Negative Binomial

In contrast to the Binomial Experiment, a negative binomial experiment is the number of $S$ 's desired to be fixed in advance and the required number of trials is random. More precisely, if the following are satisfied:

The experiment consists of independent trials
Each trial is a $S$ success or fail $F$
The probability of success is constant from trial to trial, so $P (S on trial i) = p$ for $i = 1, 2, 3, . . .$
The experiment continues until a total of $r$ successes has been observed, where $r$ is a specified positive integer.

The random variable of interest is $X =$ the number of trials to achieve the $r$ -th success, and $X$ is the negative binomial random variable. $X = r, r + 1, . . .$

Let $nb (x; r, p)$ denote the pmf of $X$ . Then:

nb (x; r, p) = P (X = x) = P (r - 1 S ’s on the first x - 1 trials) \cdot P (S)

The first probability on the far right is just $p$ , and the left probability is just the binomial probability. Thus:

nb (x; r, p) = (\binom{x - 1}{r - 1}) p^{r - 1} (1 - p)^{(x - 1) - (r - 1)}

Proposition

The pmf of the negative binomial rv $X$ with parameters $r =$ desired number of $S$ 's and $p = P (S)$ is:

nb (x; r, p) = (\binom{x - 1}{r - 1}) p^{r} (1 - p)^{x - r}

where $x = r, r + 1, \dots$

Mean, Variance, and Moment Generating Function

Proposition

If $X$ is a negative binomial rv with parameters $r$ and $p$ , then:

E (X) = \frac{r}{p}, Var (X) = \frac{r (1 - p)}{p^{2}}