PMF Negative Binomial Distribution 2 — Questions and Answers
Question 1: For a Negative Binomial with r=2 and p=0.5, what is E[X] (failures before 2nd success)?
- 2 (Correct answer)
- 4
- 1
- 0.5
Correct answer: 2
E[X] = r(1-p)/p = 2 × 0.5/0.5 = 2 failures expected before the 2nd success.
Question 2: How does the Negative Binomial PMF relate to the Binomial PMF?
- They count different things: Binomial counts successes in fixed trials; Negative Binomial counts trials to fixed successes (Correct answer)
- They are the same distribution
- Negative Binomial is the inverse of Binomial
- Binomial is a special case of Negative Binomial
Correct answer: They count different things: Binomial counts successes in fixed trials; Negative Binomial counts trials to fixed successes
The Binomial fixes n trials and counts k successes, while the Negative Binomial fixes r successes and counts the number of trials needed.
Question 3: For the Negative Binomial (total trials until r-th success), what is the PMF?
- P(X=k) = C(k-1, r-1) × p^r × (1-p)^(k-r) (Correct answer)
- P(X=k) = C(r+k-1, k) × p^r × (1-p)^k
- P(X=k) = C(k,r) × p^r × (1-p)^(k-r)
- P(X=k) = (1-p)^(k-r) × p^r
Correct answer: P(X=k) = C(k-1, r-1) × p^r × (1-p)^(k-r)
P(X=k) = C(k-1, r-1) × p^r × (1-p)^(k-r) for total trials needed, where the last trial is always the r-th success.
Question 4: A researcher needs 5 volunteer sign-ups. Each person contacted has a 30% chance of volunteering. What is E[total people contacted]?
- 5/0.3 ≈ 16.67 (Correct answer)
- 5 × 0.3 = 1.5
- 5 × 0.7 = 3.5
- 0.7/0.3 × 5 ≈ 11.67
Correct answer: 5/0.3 ≈ 16.67
For total trials until r successes, E[X] = r/p = 5/0.3 ≈ 16.67 people need to be contacted.
Question 5: What is the support of the Negative Binomial (total trials until r-th success)?
- {r, r+1, r+2, …} (Correct answer)
- {0, 1, 2, …}
- {1, 2, …, r}
- {r, r+1, …, n}
Correct answer: {r, r+1, r+2, …}
The minimum number of trials to get r successes is r (all trials are successes), so the support starts at k = r.
Question 6: If X counts failures before the r-th success and Y counts total trials, how are X and Y related?
- Y = X + r (Correct answer)
- Y = X
- Y = r × X
- Y = X - r
Correct answer: Y = X + r
Total trials Y = failures X + successes r, so Y = X + r.
PMF Negative Binomial Distribution 2
For a Negative Binomial with r=2 and p=0.5, what is E[X] (failures before 2nd success)?