- Parameters :
`r`(number of successes) is an integer where 1 ≤`r`; the special case`r`= 1 creates the geometric distribution. -
`p`= probability of success on each trial is a real number where 0 <`p`< 1. - Support (domain where probability mass > 0) = set of all integers ≥
*r*. - Probability mass function
*f*(*x*) = P(*X*=*x*) = probability that*r*th success occurs on the*x*th trial- = C(
*x*− 1,*r*− 1)*p*^{r}(1 −*p*)^{x − r}(see binomial coefficient).

- = C(
- Cumulative distribution function
*F*(*x*) = P(*X*≤*x*) = probability that*r*th success occurs on or before the*x*th trial : No simple closed form solution exists, but this can be computed via the regularized incomplete Beta function as with the binomial distribution. - Expected value E[
`X`] =*r*/*p*. - Variance var(
*X*) = σ^{2}=*r*(1 −*p*)/*p*^{2}.

*(After a problem by Dr. Diane Evans, professor of mathematics at **Rose-Hulman Institute of Technology)*

Johnny, a sixth grader at Honey Creek Middle School in Terre Haute, Indiana, is required to sell candy bars in his neighborhood to raise money for the 6th grade field trip. There are thirty homes in his neighborhood, and his father has told him not to return home until he has sold five candy bars. So the boy goes door to door, selling candy bars. At each home he visits, he has an 0.4 probability of selling one candy bar and an 0.6 probability of selling nothing.

*f*(10) = 0.100

Answer: To finish on or before the eighth house, he must finish at the fifth, sixth, seventh, or eighth house. Sum those probabilities:

- f(5) = 0.0102; f(6) = .0307, f(7) = .0553; f(8) = .0774; sum(f(j), j=5..8) =
**0.1737**

If *X _{r}* is a random variable following the negative binomial distribution with parameters

Furthermore, if *Y _{s}* is a random variable following the binomial distribution with parameters

- Pr[
*X*_{r}≤*s*] = Pr[*Y*_{s}≥*r*] = Pr["after*s*trials, there are at least*r*successes"]

The negative binomial distribution also arises as a continuous mixture of Poisson distributions for which the Poisson parameter `λ` was generated by a Gamma distribution.

Suppose *X* is a random variable with a negative binomial distribution with parameters *r* and *p*.
The statement that the sum from `x` = `r` to infinity, of the probability Pr[*X* = *x*], is equal to 1, can be shown by a bit of algebra to be equivalent to the statement that (1 − *p*)^{− r} is what Newton's binomial theorem says it should be.

Suppose *Y* is a random variable with a binomial distribution with parameters *n* and *p*.
The statement that the sum from *y* = 0 to *n*, of the probability Pr[*Y* = *y*], is equal to 1, says that that 1 = (*p* + (1 − *p*))^{n} is what the strictly finitary binomial theorem of high-school algebra says it should be.

Thus the negative binomial distribution bears the same relationship to the negative-integer-exponent case of the binomial theorem that the binomial distribution bears to the positive-integer-exponent case.