Negative Binomial Distribution¶

Definition¶

Let $r$ be a positive integer and $p\in(0,1]$ $(0, 1]$ . A random variable $X$ has a negative binomial distribution if $X$ counts the number of trials needed to obtain the $r$ th success in independent Bernoulli trials with success probability $p$ .

Interpretation¶

In this trial-count convention, a negative binomial distribution generalizes the first success distribution. Instead of waiting for the first success, it waits for a fixed later success.

Probability bars for the trial on which the third success occurs with success probability 0.4, starting at trial 3, peaking at 5 and 6, with mean 7.5.

The diagram uses $r=3$ and $p=0.4$ $0.4$ . For $k\ge r$ $r$ , $\mathbb{P}(X=k)=\binom{k-1}{r-1}p^r(1-p)^{k-r}$ $p)^{k - r}$ : the first $k-1$ trials have $r-1$ successes, and trial $k$ is the $r$ th success.

Independent random variables describe the successive trial results. Their probabilities multiply for any fixed success-and-failure sequence, after which counting the sequences that place the final success at the specified trial gives the mass function.

Key Results¶

It counts waiting time until a fixed number of successes.
For $r=1$ $1$ , $X$ has the first success distribution. The shifted variable $X-1$ has the geometric distribution used in this course.
The failure-count convention uses $X-r$ instead of $X$ . For $k=r,r+1,\ldots$ $1, \dots$ , the trial-count probability is $\mathbb{P}(X=k)=\binom{k-1}{r-1}p^r(1-p)^{k-r}$ $p)^{k - r}$ ; the final trial is a success, and the preceding $k-1$ trials contain exactly $r-1$ successes.

Examples¶

The number of sales calls needed to close five sales can be negative binomial.
The number of coin tosses needed to obtain three heads follows a negative binomial model.
The number of inspected items until the tenth defective item can be modelled this way under independent constant defect probability.

Prerequisites¶

Sources¶

Probability Theory for EOR · Week 3

Aliases: Pascal distribution Keywords: NBin(r,p), negative binomial random variable Symbols: element of

Last updated: 20 October 2026 03:02