Editing Multimodal distribution (section)

==General properties==

A mixture of two unimodal distributions with differing means is not necessarily bimodal. The combined distribution of heights of men and women is sometimes used as an example of a bimodal distribution, but in fact the difference in mean heights of men and women is too small relative to their [[standard deviation]]s to produce bimodality when the two distribution curves are combined.<ref name="Schilling2002">{{Cite journal|title=Is Human Height Bimodal?|first1=Mark F. |last1=Schilling |first2= Ann E.| last2=Watkins | author2-link = Ann E. Watkins |first3=William |last3=Watkins| journal=[[The American Statistician]]| doi=10.1198/00031300265 |volume=56 |year=2002| pages=223–229 |issue=3|s2cid=53495657 }}</ref>

Bimodal distributions have the peculiar property that – unlike the unimodal distributions – the mean may be a more robust sample estimator than the median.<ref name=Mosteller1977>{{cite book |last1=Mosteller |first1=F. |last2=Tukey |first2=J. W. |year=1977 |title=Data Analysis and Regression: A Second Course in Statistics |location=Reading, Mass |publisher=Addison-Wesley |isbn=0-201-04854-X }}</ref> This is clearly the case when the distribution is U-shaped like the arcsine distribution. It may not be true when the distribution has one or more long tails.

===Moments of mixtures===

Let

<math display="block"> f( x ) = p g_1( x ) + ( 1 - p ) g_2( x ) \, </math>

where {{math|''g''<sub>''i''</sub>}} is a probability distribution and {{math|''p''}} is the mixing parameter.

The moments of {{math|''f''(''x'')}} are<ref name=Kim2003>{{cite web |last1=Kim |first1=T.-H. |last2=White |first2=H. |author-link2=Halbert White |year=2003 |url=https://escholarship.org/uc/item/7b52v07p |title=On more robust estimation of skewness and kurtosis: Simulation and application to the S & P 500 index }}</ref>

<math display="block">\begin{align}
\mu &= p \mu_1 + ( 1 - p ) \mu_2 \\[1ex]
\nu_2 &= p \left[ \sigma_1^2 + \delta_1^2 \right] + ( 1 - p ) \left[ \sigma_2^2 + \delta_2^2 \right] \\[1ex]
\nu_3 &= p \left[ S_1 \sigma_1^3 + 3 \delta_1 \sigma_1^2 + \delta_1^3 \right] + ( 1 - p ) \left[ S_2 \sigma_2^3 + 3 \delta_2 \sigma_2^2 + \delta_2^3 \right] \\[1ex]
\nu_4 &= p \left[ K_1 \sigma_1^4 + 4 S_1 \delta_1 \sigma_1^3 + 6 \delta_1^2 \sigma_1^2 + \delta_1^4 \right] + ( 1 - p ) \left[ K_2 \sigma_2^4 + 4 S_2 \delta_2 \sigma_2^3 + 6 \delta_2^2 \sigma_2^2 + \delta_2^4 \right] \\
\end{align}</math>

where 
* <math> \mu = \int x f( x ) \, dx </math>
* <math> \delta_i = \mu_i - \mu </math>
* <math> \nu_r = \int ( x - \mu )^r f( x ) \, dx </math>

and {{math|''S''<sub>''i''</sub>}} and {{math|''K''<sub>''i''</sub>}} are the [[skewness]] and [[kurtosis]] of the {{mvar|i}}-th distribution.