Editing Negative binomial distribution (section)

{{short description|Probability distribution}}
{{Negative binomial distribution}}

In [[probability theory]] and [[statistics]], the '''negative binomial distribution''' is a [[discrete probability distribution]] that models the number of failures in a sequence of independent and identically distributed [[Bernoulli trial]]s before a specified/constant/fixed number of successes <math>r</math> occur.<ref name="Wolfram">{{cite web |last1=Weisstein |first1=Eric |title=Negative Binomial Distribution |url=https://mathworld.wolfram.com/NegativeBinomialDistribution.html |website=Wolfram MathWorld |publisher=Wolfram Research |access-date=11 October 2020}}</ref> For example, we can define rolling a 6 on some dice as a success, and rolling any other number as a failure, and ask how many failure rolls will occur before we see the third success (<math>r=3</math>). In such a case, the probability distribution of the number of failures that appear will be a negative binomial distribution.

An alternative formulation is to model the number of total trials (instead of the number of failures). In fact, for a specified (non-random) number of successes {{math|(''r'')}}, the number of failures {{math|(''n'' − ''r'')}} is random because the number of total trials {{math|(''n'')}} is random. For example, we could use the negative binomial distribution to model the number of days {{mvar|n}} (random) a certain machine works (specified by {{mvar|r}}) before it breaks down.

The negative binomial distribution has a variance <math>\mu /p</math>, with the distribution becoming identical to Poisson in the limit <math>p\to 1</math> for a given mean <math>\mu</math> (i.e. when the failures are increasingly rare). Here <math>p\in [0,1]</math> is the success probability of each Bernoulli trial. This can make the distribution a useful [[overdispersion|overdispersed]] alternative to the Poisson distribution, for example for a [[robust regression|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.<ref>e.g. {{cite journal|journal=[[Nature (journal)|Nature]] |doi=10.1038/nature04153 |title=Superspreading and the effect of individual variation on disease emergence |date=2005 |last1=Lloyd-Smith |first1=J. O. |last2=Schreiber |first2=S. J. |last3=Kopp |first3=P. E. |last4=Getz |first4=W. M. |volume=438 |issue=7066 |pages=355–359 |doi-access=free |pmid=16292310 |pmc=7094981 |bibcode=2005Natur.438..355L }}<br />The overdispersion parameter is usually denoted by the letter <math>k</math> in epidemiology, rather than <math>r</math> as here.</ref> 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.<ref>{{cite book |last1=Casella |first1=George |last2=Berger |first2=Roger L. |title=Statistical inference |year=2002 |url=https://archive.org/details/statisticalinfer00case_045|url-access=limited |publisher=Thomson Learning |isbn=0-534-24312-6 |page=[https://archive.org/details/statisticalinfer00case_045/page/n59 95] |edition=2nd}}</ref>