Editing Multimodal distribution (section)

==Origins==

{{main article|Mixture distribution}}

===Mathematical===

A bimodal distribution commonly arises as a mixture of two different [[unimodal]] distributions (i.e. distributions having only one mode).  In other words, the bimodally distributed random variable X is defined as <math> Y </math> with probability <math> \alpha </math> or <math> Z </math> with probability <math> (1-\alpha), </math> where ''Y'' and ''Z'' are unimodal random variables and <math>0 < \alpha < 1</math> is a mixture coefficient.

Mixtures with two distinct components need not be bimodal and two component mixtures of unimodal component densities can have more than two modes. There is no immediate connection between the number of components in a mixture and the number of modes of the resulting density.

===Particular distributions===

Bimodal distributions, despite their frequent occurrence in data sets, have only rarely been studied{{citation needed|date=March 2019}}. This may be because of the difficulties in estimating their parameters either with frequentist or Bayesian methods. Among those that have been studied are

* Bimodal exponential distribution.<ref name=Hassan2010>{{cite journal | last1 = Hassan | first1 = MY | last2 = Hijazi | first2 = RH | year = 2010 | title = A bimodal exponential power distribution | journal = Pakistan Journal of Statistics | volume = 26 | issue = 2| pages = 379–396 }}</ref>
* Alpha-skew-normal distribution.<ref name=Elal-Olivero2010>{{cite journal | last1 = Elal-Olivero | first1 = D | year = 2010 | title = Alpha-skew-normal distribution | journal = Proyecciones Journal of Mathematics | volume = 29 | issue = 3| pages = 224–240 | doi=10.4067/s0716-09172010000300006| doi-access = free }}</ref>
* Bimodal skew-symmetric normal distribution.<ref name=Hassan2013>{{cite journal |last1=Hassan |first1=M. Y. |last2=El-Bassiouni |first2=M. Y. |year=2016 |title=Bimodal skew-symmetric normal distribution |journal=Communications in Statistics - Theory and Methods |volume=45 |issue=5 |pages=1527–1541 |doi=10.1080/03610926.2014.882950 |s2cid=124087015 }}</ref>
* A mixture of [[Conway-Maxwell-Poisson distribution]]s has been fitted to bimodal count data.<ref name=Bosea2013>{{cite book |last1=Bosea |first1=S. |last2=Shmuelib |first2=G. |last3=Sura |first3=P. |last4=Dubey |first4=P. |year=2013 |chapter=Fitting Com-Poisson mixtures to bimodal count data |title=Proceedings of the 2013 International Conference on Information, Operations Management and Statistics (ICIOMS2013), Kuala Lumpur, Malaysia |pages=1–8 |chapter-url=https://www.galitshmueli.com/system/files/ICIOMS%202013%20Malaysia%20Paper%20ID%2028.pdf }}</ref>

Bimodality also naturally arises in the [[Catastrophe theory#Cusp catastrophe|cusp catastrophe distribution]].

===Biology===

In biology, several factors are known to contribute to bimodal distributions of population sizes{{citation needed|date=March 2019}}:

*the initial distribution of individual sizes
*the distribution of growth rates among the individuals
*the size and time dependence of the growth rate of each individual
* mortality rates that may affect each size class differently
* the DNA methylation in human and mouse genome.
* the dynamics of transcription at the promoter region.

The bimodal distribution of sizes of [[weaver ant]] workers arises due to existence of two distinct classes of workers, namely major workers and minor workers.<ref name="Weber1946">{{cite journal|author=Weber, NA|year=1946| title=Dimorphism in the African ''Oecophylla'' worker and an anomaly (Hym.: Formicidae)| journal=Annals of the Entomological Society of America| volume=39| pages=7–10| url=http://antbase.org/ants/publications/10434/10434.pdf| doi=10.1093/aesa/39.1.7}}</ref>

The [[distribution of fitness effects]] of mutations for both whole [[genome]]s<ref>{{cite journal|last=Sanjuán|first=R|title=Mutational fitness effects in RNA and single-stranded DNA viruses: common patterns revealed by site-directed mutagenesis studies.|journal=Philosophical Transactions of the Royal Society of London B: Biological Sciences|date=Jun 27, 2010|volume=365|issue=1548|pages=1975–82|pmid=20478892|doi=10.1098/rstb.2010.0063|pmc=2880115}}</ref><ref>{{cite journal|last=Eyre-Walker|first=A|author2=Keightley, PD|title=The distribution of fitness effects of new mutations.|journal=Nature Reviews Genetics|date=Aug 2007|volume=8|issue=8|pages=610–8|pmid=17637733|doi=10.1038/nrg2146|s2cid=10868777}}</ref> and individual [[gene]]s<ref>{{cite journal|last=Hietpas|first=RT|author2=Jensen, JD |author3=Bolon, DN |title=Experimental illumination of a fitness landscape.|journal=Proceedings of the National Academy of Sciences of the United States of America|date=May 10, 2011|volume=108|issue=19|pages=7896–901|pmid=21464309|doi=10.1073/pnas.1016024108|pmc=3093508|bibcode = 2011PNAS..108.7896H |doi-access=free}}</ref> is also frequently found to be bimodal with most [[mutations]] being either neutral or lethal with relatively few having intermediate effect.