Negative binomial distribution

Definition

Let $r \ge 1$ be an integer and let $p \in (0, 1]$. The negative binomial distribution with parameters $r$ and $p$ is the distribution on $\{0, 1, 2, \dots\}$ with the probability mass function $$p(k) = \binom{k + r - 1}{k} p^r (1 - p)^k \quad \text{for } k = 0, 1, 2, \dots$$

Remarks

It is the distribution of the number of failures before the $r$-th success in independent Bernoulli trials with success probability $p$. The last trial is a success, and the $k$ failures are among the other $k + r - 1$ trials. For $r = 1$ it is the geometric distribution. The sum of $r$ independent geometric random variables with parameter $p$ has the negative binomial distribution with parameters $r$ and $p$. The mean is $r(1 - p)/p$ and the variance is $r(1 - p)/p^2$. Some authors count the trials, or the successes before a fixed number of failures. Check the convention of each source.

Explanations

L1 human

Toss a coin until you have seen $r$ heads, and count the tails on the way. This count has the negative binomial distribution. For $r = 1$ you wait only for the first head.