PMF PMF Properties and Axioms 1 — Questions and Answers
Question 1: Which two conditions must every valid PMF satisfy?
- P(X=x) ≥ 0 for all x, and Σ P(X=x) = 1 (Correct answer)
- P(X=x) > 0 for all x, and Σ P(X=x) = 1
- P(X=x) ≥ 0 for all x, and Σ P(X=x) ≤ 1
- 0 < P(X=x) < 1 for all x
Correct answer: P(X=x) ≥ 0 for all x, and Σ P(X=x) = 1
A valid PMF requires all probabilities to be non-negative and all probabilities to sum to exactly 1.
Question 2: Which of the following is NOT a valid PMF for X ∈ {1, 2, 3}?
- P(1)=0.4, P(2)=0.4, P(3)=0.3 (Correct answer)
- P(1)=0.5, P(2)=0.3, P(3)=0.2
- P(1)=1, P(2)=0, P(3)=0
- P(1)=0.1, P(2)=0.4, P(3)=0.5
Correct answer: P(1)=0.4, P(2)=0.4, P(3)=0.3
P(1)+P(2)+P(3) = 0.4+0.4+0.3 = 1.1 ≠ 1, so this is not a valid PMF.
Question 3: What is the key difference between a PMF and a PDF (probability density function)?
- PMF applies to discrete random variables; PDF applies to continuous random variables (Correct answer)
- PMF integrates to 1; PDF sums to 1
- PMF values can exceed 1; PDF values cannot
- PMF and PDF are identical
Correct answer: PMF applies to discrete random variables; PDF applies to continuous random variables
PMF gives exact probabilities for discrete outcomes; PDF gives probability density for continuous variables (probabilities require integration).
Question 4: For a valid PMF, can P(X = x₀) = 0 for some specific value x₀ in the support?
- Yes, a PMF can assign zero probability to values in its defined domain (Correct answer)
- No, all defined values must have positive probability
- Only if x₀ is not in the support
- Only if there are infinitely many values
Correct answer: Yes, a PMF can assign zero probability to values in its defined domain
P(X=x) ≥ 0 allows values of zero; only values in the support (where P > 0) are typically listed.
Question 5: What is the support of a PMF?
- The set of all values x where P(X=x) > 0 (Correct answer)
- The set of all real numbers
- The range from min to max value
- The set of all values where P(X=x) = 1
Correct answer: The set of all values x where P(X=x) > 0
The support is the set of values that have positive probability under the PMF.
Question 6: If P(X = 2) = 0.3, what is P(X ≠ 2)?
- 0.7 (Correct answer)
- 0.3
- 1.3
- 0
Correct answer: 0.7
P(X ≠ 2) = 1 - P(X = 2) = 1 - 0.3 = 0.7 by the complement rule.
PMF PMF Properties and Axioms 1
Which two conditions must every valid PMF satisfy?