# About:Negative binomial distribution

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In probability theory and statistics, the negative binomial distribution is a discrete probability distribution that models the number of successes in a sequence of independent and identically distributed Bernoulli trials before a specified (non-random) number of failures (denoted r) occurs. For example, we can define rolling a 6 on a die as a failure, and rolling any other number as a success, and ask how many successful rolls will occur before we see the third failure (r = 3). In such a case, the probability distribution of the number of non-6s that appear will be a negative binomial distribution. We could similarly use the negative binomial distribution to model the number of days a certain machine works before it breaks down (r = 1).

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• In probability theory and statistics, the negative binomial distribution is a discrete probability distribution that models the number of successes in a sequence of independent and identically distributed Bernoulli trials before a specified (non-random) number of failures (denoted r) occurs. For example, we can define rolling a 6 on a die as a failure, and rolling any other number as a success, and ask how many successful rolls will occur before we see the third failure (r = 3). In such a case, the probability distribution of the number of non-6s that appear will be a negative binomial distribution. We could similarly use the negative binomial distribution to model the number of days a certain machine works before it breaks down (r = 1). "Success" and "failure" are arbitrary terms that are sometimes swapped. We could just as easily say that the negative binomial distribution is the distribution of the number of failures before r successes. When applied to real-world problems, outcomes of success and failure may or may not be outcomes we ordinarily view as good and bad, respectively. This article is inconsistent in its use of these terms, so the reader should be careful to identify which outcome can vary in number of occurrences and which outcome stops the sequence of trials. The article may also use p (the probability of one of the outcomes in any given Bernoulli trial) inconsistently. The Pascal distribution (after Blaise Pascal) and Polya distribution (for George Pólya) are special cases of the negative binomial distribution. A convention among engineers, climatologists, and others is to use "negative binomial" or "Pascal" for the case of an integer-valued stopping-time parameter r, and use "Polya" for the real-valued case. For occurrences of associated discrete events, like tornado outbreaks, the Polya distributions can be used to give more accurate models than the Poisson distribution by allowing the mean and variance to be different, unlike the Poisson. The negative binomial distribution has a variance , with the distribution becoming identical to Poisson in the limit for a given mean . This can make the distribution a useful overdispersed alternative to the Poisson distribution, for example for a robust modification of Poisson regression. In epidemiology it has been used to model disease transmission for infectious diseases where the likely number of onward infections may vary considerably from individual to individual and from setting to setting. More generally it may be appropriate where events have positively correlated occurrences causing a larger variance than if the occurrences were independent, due to a positive covariance term. The term "negative binomial" is likely due to the fact that a certain binomial coefficient that appears in the formula for the probability mass function of the distribution can be written more simply with negative numbers. (en)
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• 45177 (xsd:integer)
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• May 2021 (en)
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• The chosen parameterization throughout the article is inconsistent and therefore makes the article unnecessarily confusing. Compare this with the definition of the PMF given in the box at the top of the page. (en)
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• In probability theory and statistics, the negative binomial distribution is a discrete probability distribution that models the number of successes in a sequence of independent and identically distributed Bernoulli trials before a specified (non-random) number of failures (denoted r) occurs. For example, we can define rolling a 6 on a die as a failure, and rolling any other number as a success, and ask how many successful rolls will occur before we see the third failure (r = 3). In such a case, the probability distribution of the number of non-6s that appear will be a negative binomial distribution. We could similarly use the negative binomial distribution to model the number of days a certain machine works before it breaks down (r = 1). (en)
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• Negative binomial distribution (en)
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