The Negative Binomial Distribution

Definition In a sequence of trials, a stopping rule decides after each trial whether to stop, using only the outcomes observed so far. A binomial experiment stops after a fixed number $n$ of trials whatever the outcomes, so the number of trials is fixed and the number of successes is random. The negative binomial experiment stops as soon as the number of successes reaches a fixed positive integer $r$, so the number of successes is fixed at $r$ and the number of trials is random.

Definition Let $r$ be a positive integer and $0 < p \le 1$. Trials are independent, each a success with probability $p$, and the waiting time for the $r$th success, $T$, is the number of trials up to and including the $r$th success. The event $\{T = k\}$ happens exactly when trial $k$ is a success and the first $k - 1$ trials contain exactly $r - 1$ successes. Choosing the positions of those earlier successes gives

for every integer $k \ge r$, and $P(T = x) = 0$ for every other real $x$. This is the negative binomial distribution with parameters $r$ and $p$ in the trial-count convention, and its support is $\{r, r + 1, r + 2, \dots\}$ when $p < 1$. For $r = 1$ the coefficient is $1$ and the formula is the geometric PMF. When $p = 1$ the formula, with the usual convention $(1 - p)^0 = 1$, gives $T = r$ with certainty, so the support is $\{r\}$. The value $p = 0$ is excluded, because then the target is reached with probability $0$.

Definition With the same $r$ and $p$, let $F$ be the number of failures before the $r$th success. Then $F = T - r$, its support is $\{0, 1, 2, \dots\}$ when $p < 1$ and $\{0\}$ when $p = 1$, and substituting $k = f + r$ in the trial-count formula gives

for every integer $f \ge 0$. Some references define the negative binomial distribution by this form. Shifting by the constant $r$ lowers the mean by $r$ and leaves the variance unchanged, so quote a probability or a mean together with its convention.

Definition Let $G_1$ be the number of trials up to and including the first success, and for $i = 2, \dots, r$ let $G_i$ be the number of further trials after the $(i - 1)$th success up to and including the $i$th. Then $T = G_1 + \dots + G_r$. For positive integers $g_1, \dots, g_r$, the event that $G_i = g_i$ for every $i$ fixes the outcome of each of the first $g_1 + \dots + g_r$ trials, so with $q = 1 - p$ its probability is

This product of geometric masses shows that $G_1, \dots, G_r$ are independent and each is geometric with parameter $p$. Expectations add, and variances of independent variables add, so

The failure count $F = T - r$ has $E[F] = r(1 - p)/p$ and the same variance.

Definition Let $T$ be the waiting time for the $r$th success, and for a positive integer $n$ let $X_n$ be the number of successes among the first $n$ trials of the same sequence, so $X_n \sim \operatorname{Bin}(n, p)$. The $r$th success has occurred by trial $n$ exactly when the first $n$ trials contain at least $r$ successes. Hence

For $n < r$ the sum is empty and the probability is $0$. The model is the whole sequence of independent trials, and stopping at the $r$th success only means that later trials are not needed. Both events depend only on the first $n$ trials. The complement gives $P(T > n) = P(X_n \le r - 1)$.