Finite Random Variable Foundations
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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 explanation
0
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
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
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 explanation
4
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
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 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
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
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
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 explanation
8
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
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
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
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
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 explanation
12
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
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
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
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
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 explanation
16
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
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 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
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
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
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 explanation
20
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
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
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
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
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 explanation
24
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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 explanation
1/2
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
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 explanation
3/7
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
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 explanation
0/1
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
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 explanation
0/1
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
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 explanation
7/10
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
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 explanation
3/11
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
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 explanation
0/1
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
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 explanation
0/1
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
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 explanation
11/14
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
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 explanation
1/5
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
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 explanation
0/1
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
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 explanation
0/1
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
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 explanation
5/6
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.
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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.
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3/19
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.
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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.
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0/1
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.
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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.
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0/1
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.
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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.
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19/22
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.
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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.
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3/23
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.
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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.
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0/1
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.
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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.
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0/1
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.
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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.
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23/26
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.
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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.
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1/9
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.
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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.
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0/1
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.
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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.
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0/1
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
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.
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9/10
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.
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Real random variables on finite sample spaces · qf-random-variable-point-mass · candidate
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