Negative Binomial and Hypergeometric

Waiting for k successes, and sampling without replacement.

Two more discrete distributions for problems where the simple Bernoulli/Binomial model isn't quite right.

The Negative Binomial $(r, p)$ counts the number of trials until the $r$ -th success in independent Bernoulli $(p)$ trials:

P(X = k) = \binom{k-1}{r-1} p^r (1-p)^{k-r}, \quad k = r, r+1, \dots

with $E[X] = r/p$ . It generalizes the Geometric, which is the $r = 1$ case. Useful whenever you care about the time to the $r$ -th event rather than just the first.

The Hypergeometric models drawing $n$ items without replacement from a population of $N$ containing $K$ successes:

P(X = k) = \frac{\binom{K}{k}\binom{N-K}{n-k}}{\binom{N}{n}}

Without replacement matters when the population is small. As $N \to \infty$ with $K/N$ fixed, the Hypergeometric converges to the Binomial $(n, K/N)$ — large populations look effectively independent.

Use Hypergeometric for card hands, ball-and-urn problems, audit sampling, and survey sampling from finite populations.