Modeling With the Negative Binomial Distribution

Use the negative binomial distribution to model real-world scenarios involving repeated independent Bernoulli trials. Compute single-value probabilities, range probabilities, and expected value and variance.

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Tutorial

Setting Up a Negative Binomial Model

The negative binomial distribution models $X,$ the number of independent Bernoulli trials needed to obtain the $r$ -th success, where each trial succeeds with probability $p.$ Its PMF is

P(X = k) = \binom{k-1}{r-1}\, p^r (1-p)^{k-r}, \qquad k = r,\, r+1,\, r+2, \ldots

To model a real-world scenario with the negative binomial, we identify three quantities:

The Bernoulli trial and its success event.
The success probability $p.$
The number of successes $r$ being awaited.

For instance, suppose a dart player hits the bullseye on each throw independently with probability $p = \dfrac{1}{4}.$ Let $X$ be the number of throws needed to land the $3$ rd bullseye. Then $X$ is negative binomial with $r = 3$ and $p = \dfrac{1}{4}.$ The probability that the $3$ rd bullseye occurs on the $6$ th throw is

\begin{align*} P(X = 6) &= \binom{5}{2}\left(\dfrac{1}{4}\right)^{\!3}\left(\dfrac{3}{4}\right)^{\!3} \\[3pt] &= 10 \cdot \dfrac{1}{64} \cdot \dfrac{27}{64} \\[3pt] &= \dfrac{270}{4096} = \dfrac{135}{2048}. \end{align*}