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Negative Binomial and Hypergeometric

These two distributions each fix a different assumption of the binomial. The binomial fixes the number of trials and counts successes; the negative binomial fixes the successes and counts trials. The binomial assumes constant pp; the hypergeometric handles the case where each draw changes the odds.

Negative binomial: waiting for the rr-th success

The probability that the rr-th success arrives on trial kk:

The negative binomial
P(X=k)=(k−1r−1)pr(1−p)k−r,k≥rP(X = k) = \binom{k-1}{r-1} p^r (1-p)^{k-r}, \qquad k \geq r

How long you wait for the rth success, rather than how many successes arrive in n trials.

The (k−1r−1)\binom{k-1}{r-1} counts where the earlier r−1r-1 successes fell among the first k−1k-1 trials. The final trial is pinned to be a success, which is what distinguishes this from a binomial coefficient over all kk trials.

E[X]=rpVar(X)=r(1−p)p2E[X] = \frac{r}{p} \qquad \text{Var}(X) = \frac{r(1-p)}{p^2}

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