# Negative Binomial Mass Function

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A Negative Binomial Mass Function is a discrete probability function where *p* is the probability of a success in a sequence independent Bernoulli trials before *r* failures.

**AKA:**Negative Binomial Distribution, [math]\displaystyle{ NB(r,p) }[/math].- …

**Counter-Example(s):****See:**Binomial Mass Function, Positive Binomial Mass Function.

## References

### 2011

- (Wikipedia, 2011) ⇒ http://en.wikipedia.org/wiki/Negative_binomial_distribution
- QUOTE: In probability theory and statistics, the
**negative binomial distribution**is a discrete probability distribution of the number of successes in a sequence of Bernoulli trials before a specified (non-random) number [math]\displaystyle{ r }[/math] of failures occurs. For example, if one throws a die repeatedly until the third time “1” appears, then the probability distribution of the number of non-“1”s that had appeared will be negative binomial. The Pascal distribution (after Blaise Pascal) and**Polya distribution**(for George Pólya) 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” for the case of an integer-valued stopping-time parameter [math]\displaystyle{ r }[/math], and use “Polya” for the real-valued case. The Polya distribution more accurately models occurrences of “contagious” discrete events, like tornado outbreaks, than the Poisson distribution. - QUOTE: Suppose there is a sequence of independent Bernoulli trials, each trial having two potential outcomes called “success” and “failure”. In each trial the probability of success is [math]\displaystyle{ p }[/math] and of failure is (1 −
*p*). We are observing this sequence until a predefined number [math]\displaystyle{ r }[/math] of failures has occurred. Then the random number of successes we have seen, [math]\displaystyle{ X }[/math], will have the**negative binomial**(or Pascal) distribution: [math]\displaystyle{ X\ \sim\ \text{NB}(r,\, p) }[/math]

- QUOTE: In probability theory and statistics, the

### 2006

- (Dubnicka, 2006f) ⇒ Suzanne R. Dubnicka. (2006). “Special Discrete Distributions - Handout 6." Kansas State University, Introduction to Probability and Statistics I, STAT 510 - Fall 2006.
- TERMINOLOGY : Imagine an experiment where Bernoulli trials are observed. If X denotes the trial on which the r
^{th}success occurs, r 1, then X has a negative binomial distribution with parameters r and p, where p denotes the probability of success on any one trial, 0 < p < 1. This is sometimes written as X ~ NB(r, p)

- TERMINOLOGY : Imagine an experiment where Bernoulli trials are observed. If X denotes the trial on which the r

### 1941

- (Fisher, 1941) ⇒ R. A. Fisher. (1941). “[http://digital.library.adelaide.edu.au/dspace/bitstream/2440/15242/1/182.pdf The negative binomial distribution.” In: Annals of Human Genetics, 1.