Editing Negative binomial distribution (section)

==Occurrence and applications==

===Waiting time in a Bernoulli process===

Let {{mvar|k}} and {{mvar|r}} be integers with {{mvar|k}} non-negative and {{mvar|r}} positive.  In a sequence of independent [[Bernoulli trial]]s with success probability {{mvar|p}}, the negative binomial gives the probability of {{mvar|k}} successes and {{mvar|r}} failures, with a failure on the last trial. Therefore, the negative binomial distribution represents the probability distribution of the number of successes before the {{mvar|r}}-th failure in a [[Bernoulli process]], with probability {{mvar|p}} of successes on each trial.

Consider the following example. Suppose we repeatedly throw a die, and consider a 1 to be a failure.  The probability of success on each trial is 5/6. The number of successes before the third failure belongs to the infinite set {{math|{{mset| 0, 1, 2, 3, ... }}}}. That number of successes is a negative-binomially distributed random variable.

When {{math|1=''r'' = 1}} we get the probability distribution of number of successes before the first failure (i.e. the probability of the first failure occurring on the {{math|(''k'' + 1)}}-st trial), which is a [[geometric distribution]]:
: <math>
    f(k; r, p) = (1-p) \cdot p^k \!
  </math>

===Overdispersed Poisson===

The negative binomial distribution, especially in its alternative parameterization described above, can be used as an alternative to the Poisson distribution.  It is especially useful for discrete data over an unbounded positive range whose sample [[variance]] exceeds the sample [[mean]]. In such cases, the observations are [[Overdispersion|overdispersed]] with respect to a Poisson distribution, for which the mean is equal to the variance. Hence a Poisson distribution is not an appropriate model.  Since the negative binomial distribution has one more parameter than the Poisson, the second parameter can be used to adjust the variance independently of the mean. See [[Cumulant#Cumulants of some discrete probability distributions|Cumulants of some discrete probability distributions]].

An application of this is to annual counts of [[tropical cyclone]]s in the [[Atlantic Ocean|North Atlantic]] or to monthly to 6-monthly counts of wintertime [[extratropical cyclone]]s over Europe, for which the variance is greater than the mean.<ref>{{cite journal|last=Villarini |first=G. |author2=Vecchi, G.A. |author3=Smith, J.A.|year=2010 |title=Modeling of the dependence of tropical storm counts in the North Atlantic Basin on climate indices |journal=[[Monthly Weather Review]] |volume=138 |issue=7 |pages=2681–2705 |doi=10.1175/2010MWR3315.1  |bibcode=2010MWRv..138.2681V |doi-access=free }}</ref><ref>{{cite journal|last=Mailier |first=P.J. |author2=Stephenson, D.B. |author3=Ferro, C.A.T. |author4= Hodges, K.I. |year=2006 |title=Serial Clustering of Extratropical Cyclones |journal=[[Monthly Weather Review]] |volume=134 |issue=8 |pages=2224–2240 |doi=10.1175/MWR3160.1 |bibcode=2006MWRv..134.2224M |doi-access=free }}</ref><ref>{{cite journal|last=Vitolo |first=R. |author2=Stephenson, D.B. |author3=Cook, Ian M. |author4= Mitchell-Wallace, K. |year=2009 |title=Serial clustering of intense European storms |journal=[[Meteorologische Zeitschrift]] |volume=18 |issue=4 |pages=411–424 |doi=10.1127/0941-2948/2009/0393 |bibcode=2009MetZe..18..411V |s2cid=67845213 }}</ref>  In the case of modest overdispersion, this may produce substantially similar results to an overdispersed Poisson distribution.<ref>{{cite book  | last = McCullagh | first = Peter | author-link= Peter McCullagh |author2=Nelder, John |author-link2=John Nelder  | title = Generalized Linear Models |edition=Second | publisher = Boca Raton: Chapman and Hall/CRC | year = 1989 | isbn = 978-0-412-31760-6 |ref=McCullagh1989}}</ref><ref>{{cite book | last = Cameron | first = Adrian C. | author2 = Trivedi, Pravin K. | title = Regression analysis of count data | publisher = Cambridge University Press | year = 1998 | isbn = 978-0-521-63567-7 | ref = Cameron1998 | url-access = registration | url = https://archive.org/details/regressionanalys00came }}</ref>

Negative binomial modeling is widely employed in ecology and biodiversity research for analyzing count data where overdispersion is very common. This is because overdispersion is indicative of biological aggregation, such as species or communities forming clusters. Ignoring overdispersion can lead to significantly inflated model parameters, resulting in misleading statistical inferences. The negative binomial distribution effectively addresses overdispersed counts by permitting the variance to vary quadratically with the mean. An additional dispersion parameter governs the slope of the quadratic term, determining the severity of overdispersion. The model's quadratic mean-variance relationship proves to be a realistic approach for handling overdispersion, as supported by empirical evidence from many studies. Overall, the NB model offers two attractive features: (1) the convenient interpretation of the dispersion parameter as an index of clustering or aggregation, and (2) its tractable form, featuring a closed expression for the probability mass function.<ref>
{{cite journal|last=Stoklosa |first=J. |author2=Blakey, R.V. |author3=Hui, F.K.C. |year=2022 |title=An Overview of Modern Applications of Negative Binomial Modelling in Ecology and Biodiversity |journal=[[Diversity (journal)|Diversity]] |volume=14 |issue=5 |pages=320 |doi=10.3390/d14050320 |doi-access=free |bibcode=2022Diver..14..320S }}
</ref>

In genetics, the negative binomial distribution is commonly used to model data in the form of discrete sequence read counts from high-throughput RNA and DNA sequencing experiments.<ref>
{{cite journal|last=Robinson |first=M.D. |author2=Smyth, G.K. |year=2007 |title=Moderated statistical tests for assessing differences in tag abundance. |journal=[[Bioinformatics]] |volume=23 |issue=21 |pages=2881–2887 |doi=10.1093/bioinformatics/btm453 |pmid=17881408|doi-access=free }}
</ref><ref>
{{cite web |url=http://www.bioconductor.org/packages/release/bioc/vignettes/DESeq2/inst/doc/DESeq2.pdf |title=Differential analysis of count data – the}}</ref><ref>
{{cite conference|last=Airoldi |first=E. M. |author2=Cohen, W. W. |author3=Fienberg, S. E. |date=June 2005 |title=Bayesian Models for Frequent Terms in Text |book-title=Proceedings of the Classification Society of North America and INTERFACE Annual Meetings |volume=990 |pages=991 |location=St. Louis, MO, USA }}
</ref><ref>
{{cite web |url=http://www.bioconductor.org/packages/release/bioc/vignettes/edgeR/inst/doc/edgeRUsersGuide.pdf |title=edgeR: differential expression analysis of digital gene expression data |last1=Chen |first1=Yunshun |last2=Davis |first2=McCarthy |date=September 25, 2014 |access-date=October 14, 2014}}</ref>

In epidemiology of infectious diseases, the negative binomial has been used as a better option than the Poisson distribution to model overdispersed counts of secondary infections from one infected case (super-spreading events).<ref>{{cite journal|last=Lloyd-Smith|first=J. O. |author2= Schreiber, S. J. |author3= Kopp, P. E. |author4= Getz, W. M. |year=2005 |title=Superspreading and the effect of individual variation on disease emergence |journal=[[Nature (journal)|Nature]] |volume=438 |issue=7066 |pages=355–359 |doi=10.1038/nature04153|pmid=16292310 |pmc=7094981 |bibcode=2005Natur.438..355L }}</ref>

===Multiplicity observations (physics)===

The negative binomial distribution has been the most effective statistical model for a broad range of multiplicity observations in [[particle collision]] experiments, e.g., <math>p\bar p,\ hh,\ hA,\ AA,\ e^{+}e^-</math> <ref>{{Cite journal |last1=Grosse-Oetringhaus |first1=Jan Fiete |last2=Reygers |first2=Klaus |date=2010-08-01 |title=Charged-particle multiplicity in proton–proton collisions |url=https://iopscience.iop.org/article/10.1088/0954-3899/37/8/083001 |journal=Journal of Physics G: Nuclear and Particle Physics |volume=37 |issue=8 |pages=083001 |doi=10.1088/0954-3899/37/8/083001 |issn=0954-3899|arxiv=0912.0023 |s2cid=119233810 }}</ref><ref>{{Cite journal |last1=Rybczyński |first1=Maciej |last2=Wilk |first2=Grzegorz |last3=Włodarczyk |first3=Zbigniew |date=2019-05-31 |title=Intriguing properties of multiplicity distributions |journal=Physical Review D |language=en |volume=99 |issue=9 |page=094045 |doi=10.1103/PhysRevD.99.094045 |arxiv=1811.07197 |bibcode=2019PhRvD..99i4045R |issn=2470-0010|doi-access=free }}</ref><ref>{{Cite journal |last1=Tarnowsky |first1=Terence J. |last2=Westfall |first2=Gary D. |date=2013-07-09 |title=First study of the negative binomial distribution applied to higher moments of net-charge and net-proton multiplicity distributions |journal=Physics Letters B |volume=724 |issue=1 |pages=51–55 |doi=10.1016/j.physletb.2013.05.064 |arxiv=1210.8102 |bibcode=2013PhLB..724...51T |issn=0370-2693|doi-access=free }}</ref><ref>{{Cite journal |last1=Derrick |first1=M. |last2=Gan |first2=K. K. |last3=Kooijman |first3=P. |last4=Loos |first4=J. S. |last5=Musgrave |first5=B. |last6=Price |first6=L. E. |last7=Repond |first7=J. |last8=Schlereth |first8=J. |last9=Sugano |first9=K. |last10=Weiss |first10=J. M. |last11=Wood |first11=D. E. |last12=Baranko |first12=G. |last13=Blockus |first13=D. |last14=Brabson |first14=B. |last15=Brom |first15=J. M. |date=1986-12-01 |title=<nowiki>Study of quark fragmentation in ${e}^{+}$${e}^{\mathrm{\ensuremath{-}}}$ annihilation at 29 GeV: Charged-particle multiplicity and single-particle rapidity distributions</nowiki> |url=https://link.aps.org/doi/10.1103/PhysRevD.34.3304 |journal=Physical Review D |volume=34 |issue=11 |pages=3304–3320 |doi=10.1103/PhysRevD.34.3304|pmid=9957066 |hdl=1808/15222 |hdl-access=free }}</ref><ref>{{Cite journal |last=Zborovský |first=I. |date=2018-10-10 |title=Three-component multiplicity distribution, oscillation of combinants and properties of clans in pp collisions at the LHC |journal=The European Physical Journal C |language=en |volume=78 |issue=10 |pages=816 |doi=10.1140/epjc/s10052-018-6287-x |arxiv=1811.11230 |bibcode=2018EPJC...78..816Z |issn=1434-6052|doi-access=free }}</ref> (See <ref>{{Cite book |last1=Kittel |first1=Wolfram |title=Soft multihardon dynamics |last2=De Wolf |first2=Eddi A |publisher=World Scientific |year=2005}}</ref> for an overview), and is argued to be a [[scale-invariant]] property of matter,<ref>{{Cite journal |last=Schaeffer |first=R |date=1984 |title=Determination of the galaxy N-point correlation function |journal=Astronomy and Astrophysics |volume=134 |issue=2 |pages=L15|bibcode=1984A&A...134L..15S }}</ref><ref>{{Cite journal |last=Schaeffer |first=R |date=1985 |title=The probability generating function for galaxy clustering |journal=Astronomy and Astrophysics |volume=144 |issue=1 |pages=L1–L4|bibcode=1985A&A...144L...1S }}</ref> providing the best fit for astronomical observations, where it predicts the number of galaxies in a region of space.<ref>{{Cite journal |last1=Perez |first1=Lucia A. |last2=Malhotra |first2=Sangeeta |last3=Rhoads |first3=James E. |last4=Tilvi |first4=Vithal |date=2021-01-07 |title=Void Probability Function of Simulated Surveys of High-redshift Ly α Emitters |journal=The Astrophysical Journal |volume=906 |issue=1 |pages=58 |doi=10.3847/1538-4357/abc88b |arxiv=2011.03556 |bibcode=2021ApJ...906...58P |issn=1538-4357 |doi-access=free }}</ref><ref>{{Cite journal |last1=Hurtado-Gil |first1=Lluís |last2=Martínez |first2=Vicent J. |last3=Arnalte-Mur |first3=Pablo |last4=Pons-Bordería |first4=María-Jesús |last5=Pareja-Flores |first5=Cristóbal |last6=Paredes |first6=Silvestre |date=2017-05-01 |title=The best fit for the observed galaxy counts-in-cell distribution function |url=https://www.aanda.org/articles/aa/abs/2017/05/aa29097-16/aa29097-16.html |journal=Astronomy & Astrophysics |language=en |volume=601 |pages=A40 |doi=10.1051/0004-6361/201629097 |arxiv=1703.01087 |bibcode=2017A&A...601A..40H |issn=0004-6361|doi-access=free }}</ref><ref>{{Cite journal |last1=Elizalde |first1=E. |last2=Gaztanaga |first2=E. |date=January 1992 |title=Void probability as a function of the void's shape and scale-invariant models |journal=Monthly Notices of the Royal Astronomical Society |volume=254 |issue=2 |pages=247–256 |doi=10.1093/mnras/254.2.247 |issn=0035-8711|doi-access=free |hdl=2060/19910019799 |hdl-access=free }}</ref><ref>{{Cite journal |last1=Hameeda |first1=M |last2=Plastino |first2=Angelo |last3=Rocca |first3=M C |date=2021-03-01 |title=Generalized Poisson distributions for systems with two-particle interactions |journal=IOP SciNotes |volume=2 |issue=1 |pages=015003 |doi=10.1088/2633-1357/abec9f |bibcode=2021IOPSN...2a5003H |issn=2633-1357|doi-access=free |hdl=11336/181371 |hdl-access=free }}</ref> The phenomenological justification for the effectiveness of the negative binomial distribution in these contexts remained unknown for fifty years, since their first observation in 1973.<ref>{{Cite journal |last=Giovannini |first=A. |date=June 1973 |title="Thermal chaos" and "coherence" in multiplicity distributions at high energies |url=http://dx.doi.org/10.1007/bf02734689 |journal=Il Nuovo Cimento A |volume=15 |issue=3 |pages=543–551 |doi=10.1007/bf02734689 |bibcode=1973NCimA..15..543G |s2cid=118805136 |issn=0369-3546|url-access=subscription }}</ref> In 2023, a proof from [[first principle]]s was eventually demonstrated by Scott V. Tezlaf, where it was shown that the negative binomial distribution emerges from [[Spacetime symmetries|symmetries]] in the [[Dynamics (mechanics)|dynamical equations]] of a [[canonical ensemble]] of particles in [[Minkowski space]].<ref name=":1">{{Cite journal |last=Tezlaf |first=Scott V. |date=2023-09-29 |title=Significance of the negative binomial distribution in multiplicity phenomena |url=https://iopscience.iop.org/article/10.1088/1402-4896/acfead |journal=Physica Scripta |volume=98 |issue=11 |doi=10.1088/1402-4896/acfead |arxiv=2310.03776 |bibcode=2023PhyS...98k5310T |s2cid=263300385 |issn=0031-8949}}</ref> Roughly, given an expected number of trials <math>\langle n \rangle</math> and expected number of successes <math>\langle r \rangle</math>, where

: <math>\langle \mathcal{n} \rangle - \langle r \rangle = k,  \quad \quad  \langle p \rangle  = \frac{\langle r \rangle}{\langle \mathcal{n} \rangle}  \quad\quad \quad \implies \quad\quad \quad
\langle \mathcal{n} \rangle = \frac{k}{1-\langle p \rangle},  \quad \quad \langle {r} \rangle = \frac{k\langle p \rangle}{1 - \langle p \rangle},</math>

an [[Isomorphism|isomorphic]] set of equations can be identified with the parameters of a [[Special relativity|relativistic]] [[current density]] of a canonical ensemble of massive particles, via

: <math>c^2\langle \rho^2 \rangle  - \langle j^2 \rangle = c^2\rho_0^2, \quad \quad  \quad \langle \beta^2_v \rangle = \frac{\langle j^2 \rangle}{c^2\langle \rho^2 \rangle}  \quad \quad \implies \quad \quad
c^2\langle \rho^2 \rangle = \frac{c^2\rho_0^2}{1-\langle \beta^2_v \rangle},  \quad \quad  \quad \langle j^2 \rangle = \frac{c^2\rho_0^2  \langle \beta^2_v \rangle}{1-\langle \beta^2_v \rangle},</math>

where <math>\rho_0</math> is the rest [[density]], <math>\langle \rho ^2 \rangle</math> is the relativistic mean square density, <math>\langle j ^2 \rangle</math> is the relativistic mean square current density, and <math>\langle \beta^2_v \rangle=\langle v^2 \rangle /c^2</math>, where <math>\langle v ^2 \rangle</math> is the [[Maxwell–Boltzmann distribution|mean square speed]] of the particle ensemble and <math>c</math> is the [[speed of light]]—such that one can establish the following [[Bijection|bijective map]]:

: <math>c^2\rho_0^2 \mapsto k, \quad \quad \langle \beta^2_v \rangle \mapsto \langle p \rangle, \quad \quad c^2\langle\rho^2 \rangle  \mapsto \langle \mathcal{n} \rangle, \quad \quad \langle j^2 \rangle \mapsto \langle r \rangle.</math>

A rigorous alternative proof of the above correspondence has also been demonstrated through [[quantum mechanics]] via the Feynman [[Path integral formulation|path integral]].<ref name=":1" />