · qf-random-variable-range · candidate





## 56. Attained Range of a Random Variable

The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -5), (1, -5), (2, 7), (3, -3)}. List the attained range X[S] in increasing order with each output value once.

View answer and explanation{-5, -3, 7}

The range is {-5, -3, 7}, a set of attained output values rather than elementary outcomes or probability weights. Removing duplicate output values does not remove their preimage weights from a probability calculation.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r05-random-variable-0056/)

Real random variables on finite sample spaces · qf-random-variable-range · candidate





## 57. Attained Range of a Random Variable

The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -6), (1, -6), (2, 8), (3, 8)}. List the attained range X[S] in increasing order with each output value once.

View answer and explanation{-6, 8}

The range is {-6, 8}, a set of attained output values rather than elementary outcomes or probability weights. Removing duplicate output values does not remove their preimage weights from a probability calculation.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r05-random-variable-0057/)

Real random variables on finite sample spaces · qf-random-variable-range · candidate





## 58. Attained Range of a Random Variable

The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -7), (1, -7), (2, 9), (3, -5)}. List the attained range X[S] in increasing order with each output value once.

View answer and explanation{-7, -5, 9}

The range is {-7, -5, 9}, a set of attained output values rather than elementary outcomes or probability weights. Removing duplicate output values does not remove their preimage weights from a probability calculation.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r05-random-variable-0058/)

Real random variables on finite sample spaces · qf-random-variable-range · candidate





## 59. Attained Range of a Random Variable

The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -8), (1, -8), (2, 10), (3, -6)}. List the attained range X[S] in increasing order with each output value once.

View answer and explanation{-8, -6, 10}

The range is {-8, -6, 10}, a set of attained output values rather than elementary outcomes or probability weights. Removing duplicate output values does not remove their preimage weights from a probability calculation.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r05-random-variable-0059/)

Real random variables on finite sample spaces · qf-random-variable-range · candidate





## 60. Attained Range of a Random Variable

The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -9), (1, -9), (2, 11), (3, 11)}. List the attained range X[S] in increasing order with each output value once.

View answer and explanation{-9, 11}

The range is {-9, 11}, a set of attained output values rather than elementary outcomes or probability weights. Removing duplicate output values does not remove their preimage weights from a probability calculation.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r05-random-variable-0060/)

Real random variables on finite sample spaces · qf-random-variable-range · candidate





## 61. Attained Range of a Random Variable

The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -10), (1, -10), (2, 12), (3, -8)}. List the attained range X[S] in increasing order with each output value once.

View answer and explanation{-10, -8, 12}

The range is {-10, -8, 12}, a set of attained output values rather than elementary outcomes or probability weights. Removing duplicate output values does not remove their preimage weights from a probability calculation.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r05-random-variable-0061/)

Real random variables on finite sample spaces · qf-random-variable-range · candidate





## 62. Attained Range of a Random Variable

The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -11), (1, -11), (2, 13), (3, -9)}. List the attained range X[S] in increasing order with each output value once.

View answer and explanation{-11, -9, 13}

The range is {-11, -9, 13}, a set of attained output values rather than elementary outcomes or probability weights. Removing duplicate output values does not remove their preimage weights from a probability calculation.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r05-random-variable-0062/)

Real random variables on finite sample spaces · qf-random-variable-range · candidate





## 63. Attained Range of a Random Variable

The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -12), (1, -12), (2, 14), (3, 14)}. List the attained range X[S] in increasing order with each output value once.

View answer and explanation{-12, 14}

The range is {-12, 14}, a set of attained output values rather than elementary outcomes or probability weights. Removing duplicate output values does not remove their preimage weights from a probability calculation.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r05-random-variable-0063/)

Real random variables on finite sample spaces · qf-random-variable-range · candidate





## 64. Attained Range of a Random Variable

The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -13), (1, -13), (2, 15), (3, -11)}. List the attained range X[S] in increasing order with each output value once.

View answer and explanation{-13, -11, 15}

The range is {-13, -11, 15}, a set of attained output values rather than elementary outcomes or probability weights. Removing duplicate output values does not remove their preimage weights from a probability calculation.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r05-random-variable-0064/)

Real random variables on finite sample spaces · qf-random-variable-range · candidate





## 65. Attained Range of a Random Variable

The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -14), (1, -14), (2, 16), (3, -12)}. List the attained range X[S] in increasing order with each output value once.

View answer and explanation{-14, -12, 16}

The range is {-14, -12, 16}, a set of attained output values rather than elementary outcomes or probability weights. Removing duplicate output values does not remove their preimage weights from a probability calculation.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r05-random-variable-0065/)

Real random variables on finite sample spaces · qf-random-variable-range · candidate





## 66. Attained Range of a Random Variable

The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -15), (1, -15), (2, 17), (3, 17)}. List the attained range X[S] in increasing order with each output value once.

View answer and explanation{-15, 17}

The range is {-15, 17}, a set of attained output values rather than elementary outcomes or probability weights. Removing duplicate output values does not remove their preimage weights from a probability calculation.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r05-random-variable-0066/)

Real random variables on finite sample spaces · qf-random-variable-range · candidate





## 67. Attained Range of a Random Variable

The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -16), (1, -16), (2, 18), (3, -14)}. List the attained range X[S] in increasing order with each output value once.

View answer and explanation{-16, -14, 18}

The range is {-16, -14, 18}, a set of attained output values rather than elementary outcomes or probability weights. Removing duplicate output values does not remove their preimage weights from a probability calculation.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r05-random-variable-0067/)

Real random variables on finite sample spaces · qf-random-variable-range · candidate





## 68. Attained Range of a Random Variable

The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -17), (1, -17), (2, 19), (3, -15)}. List the attained range X[S] in increasing order with each output value once.

View answer and explanation{-17, -15, 19}

The range is {-17, -15, 19}, a set of attained output values rather than elementary outcomes or probability weights. Removing duplicate output values does not remove their preimage weights from a probability calculation.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r05-random-variable-0068/)

Real random variables on finite sample spaces · qf-random-variable-range · candidate





## 69. Attained Range of a Random Variable

The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -18), (1, -18), (2, 20), (3, 20)}. List the attained range X[S] in increasing order with each output value once.

View answer and explanation{-18, 20}

The range is {-18, 20}, a set of attained output values rather than elementary outcomes or probability weights. Removing duplicate output values does not remove their preimage weights from a probability calculation.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r05-random-variable-0069/)

Real random variables on finite sample spaces · qf-random-variable-range · candidate





## 70. Attained Range of a Random Variable

The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -19), (1, -19), (2, 21), (3, -17)}. List the attained range X[S] in increasing order with each output value once.

View answer and explanation{-19, -17, 21}

The range is {-19, -17, 21}, a set of attained output values rather than elementary outcomes or probability weights. Removing duplicate output values does not remove their preimage weights from a probability calculation.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r05-random-variable-0070/)

Real random variables on finite sample spaces · qf-random-variable-range · candidate





## 71. Attained Range of a Random Variable

The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -20), (1, -20), (2, 22), (3, -18)}. List the attained range X[S] in increasing order with each output value once.

View answer and explanation{-20, -18, 22}

The range is {-20, -18, 22}, a set of attained output values rather than elementary outcomes or probability weights. Removing duplicate output values does not remove their preimage weights from a probability calculation.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r05-random-variable-0071/)

Real random variables on finite sample spaces · qf-random-variable-range · candidate





## 72. Attained Range of a Random Variable

The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -21), (1, -21), (2, 23), (3, 23)}. List the attained range X[S] in increasing order with each output value once.

View answer and explanation{-21, 23}

The range is {-21, 23}, a set of attained output values rather than elementary outcomes or probability weights. Removing duplicate output values does not remove their preimage weights from a probability calculation.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r05-random-variable-0072/)

Real random variables on finite sample spaces · qf-random-variable-range · candidate





## 73. Attained Range of a Random Variable

The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -22), (1, -22), (2, 24), (3, -20)}. List the attained range X[S] in increasing order with each output value once.

View answer and explanation{-22, -20, 24}

The range is {-22, -20, 24}, a set of attained output values rather than elementary outcomes or probability weights. Removing duplicate output values does not remove their preimage weights from a probability calculation.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r05-random-variable-0073/)

Real random variables on finite sample spaces · qf-random-variable-range · candidate





## 74. Attained Range of a Random Variable

The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -23), (1, -23), (2, 25), (3, -21)}. List the attained range X[S] in increasing order with each output value once.

View answer and explanation{-23, -21, 25}

The range is {-23, -21, 25}, a set of attained output values rather than elementary outcomes or probability weights. Removing duplicate output values does not remove their preimage weights from a probability calculation.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r05-random-variable-0074/)

Real random variables on finite sample spaces · qf-random-variable-range · candidate





## 75. Attained Range of a Random Variable

The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -24), (1, -24), (2, 26), (3, 26)}. List the attained range X[S] in increasing order with each output value once.

View answer and explanation{-24, 26}

The range is {-24, 26}, a set of attained output values rather than elementary outcomes or probability weights. Removing duplicate output values does not remove their preimage weights from a probability calculation.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r05-random-variable-0075/)

Real random variables on finite sample spaces · qf-random-variable-range · candidate





## 76. Finite Point Probability of a Repeated Value

The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -1), (1, -1), (2, 2), (3, 999)}. In the displayed order of S, the nonnegative probability weights are [1/(6), 2/(6), 3/(6), 0/(6)], summing exactly to one. Find P(X=-1) exactly as a reduced fraction p/q with positive denominator.

View answer and explanation1/2

Add the weights of every outcome whose value equals -1, giving P(X=-1)=1/2. Zero-weight preimages add no mass; an unattained value has probability 0/1. Range cardinality is not a denominator.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r05-random-variable-0076/)

Real random variables on finite sample spaces · qf-random-variable-point-mass · candidate





## 77. Finite Point Probability of a Repeated Value

The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -2), (1, -2), (2, 3), (3, 1000)}. In the displayed order of S, the nonnegative probability weights are [1/(7), 3/(7), 3/(7), 0/(7)], summing exactly to one. Find P(X=3) exactly as a reduced fraction p/q with positive denominator.

View answer and explanation3/7

Add the weights of every outcome whose value equals 3, giving P(X=3)=3/7. Zero-weight preimages add no mass; an unattained value has probability 0/1. Range cardinality is not a denominator.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r05-random-variable-0077/)

Real random variables on finite sample spaces · qf-random-variable-point-mass · candidate





## 78. Finite Point Probability of a Repeated Value

The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -3), (1, -3), (2, 4), (3, 1001)}. In the displayed order of S, the nonnegative probability weights are [1/(8), 4/(8), 3/(8), 0/(8)], summing exactly to one. Find P(X=1001) exactly as a reduced fraction p/q with positive denominator.

View answer and explanation0/1

Add the weights of every outcome whose value equals 1001, giving P(X=1001)=0/1. Zero-weight preimages add no mass; an unattained value has probability 0/1. Range cardinality is not a denominator.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r05-random-variable-0078/)

Real random variables on finite sample spaces · qf-random-variable-point-mass · candidate





## 79. Finite Point Probability of a Repeated Value

The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -4), (1, -4), (2, 5), (3, 1002)}. In the displayed order of S, the nonnegative probability weights are [1/(9), 5/(9), 3/(9), 0/(9)], summing exactly to one. Find P(X=0) exactly as a reduced fraction p/q with positive denominator.

View answer and explanation0/1

Add the weights of every outcome whose value equals 0, giving P(X=0)=0/1. Zero-weight preimages add no mass; an unattained value has probability 0/1. Range cardinality is not a denominator.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r05-random-variable-0079/)

Real random variables on finite sample spaces · qf-random-variable-point-mass · candidate





## 80. Finite Point Probability of a Repeated Value

The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -5), (1, -5), (2, 6), (3, 1003)}. In the displayed order of S, the nonnegative probability weights are [1/(10), 6/(10), 3/(10), 0/(10)], summing exactly to one. Find P(X=-5) exactly as a reduced fraction p/q with positive denominator.

View answer and explanation7/10

Add the weights of every outcome whose value equals -5, giving P(X=-5)=7/10. Zero-weight preimages add no mass; an unattained value has probability 0/1. Range cardinality is not a denominator.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r05-random-variable-0080/)

Real random variables on finite sample spaces · qf-random-variable-point-mass · candidate





## 81. Finite Point Probability of a Repeated Value

The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -6), (1, -6), (2, 7), (3, 1004)}. In the displayed order of S, the nonnegative probability weights are [1/(11), 7/(11), 3/(11), 0/(11)], summing exactly to one. Find P(X=7) exactly as a reduced fraction p/q with positive denominator.

View answer and explanation3/11

Add the weights of every outcome whose value equals 7, giving P(X=7)=3/11. Zero-weight preimages add no mass; an unattained value has probability 0/1. Range cardinality is not a denominator.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r05-random-variable-0081/)

Real random variables on finite sample spaces · qf-random-variable-point-mass · candidate





## 82. Finite Point Probability of a Repeated Value

The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -7), (1, -7), (2, 8), (3, 1005)}. In the displayed order of S, the nonnegative probability weights are [1/(12), 8/(12), 3/(12), 0/(12)], summing exactly to one. Find P(X=1005) exactly as a reduced fraction p/q with positive denominator.

View answer and explanation0/1

Add the weights of every outcome whose value equals 1005, giving P(X=1005)=0/1. Zero-weight preimages add no mass; an unattained value has probability 0/1. Range cardinality is not a denominator.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r05-random-variable-0082/)

Real random variables on finite sample spaces · qf-random-variable-point-mass · candidate





## 83. Finite Point Probability of a Repeated Value

The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -8), (1, -8), (2, 9), (3, 1006)}. In the displayed order of S, the nonnegative probability weights are [1/(13), 9/(13), 3/(13), 0/(13)], summing exactly to one. Find P(X=0) exactly as a reduced fraction p/q with positive denominator.

View answer and explanation0/1

Add the weights of every outcome whose value equals 0, giving P(X=0)=0/1. Zero-weight preimages add no mass; an unattained value has probability 0/1. Range cardinality is not a denominator.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r05-random-variable-0083/)

Real random variables on finite sample spaces · qf-random-variable-point-mass · candidate





## 84. Finite Point Probability of a Repeated Value

The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -9), (1, -9), (2, 10), (3, 1007)}. In the displayed order of S, the nonnegative probability weights are [1/(14), 10/(14), 3/(14), 0/(14)], summing exactly to one. Find P(X=-9) exactly as a reduced fraction p/q with positive denominator.

View answer and explanation11/14

Add the weights of every outcome whose value equals -9, giving P(X=-9)=11/14. Zero-weight preimages add no mass; an unattained value has probability 0/1. Range cardinality is not a denominator.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r05-random-variable-0084/)

Real random variables on finite sample spaces · qf-random-variable-point-mass · candidate





## 85. Finite Point Probability of a Repeated Value

The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -10), (1, -10), (2, 11), (3, 1008)}. In the displayed order of S, the nonnegative probability weights are [1/(15), 11/(15), 3/(15), 0/(15)], summing exactly to one. Find P(X=11) exactly as a reduced fraction p/q with positive denominator.

View answer and explanation1/5

Add the weights of every outcome whose value equals 11, giving P(X=11)=1/5. Zero-weight preimages add no mass; an unattained value has probability 0/1. Range cardinality is not a denominator.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r05-random-variable-0085/)

Real random variables on finite sample spaces · qf-random-variable-point-mass · candidate





## 86. Finite Point Probability of a Repeated Value

The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -11), (1, -11), (2, 12), (3, 1009)}. In the displayed order of S, the nonnegative probability weights are [1/(16), 12/(16), 3/(16), 0/(16)], summing exactly to one. Find P(X=1009) exactly as a reduced fraction p/q with positive denominator.

View answer and explanation0/1

Add the weights of every outcome whose value equals 1009, giving P(X=1009)=0/1. Zero-weight preimages add no mass; an unattained value has probability 0/1. Range cardinality is not a denominator.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r05-random-variable-0086/)

Real random variables on finite sample spaces · qf-random-variable-point-mass · candidate





## 87. Finite Point Probability of a Repeated Value

The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -12), (1, -12), (2, 13), (3, 1010)}. In the displayed order of S, the nonnegative probability weights are [1/(17), 13/(17), 3/(17), 0/(17)], summing exactly to one. Find P(X=0) exactly as a reduced fraction p/q with positive denominator.

View answer and explanation0/1

Add the weights of every outcome whose value equals 0, giving P(X=0)=0/1. Zero-weight preimages add no mass; an unattained value has probability 0/1. Range cardinality is not a denominator.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r05-random-variable-0087/)

Real random variables on finite sample spaces · qf-random-variable-point-mass · candidate





## 88. Finite Point Probability of a Repeated Value

The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -13), (1, -13), (2, 14), (3, 1011)}. In the displayed order of S, the nonnegative probability weights are [1/(18), 14/(18), 3/(18), 0/(18)], summing exactly to one. Find P(X=-13) exactly as a reduced fraction p/q with positive denominator.

View answer and explanation5/6

Add the weights of every outcome whose value equals -13, giving P(X=-13)=5/6. Zero-weight preimages add no mass; an unattained value has probability 0/1. Range cardinality is not a denominator.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r05-random-variable-0088/)

Real random variables on finite sample spaces · qf-random-variable-point-mass · candidate





## 89. Finite Point Probability of a Repeated Value

The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -14), (1, -14), (2, 15), (3, 1012)}. In the displayed order of S, the nonnegative probability weights are [1/(19), 15/(19), 3/(19), 0/(19)], summing exactly to one. Find P(X=15) exactly as a reduced fraction p/q with positive denominator.

View answer and explanation3/19

Add the weights of every outcome whose value equals 15, giving P(X=15)=3/19. Zero-weight preimages add no mass; an unattained value has probability 0/1. Range cardinality is not a denominator.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r05-random-variable-0089/)

Real random variables on finite sample spaces · qf-random-variable-point-mass · candidate





## 90. Finite Point Probability of a Repeated Value

The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -15), (1, -15), (2, 16), (3, 1013)}. In the displayed order of S, the nonnegative probability weights are [1/(20), 16/(20), 3/(20), 0/(20)], summing exactly to one. Find P(X=1013) exactly as a reduced fraction p/q with positive denominator.

View answer and explanation0/1

Add the weights of every outcome whose value equals 1013, giving P(X=1013)=0/1. Zero-weight preimages add no mass; an unattained value has probability 0/1. Range cardinality is not a denominator.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r05-random-variable-0090/)

Real random variables on finite sample spaces · qf-random-variable-point-mass · candidate





## 91. Finite Point Probability of a Repeated Value

The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -16), (1, -16), (2, 17), (3, 1014)}. In the displayed order of S, the nonnegative probability weights are [1/(21), 17/(21), 3/(21), 0/(21)], summing exactly to one. Find P(X=0) exactly as a reduced fraction p/q with positive denominator.

View answer and explanation0/1

Add the weights of every outcome whose value equals 0, giving P(X=0)=0/1. Zero-weight preimages add no mass; an unattained value has probability 0/1. Range cardinality is not a denominator.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r05-random-variable-0091/)

Real random variables on finite sample spaces · qf-random-variable-point-mass · candidate





## 92. Finite Point Probability of a Repeated Value

The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -17), (1, -17), (2, 18), (3, 1015)}. In the displayed order of S, the nonnegative probability weights are [1/(22), 18/(22), 3/(22), 0/(22)], summing exactly to one. Find P(X=-17) exactly as a reduced fraction p/q with positive denominator.

View answer and explanation19/22

Add the weights of every outcome whose value equals -17, giving P(X=-17)=19/22. Zero-weight preimages add no mass; an unattained value has probability 0/1. Range cardinality is not a denominator.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r05-random-variable-0092/)

Real random variables on finite sample spaces · qf-random-variable-point-mass · candidate





## 93. Finite Point Probability of a Repeated Value

The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -18), (1, -18), (2, 19), (3, 1016)}. In the displayed order of S, the nonnegative probability weights are [1/(23), 19/(23), 3/(23), 0/(23)], summing exactly to one. Find P(X=19) exactly as a reduced fraction p/q with positive denominator.

View answer and explanation3/23

Add the weights of every outcome whose value equals 19, giving P(X=19)=3/23. Zero-weight preimages add no mass; an unattained value has probability 0/1. Range cardinality is not a denominator.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r05-random-variable-0093/)

Real random variables on finite sample spaces · qf-random-variable-point-mass · candidate





## 94. Finite Point Probability of a Repeated Value

The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -19), (1, -19), (2, 20), (3, 1017)}. In the displayed order of S, the nonnegative probability weights are [1/(24), 20/(24), 3/(24), 0/(24)], summing exactly to one. Find P(X=1017) exactly as a reduced fraction p/q with positive denominator.

View answer and explanation0/1

Add the weights of every outcome whose value equals 1017, giving P(X=1017)=0/1. Zero-weight preimages add no mass; an unattained value has probability 0/1. Range cardinality is not a denominator.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r05-random-variable-0094/)

Real random variables on finite sample spaces · qf-random-variable-point-mass · candidate





## 95. Finite Point Probability of a Repeated Value

The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -20), (1, -20), (2, 21), (3, 1018)}. In the displayed order of S, the nonnegative probability weights are [1/(25), 21/(25), 3/(25), 0/(25)], summing exactly to one. Find P(X=0) exactly as a reduced fraction p/q with positive denominator.

View answer and explanation0/1

Add the weights of every outcome whose value equals 0, giving P(X=0)=0/1. Zero-weight preimages add no mass; an unattained value has probability 0/1. Range cardinality is not a denominator.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r05-random-variable-0095/)

Real random variables on finite sample spaces · qf-random-variable-point-mass · candidate





## 96. Finite Point Probability of a Repeated Value

The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -21), (1, -21), (2, 22), (3, 1019)}. In the displayed order of S, the nonnegative probability weights are [1/(26), 22/(26), 3/(26), 0/(26)], summing exactly to one. Find P(X=-21) exactly as a reduced fraction p/q with positive denominator.

View answer and explanation23/26

Add the weights of every outcome whose value equals -21, giving P(X=-21)=23/26. Zero-weight preimages add no mass; an unattained value has probability 0/1. Range cardinality is not a denominator.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r05-random-variable-0096/)

Real random variables on finite sample spaces · qf-random-variable-point-mass · candidate





## 97. Finite Point Probability of a Repeated Value

The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -22), (1, -22), (2, 23), (3, 1020)}. In the displayed order of S, the nonnegative probability weights are [1/(27), 23/(27), 3/(27), 0/(27)], summing exactly to one. Find P(X=23) exactly as a reduced fraction p/q with positive denominator.

View answer and explanation1/9

Add the weights of every outcome whose value equals 23, giving P(X=23)=1/9. Zero-weight preimages add no mass; an unattained value has probability 0/1. Range cardinality is not a denominator.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r05-random-variable-0097/)

Real random variables on finite sample spaces · qf-random-variable-point-mass · candidate





## 98. Finite Point Probability of a Repeated Value

The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -23), (1, -23), (2, 24), (3, 1021)}. In the displayed order of S, the nonnegative probability weights are [1/(28), 24/(28), 3/(28), 0/(28)], summing exactly to one. Find P(X=1021) exactly as a reduced fraction p/q with positive denominator.

View answer and explanation0/1

Add the weights of every outcome whose value equals 1021, giving P(X=1021)=0/1. Zero-weight preimages add no mass; an unattained value has probability 0/1. Range cardinality is not a denominator.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r05-random-variable-0098/)

Real random variables on finite sample spaces · qf-random-variable-point-mass · candidate





## 99. Finite Point Probability of a Repeated Value

The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -24), (1, -24), (2, 25), (3, 1022)}. In the displayed order of S, the nonnegative probability weights are [1/(29), 25/(29), 3/(29), 0/(29)], summing exactly to one. Find P(X=0) exactly as a reduced fraction p/q with positive denominator.

View answer and explanation0/1

Add the weights of every outcome whose value equals 0, giving P(X=0)=0/1. Zero-weight preimages add no mass; an unattained value has probability 0/1. Range cardinality is not a denominator.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r05-random-variable-0099/)

Real random variables on finite sample spaces · qf-random-variable-point-mass · candidate





## 100. Finite Point Probability of a Repeated Value

The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -25), (1, -25), (2, 26), (3, 1023)}. In the displayed order of S, the nonnegative probability weights are [1/(30), 26/(30), 3/(30), 0/(30)], summing exactly to one. Find P(X=-25) exactly as a reduced fraction p/q with positive denominator.

View answer and explanation9/10

Add the weights of every outcome whose value equals -25, giving P(X=-25)=9/10. Zero-weight preimages add no mass; an unattained value has probability 0/1. Range cardinality is not a denominator.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r05-random-variable-0100/)

Real random variables on finite sample spaces · qf-random-variable-point-mass · candidate



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