PMF Negative Binomial Distribution 1 — Questions and Answers
Question 1: What does the Negative Binomial distribution model?
- The number of trials needed to achieve r successes (Correct answer)
- The number of successes in n trials
- The time until the first event
- The count of events per unit time
Correct answer: The number of trials needed to achieve r successes
The Negative Binomial distribution models the number of trials (or failures) until a specified number r of successes is achieved.
Question 2: What is the PMF of the Negative Binomial (number of failures before r-th success)?
- P(X=k) = C(r+k-1, k) × p^r × (1-p)^k (Correct answer)
- P(X=k) = C(n,k) × p^k × (1-p)^(n-k)
- P(X=k) = C(k-1,r-1) × p^r × (1-p)^(k-r)
- P(X=k) = (1-p)^(k-1) × p
Correct answer: P(X=k) = C(r+k-1, k) × p^r × (1-p)^k
P(X=k) = C(r+k-1, k) × p^r × (1-p)^k counts k failures before the r-th success.
Question 3: What is the mean of a Negative Binomial distribution (failures before r-th success)?
- r(1-p)/p (Correct answer)
- r/p
- rp
- r(1-p)
Correct answer: r(1-p)/p
E[X] = r(1-p)/p for the number of failures before the r-th success.
Question 4: Which special case of the Negative Binomial gives the Geometric distribution?
- r = 1 (Correct answer)
- r = 0
- p = 0.5
- r = n
Correct answer: r = 1
When r=1 (waiting for the first success), the Negative Binomial reduces to the Geometric distribution.
Question 5: An inspector needs to find 3 defective items. Each item has a 20% chance of being defective. What distribution models the number of items inspected until the 3rd defect?
- Negative Binomial with r=3, p=0.2 (Correct answer)
- Binomial with n=3, p=0.2
- Poisson with λ=0.6
- Geometric with p=0.2
Correct answer: Negative Binomial with r=3, p=0.2
Waiting for a specified number (r=3) of successes uses the Negative Binomial distribution.
Question 6: What is the variance of the Negative Binomial distribution (failures before r-th success)?
- r(1-p)/p² (Correct answer)
- r(1-p)/p
- r/p²
- rp(1-p)
Correct answer: r(1-p)/p²
Var(X) = r(1-p)/p² for the Negative Binomial distribution (failures before r-th success).
PMF Negative Binomial Distribution 1
What does the Negative Binomial distribution model?