Editing Multimodal distribution (section)

===Probability distributions===

Important bimodal distributions include the [[arcsine distribution]] and the [[beta distribution]] (iff both parameters ''a'' and ''b'' are less than 1). Others include the [[U-quadratic distribution]].

The ratio of two normal distributions is also bimodally distributed. Let

<math display="block"> R = \frac{ a + x }{ b + y } </math>

where ''a'' and ''b'' are constant and ''x'' and ''y'' are distributed as normal variables with a mean of 0 and a standard deviation of 1. ''R'' has a known density that can be expressed as a [[confluent hypergeometric function]].<ref name=Fieller1932>{{cite journal |author=Fieller E |date=1932 |title=The distribution of the index in a normal bivariate population |journal=Biometrika |volume=24 |issue=3–4 |pages=428–440 |doi=10.1093/biomet/24.3-4.428}}</ref>

The distribution of the [[Inverse distribution|reciprocal]] of a ''t'' distributed random variable is bimodal when the degrees of freedom are more than one. Similarly the reciprocal of a normally distributed variable is also bimodally distributed.

A ''t'' statistic generated from data set drawn from a [[Cauchy distribution]] is bimodal.<ref name=Fiorio2010>{{cite journal | last1 = Fiorio | first1 = CV | last2 = HajivassILiou | first2 = VA | last3 = Phillips | first3 = PCB | year = 2010 | title = Bimodal t-ratios: the impact of thick tails on inference | url = https://ink.library.smu.edu.sg/soe_research/1817| journal = The Econometrics Journal | volume = 13 | issue = 2| pages = 271–289 | doi = 10.1111/j.1368-423X.2010.00315.x | s2cid = 363740 }}</ref>