Editing Beta distribution (section)

===Characteristic function===
[[File:Re(CharacteristicFunction) Beta Distr alpha=beta from 0 to 25 Back - J. Rodal.jpg|325px|thumb|[[Characteristic function (probability theory)|Re(characteristic function)]] symmetric case ''α''&nbsp;=&nbsp;''β'' ranging from 25 to 0]][[File:Re(CharacteristicFunc) Beta Distr alpha=beta from 0 to 25 Front- J. Rodal.jpg|325px|thumb|[[Characteristic function (probability theory)|Re(characteristic function)]] symmetric case ''α''&nbsp;=&nbsp;''β'' ranging from 0 to 25]][[File:Re(CharacteristFunc) Beta Distr alpha from 0 to 25 and beta=alpha+0.5 Back - J. Rodal.jpg|325px|thumb|[[Characteristic function (probability theory)|Re(characteristic function)]] ''β'' = ''α''&nbsp;+&nbsp;1/2; ''α''&nbsp;ranging from 25 to 0]][[File:Re(CharacterFunc) Beta Distrib. beta from 0 to 25, alpha=beta+0.5 Back - J. Rodal.jpg|325px|thumb|[[Characteristic function (probability theory)|Re(characteristic function)]] ''α''&nbsp;=&nbsp;''β''&nbsp;+&nbsp;1/2; ''β'' ranging from 25 to 0]][[File:Re(CharacterFunc) Beta Distr. beta from 0 to 25, alpha=beta+0.5 Front - J. Rodal.jpg|325px|thumb|[[Characteristic function (probability theory)|Re(characteristic function)]] ''α''&nbsp;=&nbsp;''β''&nbsp;+&nbsp;1/2; ''β'' ranging from 0 to 25]]

The [[Characteristic function (probability theory)|characteristic function]] is the [[Fourier transform]] of the probability density function. The characteristic function of the beta distribution is [[confluent hypergeometric function|Kummer's confluent hypergeometric function]] (of the first kind):<ref name=JKB /><ref name=Abramowitz /><ref name="Zwillinger_2014">{{cite book |author-first1=Izrail Solomonovich |author-last1=Gradshteyn |author-link1=Izrail Solomonovich Gradshteyn |author-first2=Iosif Moiseevich |author-last2=Ryzhik |author-link2=Iosif Moiseevich Ryzhik |author-first3=Yuri Veniaminovich |author-last3=Geronimus |author-link3=Yuri Veniaminovich Geronimus |author-first4=Michail Yulyevich |author-last4=Tseytlin |author-link4=Michail Yulyevich Tseytlin |author-first5=Alan |author-last5=Jeffrey |editor1-first=Daniel |editor1-last=Zwillinger |editor2-first=Victor Hugo |editor2-last=Moll |editor-link2=Victor Hugo Moll |translator=Scripta Technica, Inc. |title=Table of Integrals, Series, and Products |publisher=[[Academic Press, Inc.]] |date=2015 |orig-year=October 2014 |edition=8 |language=en |isbn=978-0-12-384933-5 |lccn=2014010276 <!-- |url=https://books.google.com/books?id=NjnLAwAAQBAJ |access-date=2016-02-21-->|title-link=Gradshteyn and Ryzhik}}</ref>
:<math>\begin{align}
\varphi_X(\alpha;\beta;t)
&= \operatorname{E}\left[e^{itX}\right]\\
&= \int_0^1 e^{itx} f(x;\alpha,\beta) \, dx \\
&={}_1F_1(\alpha; \alpha+\beta; it)\!\\
&=\sum_{n=0}^\infty \frac {\alpha^{(n)} (it)^n} {(\alpha+\beta)^{(n)} n!}\\
&= 1 +\sum_{k=1}^{\infty} \left( \prod_{r=0}^{k-1} \frac{\alpha+r}{\alpha+\beta+r} \right) \frac{(it)^k}{k!}
\end{align}</math>

where

: <math>x^{(n)}=x(x+1)(x+2)\cdots(x+n-1)</math>

is the [[rising factorial]], also called the "Pochhammer symbol". The value of the characteristic function for ''t''&nbsp;=&nbsp;0, is one:

:<math> \varphi_X(\alpha;\beta;0)={}_1F_1(\alpha; \alpha+\beta; 0) = 1.</math>

Also, the real and imaginary parts of the characteristic function enjoy the following symmetries with respect to the origin of variable ''t'':

:<math> \operatorname{Re} \left [ {}_1F_1(\alpha; \alpha+\beta; it) \right ] = \operatorname{Re} \left [ {}_1F_1(\alpha; \alpha+\beta; - it) \right ]</math>
:<math> \operatorname{Im} \left [ {}_1F_1(\alpha; \alpha+\beta; it) \right ] = - \operatorname{Im} \left [ {}_1F_1(\alpha; \alpha+\beta; - it) \right ]</math>

The symmetric case ''α'' = ''β'' simplifies the characteristic function of the beta distribution to a [[Bessel function]], since in the special case ''α'' + ''β'' = 2''α'' the [[confluent hypergeometric function]] (of the first kind) reduces to a [[Bessel function]] (the modified Bessel function of the first kind <math>I_{\alpha-\frac 1 2}</math> ) using [[Ernst Kummer|Kummer's]] second transformation as follows:

:<math>\begin{align} {}_1F_1(\alpha;2\alpha; it) &= e^{\frac{it}{2}} {}_0F_1 \left(; \alpha+\tfrac{1}{2}; \frac{(it)^2}{16} \right) \\
&= e^{\frac{it}{2}} \left(\frac{it}{4}\right)^{\frac{1}{2}-\alpha} \Gamma\left(\alpha+\tfrac{1}{2}\right) I_{\alpha-\frac 1 2} \left(\frac{it}{2}\right).\end{align}</math>

In the accompanying plots, the [[Complex number|real part]] (Re) of the [[Characteristic function (probability theory)|characteristic function]] of the beta distribution is displayed for symmetric (''α'' = ''β'') and skewed (''α'' ≠ ''β'') cases.