PMF Negative Binomial Distribution

Question 1

What does the Negative Binomial distribution model?

Accepted Answer

The number of trials needed to achieve r successes

Answer

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)?

Accepted Answer

P(X=k) = C(r+k-1, k) × p^r × (1-p)^k

Answer

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)?

Accepted Answer

r(1-p)/p

Answer

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?

Accepted Answer

r = 1

Answer

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?

Accepted Answer

Negative Binomial with r=3, p=0.2

Answer

Waiting for a specified number (r=3) of successes uses the Negative Binomial distribution.

PMF - Probability Mass Function Practice Test

PMF - Probability Mass Function Practice Test

PMF Negative Binomial Distribution