Negative Binomial Distribution
Overview
- The Negative Binomial distribution models the number of successes in a sequence of independent Bernoulli trials before a specified number of failures occurs.
- Unlike the Binomial distribution, which focuses on the number of trials until a fixed number of successes, the Negative Binomial distribution focuses on the number of successes until a fixed number of failures.
Use Case
- Used when you want to model the number of successes (e.g., number of goals scored, number of defective items produced) before observing a certain number of failures (e.g., strikes in baseball before three outs, defective products in a production run).
- Probability Mass Function (PMF):
- : Number of successes
- : Number of failures
- : Probability of success in each trial
Example
- Baseball Game:
- Suppose a baseball team needs to score 3 runs () before getting 2 outs to win the game.
- Let ( p = 0.3 ) be the probability of scoring a run in each at-bat.
- We want to find the probability of scoring exactly 3 runs before getting 2 outs.
- Using the Negative Binomial PMF:
