PMF PMF Properties and Axioms 2 — Questions and Answers
Question 1: How is the CDF F(x) related to the PMF P(X=k) for discrete random variables?
- F(x) = Σ P(X=k) for all k ≤ x (Correct answer)
- F(x) = P(X=x)
- F(x) = dP(X=x)/dx
- F(x) = P(X > x)
Correct answer: F(x) = Σ P(X=k) for all k ≤ x
The CDF is the cumulative sum of the PMF: F(x) = P(X ≤ x) = Σ P(X=k) for all k ≤ x.
Question 2: For a PMF with values {1,2,3,4}, if F(3) = 0.8 and F(4) = 1, what is P(X=4)?
- 0.2 (Correct answer)
- 0.8
- 1.0
- 0.4
Correct answer: 0.2
P(X=4) = F(4) - F(3) = 1.0 - 0.8 = 0.2.
Question 3: What property of the CDF makes it easy to compute P(a < X ≤ b)?
- P(a < X ≤ b) = F(b) - F(a) (Correct answer)
- P(a < X ≤ b) = F(b) + F(a)
- P(a < X ≤ b) = 1 - F(b) + F(a)
- P(a < X ≤ b) = F(b) × F(a)
Correct answer: P(a < X ≤ b) = F(b) - F(a)
P(a < X ≤ b) = F(b) - F(a) is a direct consequence of CDF being a cumulative sum.
Question 4: A PMF assigns P(X=1)=0.2, P(X=2)=0.5, P(X=3)=0.3. What is F(2)?
- 0.7 (Correct answer)
- 0.5
- 0.2
- 1.0
Correct answer: 0.7
F(2) = P(X ≤ 2) = P(X=1) + P(X=2) = 0.2 + 0.5 = 0.7.
Question 5: For a discrete PMF, what type of function is the CDF?
- A right-continuous step function (Correct answer)
- A smooth continuous function
- A differentiable function
- A linear function
Correct answer: A right-continuous step function
The CDF of a discrete distribution is a step function that jumps at each mass point and is right-continuous.
Question 6: What does it mean for a PMF to be uniform on {1, 2, 3, 4, 5}?
- Each outcome has P(X=k) = 1/5 = 0.2 (Correct answer)
- Each outcome has probability proportional to k
- The CDF is linear everywhere
- The PMF values decrease uniformly
Correct answer: Each outcome has P(X=k) = 1/5 = 0.2
A uniform PMF assigns equal probability 1/n to each of the n outcomes; here, P(X=k) = 1/5 = 0.2 for each k.
PMF PMF Properties and Axioms 2
How is the CDF F(x) related to the PMF P(X=k) for discrete random variables?