Editing Beta distribution (section)

==Occurrence and applications==

===Order statistics===
{{Main|Order statistic}}
The beta distribution has an important application in the theory of [[order statistic]]s. A basic result is that the distribution of the ''k''th smallest of a sample of size ''n'' from a continuous [[Uniform distribution (continuous)|uniform distribution]] has a beta distribution.<ref name=David1>David, H. A., Nagaraja, H. N. (2003) ''Order Statistics'' (3rd Edition). Wiley, New Jersey pp 458. {{ISBN|0-471-38926-9}}</ref> This result is summarized as:

:<math>U_{(k)} \sim \operatorname{Beta}(k,n+1-k).</math>

From this, and application of the theory related to the [[probability integral transform]], the distribution of any individual order statistic from any [[continuous distribution]] can be derived.<ref name=David1/>

===Subjective logic===
{{Main|Subjective logic}}

In standard logic, propositions are considered to be either true or false. In contradistinction, [[subjective logic]] assumes that humans cannot determine with absolute certainty whether a proposition about the real world is absolutely true or false. In [[subjective logic]] the [[A posteriori|posteriori]] probability estimates of binary events can be represented by beta distributions.<ref name="J01">{{cite journal
 | last = Jøsang | first = Audun
 | doi = 10.1142/S0218488501000831
 | issue = 3
 | journal = International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems
 | mr = 1843261
 | pages = 279–311
 | title = A logic for uncertain probabilities
 | url = https://scholar.archive.org/work/nilorkzfvjccjir72m75zk3pgy
 | volume = 9
 | year = 2001}}</ref>

===Wavelet analysis===
{{Main|Beta wavelet}}

A [[wavelet]] is a wave-like [[oscillation]] with an [[amplitude]] that starts out at zero, increases, and then decreases back to zero. It can typically be visualized as a "brief oscillation" that promptly decays. Wavelets can be used to extract information from many different kinds of data, including&nbsp;– but certainly not limited to&nbsp;– audio signals and images. Thus, wavelets are purposefully crafted to have specific properties that make them useful for [[signal processing]]. Wavelets are localized in both time and [[frequency]] whereas the standard [[Fourier transform]] is only localized in frequency. Therefore, standard Fourier Transforms are only applicable to [[stationary process]]es, while [[wavelet]]s are applicable to non-[[stationary process]]es.  Continuous wavelets can be constructed based on the beta distribution. [[Beta wavelet]]s<ref name="wavelet oliveira">H.M. de Oliveira and G.A.A. Araújo,. Compactly Supported One-cyclic Wavelets Derived from Beta Distributions. ''Journal of Communication and Information Systems.'' vol.20, n.3, pp.27-33, 2005.</ref> can be viewed as a soft variety of [[Haar wavelet]]s whose shape is fine-tuned by two shape parameters α and β.

===Population genetics===
{{main|Balding–Nichols model}}
{{further|F-statistics|Fixation index|Coefficient of relationship}}

The [[Balding–Nichols model]] is a two-parameter [[Statistical parameter|parametrization]] of the beta distribution used in [[population genetics]].<ref name=Balding>{{cite journal |last1=Balding |first1=David J. |author-link1=David Balding |last2=Nichols |first2=Richard A. |year=1995 |title=A method for quantifying differentiation between populations at multi-allelic loci and its implications for investigating identity and paternity |journal=Genetica |volume=96 |issue=1–2 |pages=3–12 |publisher=Springer |doi=10.1007/BF01441146 |pmid=7607457|s2cid=30680826 }}</ref>  It is a statistical description of the [[allele frequencies]] in the components of a sub-divided population:

:<math>
  \begin{align}
    \alpha &= \mu \nu,\\
    \beta  &= (1 - \mu) \nu,
  \end{align}
</math>
where <math>\nu =\alpha+\beta= \frac{1-F}{F}</math> and <math>0 < F < 1</math>; here ''F'' is (Wright's) genetic distance between two populations.

===Project management: task cost and schedule modeling===
The beta distribution can be used to model events which are constrained to take place within an interval defined by a minimum and maximum value. For this reason, the beta distribution&nbsp;— along with the [[triangular distribution]]&nbsp;— is used extensively in [[PERT]], [[critical path method]] (CPM), Joint Cost Schedule Modeling (JCSM) and other [[project management]]/control systems to describe the time to completion and the cost of a task. In project management, shorthand computations are widely used to estimate the [[mean]] and [[standard deviation]] of the beta distribution:<ref name=Malcolm>{{cite journal |last1=Malcolm |first1=D. G. |last2=Roseboom |first2=J. H. |last3=Clark |first3=C. E. |last4=Fazar |first4=W. |title=Application of a Technique for Research and Development Program Evaluation |journal=Operations Research |date=September–October 1958 |volume=7 |issue=5 |pages=646–669 |doi=10.1287/opre.7.5.646 |issn=0030-364X}}</ref>

:<math> \begin{align}
  \mu(X) & = \frac{a + 4b + c}{6} \\[8pt]
  \sigma(X) & = \frac{c-a}{6}
\end{align}</math>

where ''a'' is the minimum, ''c'' is the maximum, and ''b'' is the most likely value (the [[Mode (statistics)|mode]] for ''α'' > 1 and ''β'' > 1).

The above estimate for the [[mean]] <math>\mu(X)= \frac{a + 4b + c}{6}</math> is known as the [[PERT]] [[three-point estimation]] and it is exact for either of the following values of ''β'' (for arbitrary α within these ranges):

:''β'' = ''α'' > 1 (symmetric case) with [[standard deviation]] <math>\sigma(X) = \frac{c-a}{2 \sqrt {1+2\alpha}}</math>, [[skewness]] = 0, and [[excess kurtosis]] = <math> \frac{-6}{3+2 \alpha}</math>
[[File:Beta_Distribution_beta=alpha_from_1.05_to_4.95.svg|thumb]]

or

:''β'' = 6 − ''α'' for 5 > ''α'' > 1 (skewed case) with [[standard deviation]]

:<math>\sigma(X) = \frac{(c-a)\sqrt{\alpha(6-\alpha)}}{6 \sqrt 7},</math>

[[skewness]]<math>{}=\frac{(3-\alpha) \sqrt 7}{2\sqrt{\alpha(6-\alpha)}}</math>, and [[excess kurtosis]]<math>{}=\frac{21}{\alpha (6- \alpha)} - 3</math>

[[File:Beta Distribution beta=6-alpha from 1.05 to 4.95.svg|thumb]]

The above estimate for the [[standard deviation]] ''σ''(''X'') = (''c'' − ''a'')/6 is exact for either of the following values of ''α'' and ''β'':

:''α'' = ''β'' = 4 (symmetric) with [[skewness]] = 0, and [[excess kurtosis]] = −6/11.
:''β'' = 6 − ''α'' and <math>\alpha = 3 - \sqrt2</math> (right-tailed, positive skew) with [[skewness]]<math>{}=\frac{1}{\sqrt 2}</math>, and [[excess kurtosis]] = 0
:''β'' = 6 − ''α'' and <math>\alpha = 3 + \sqrt2</math> (left-tailed, negative skew) with [[skewness]]<math>{}= \frac{-1}{\sqrt 2}</math>, and [[excess kurtosis]] = 0

[[File:Beta Distribution for conjugate alpha beta.svg|thumb]]

Otherwise, these can be poor approximations for beta distributions with other values of α and β, exhibiting average errors of 40% in the mean and 549% in the variance.<ref>Keefer, Donald L. and Verdini, William A. (1993). Better Estimation of PERT Activity Time Parameters. Management Science 39(9), p. 1086&ndash;1091.</ref><ref>Keefer, Donald L. and Bodily, Samuel E. (1983). Three-point Approximations for Continuous Random variables. Management Science 29(5), p. 595&ndash;609.</ref><ref>{{Cite web|url=https://www.nps.edu/web/drmi/|title=Defense Resource Management Institute - Naval Postgraduate School|website=www.nps.edu}}</ref>