In probability and statistics the negative binomial distribution is a discrete probability distribution. The Pascal distribution and the Polya distribution are special cases of the negative binomial. There is a convention among engineers, climatologists, and others to reserve “negative binomial” in a strict sense or “Pascal” (after Blaise Pascal) for the case of an integer-valued parameter r, and use “Polya” (for George Pólya) for the real-valued case, to the right. The Polya distribution more accurately models occurrences of “contagious” discrete events, like tornado outbreaks, than does the Poisson distribution.

The family of negative binomial distributions is a two-parameter family; several parametrizations are in common use. One very common parameterization employs two real-valued parameters p and r with 0 p 1 and r > 0. Under this parameterization, the probability mass function of a random variable with a NegBin(r, p) distribution takes the following form:

f(x,r,p) = (x+r-1) “over” (x) * p^r(1-p)^x

support x = 0,1,2,3,…