# 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.

## 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 $\displaystyle{ r }$ 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 $\displaystyle{ r }$, 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 $\displaystyle{ p }$ and of failure is (1 − p). We are observing this sequence until a predefined number $\displaystyle{ r }$ of failures has occurred. Then the random number of successes we have seen, $\displaystyle{ X }$, will have the negative binomial (or Pascal) distribution: $\displaystyle{ X\ \sim\ \text{NB}(r,\, p) }$