Editing Beta distribution (section)

== Related distributions ==

===Transformations===
* If ''X'' ~ Beta(''α'', ''β'') then 1 − ''X'' ~ Beta(''β'', ''α'') [[Mirror image|mirror-image]] symmetry
* If ''X'' ~ Beta(''α'', ''β'') then <math>\tfrac{X}{1-X} \sim {\beta'}(\alpha,\beta)</math>. The [[beta prime distribution]], also called "beta distribution of the second kind".
* If <math>X\sim\text{Beta}(\alpha,\beta)</math>, then <math>Y=\log\frac{X}{1-X}</math> has a [[generalized logistic distribution]], with density <math>\frac{\sigma(y)^\alpha\sigma(-y)^\beta}{B(\alpha,\beta)}</math>, where <math>\sigma</math> is the [[logistic sigmoid]].
* If ''X'' ~ Beta(''α'', ''β'') then <math>\tfrac{1}{X} -1 \sim {\beta'}(\beta,\alpha)</math>.
* If <math>X\sim\text{Beta}(\alpha_1,\beta_1)</math> and <math>Y\sim\text{Beta}(\alpha_2,\beta_2)</math> then <math>Z = \tfrac{X}{Y}</math> has density <math> \tfrac{B(\alpha_1 +\alpha_2, \beta_2) z^{\alpha_1 - 1} {}_2F_1(\alpha_1 + \alpha_2, 1- \beta_1; \alpha_1 +\alpha_2 + \beta_2; z) }{B(\alpha_1, \beta_1)B(\alpha_2, \beta_2)} </math> for <math>0 < z \leq 1 </math> and <math> \tfrac{B(\alpha_1 +\alpha_2, \beta_1) z^{-(\alpha_2 + 1)} {}_2F_1(\alpha_1 + \alpha_2, 1- \beta_2; \alpha_1 +\alpha_2 + \beta_1; \tfrac{1}{z})}{B(\alpha_1, \beta_1)B(\alpha_2, \beta_2)} </math> for <math> z \geq 1 </math>, where <math>{}_2F_1(a, b; c; x)</math> is the [[Hypergeometric function]].<ref name="Pham-Gia2000">{{cite journal |last1=Pham-Gia |first1=T. |title=Distributions of the ratios of independent beta variables and applications |journal=Communications in Statistics - Theory and Methods |date=January 2000 |volume=29 |issue=12 |pages=2693–2715 |doi=10.1080/03610920008832632 |url=https://doi.org/10.1080/03610920008832632 |access-date=13 November 2024 |language=en |issn=0361-0926}}</ref>
* If ''X'' ~ Beta(''n''/2, ''m''/2) then <math>\tfrac{mX}{n(1-X)} \sim F(n,m)</math> (assuming ''n'' > 0 and ''m'' > 0), the [[F-distribution|Fisher–Snedecor F distribution]].
* If <math>X \sim \operatorname{Beta}\left(1+\lambda\tfrac{m-\min}{\max-\min}, 1 + \lambda\tfrac{\max-m}{\max-\min}\right)</math> then min + ''X''(max − min) ~ PERT(min, max, ''m'', ''λ'') where ''PERT'' denotes a [[PERT distribution]] used in [[PERT]] analysis, and ''m''=most likely value.<ref name=NewPERT>Herrerías-Velasco, José Manuel and Herrerías-Pleguezuelo, Rafael and René van Dorp, Johan. (2011). Revisiting the PERT mean and Variance. European Journal of Operational Research (210), p. 448&ndash;451.</ref> Traditionally<ref name=Malcolm /> ''λ'' = 4 in PERT analysis.
* If ''X'' ~ Beta(1, ''β'') then ''X'' ~ [[Kumaraswamy distribution]] with parameters (1, ''β'')
* If ''X'' ~ Beta(''α'', 1) then ''X'' ~ [[Kumaraswamy distribution]] with parameters (''α'', 1)
* If ''X'' ~ Beta(''α'', 1) then −ln(''X'') ~ Exponential(''α'')

===Special and limiting cases===
[[File:Random Walk example.svg|thumb|Example of eight realizations of a random walk in one dimension starting at 0: the probability for the time of the last visit to the origin is distributed as Beta(1/2, 1/2)]]
[[File:Arcsin density.svg|thumb|Beta(1/2, 1/2): The [[arcsine distribution]] probability density was proposed by [[Harold Jeffreys]] to represent uncertainty for a [[Bernoulli distribution|Bernoulli]] or a [[binomial distribution]] in [[Bayesian inference]], and is now commonly referred to as [[Jeffreys prior]]: ''p''<sup>−1/2</sup>(1&nbsp;−&nbsp;''p'')<sup>−1/2</sup>. This distribution also appears in several [[random walk]] fundamental theorems]]

* Beta(1, 1) ~ [[uniform distribution (continuous)|U(0, 1)]] with density 1 on that interval.
* Beta(n, 1) ~ Maximum of ''n'' independent rvs. with [[uniform distribution (continuous)|U(0, 1)]], sometimes called a ''a standard power function distribution'' with density ''n''&nbsp;''x''<sup>''n''–1</sup> on that interval.
* Beta(1, n) ~ Minimum of ''n'' independent rvs. with [[uniform distribution (continuous)|U(0, 1)]] with density ''n''(1&nbsp;−&nbsp;''x'')<sup>''n''−1</sup> on that interval.
* If ''X'' ~ Beta(3/2, 3/2) and ''r'' > 0 then 2''rX''&nbsp;−&nbsp;''r'' ~ [[Wigner semicircle distribution]].
* Beta(1/2, 1/2) is equivalent to the [[arcsine distribution]]. This distribution is also [[Jeffreys prior]] probability for the [[Bernoulli distribution|Bernoulli]] and  [[binomial distribution]]s.
* <math>\lim_{n \to \infty} n \operatorname{Beta}(1,n) =  \operatorname{Exponential}(1)</math>  the [[exponential distribution]].
* <math>\lim_{n \to \infty} n \operatorname{Beta}(k,n) = \operatorname{Gamma}(k,1)</math> the [[gamma distribution]].
* For large <math>n</math>, <math>\operatorname{Beta}(\alpha n,\beta n) \to \mathcal{N}\left(\frac{\alpha}{\alpha+\beta},\frac{\alpha\beta}{(\alpha+\beta)^3}\frac{1}{n}\right)</math> the [[normal distribution]]. More precisely, if <math>X_n \sim \operatorname{Beta}(\alpha n,\beta n)</math> then  <math>\sqrt{n}\left(X_n -\tfrac{\alpha}{\alpha+\beta}\right)</math> converges in distribution to a normal distribution with mean 0 and variance <math>\tfrac{\alpha\beta}{(\alpha+\beta)^3}</math> as ''n'' increases.

===Derived from other distributions===
* The ''k''th [[order statistic]] of a sample of size ''n'' from the [[Uniform distribution (continuous)|uniform distribution]] is a beta random variable, ''U''<sub>(''k'')</sub> ~ Beta(''k'', ''n''+1−''k'').<ref name=David1/>
* [[Gamma distribution]]: If ''X'' ~ Gamma(α, θ) and ''Y'' ~ Gamma(β, θ) are independent, then <math>\tfrac{X}{X+Y} \sim \operatorname{Beta}(\alpha, \beta)\,</math>.
* [[Chi-squared distribution]]: If <math>X \sim \chi^2(\alpha)\,</math> and <math>Y \sim \chi^2(\beta)\,</math> are independent, then <math>\tfrac{X}{X+Y} \sim \operatorname{Beta}(\tfrac{\alpha}{2}, \tfrac{\beta}{2})</math>.
* The [[Power transformation (statistics)|power transformation]] for the uniform distribution: If ''X'' ~ U(0, 1) and ''α'' > 0 then ''X''<sup>1/''α''</sup> ~ Beta(''α'', 1).
* [[Cauchy distribution]]: If ''X'' ~ Cauchy(0, 1) then <math>\tfrac{1}{1+X^2} \sim \operatorname{Beta}\left(\tfrac12, \tfrac12\right)\,</math>

===Combination with other distributions===
* ''X'' ~ Beta(''α'', ''β'') and ''Y'' ~ F(2''β'',2''α'') then  <math>\Pr(X \leq \tfrac \alpha {\alpha+\beta x}) = \Pr(Y \geq x)\,</math> for all ''x'' > 0.

===Compounding with other distributions===
* If ''p'' ~ Beta(α, β) and ''X'' ~ Bin(''k'', ''p'') then ''X'' ~ [[beta-binomial distribution]]
* If ''p'' ~ Beta(α, β) and ''X'' ~ NB(''r'', ''p'') then ''X'' ~ [[beta negative binomial distribution]]

===Generalisations===
* The generalization to multiple variables, i.e. a [[Dirichlet distribution|multivariate Beta distribution]], is called a [[Dirichlet distribution]]. Univariate marginals of the Dirichlet distribution have a beta distribution.  The beta distribution is [[Conjugate prior|conjugate]] to the binomial and Bernoulli distributions in exactly the same way as the [[Dirichlet distribution]] is conjugate to the [[multinomial distribution]] and [[categorical distribution]].
* The [[Pearson distribution#The Pearson type I distribution|Pearson type I distribution]] is identical to the beta distribution (except for arbitrary shifting and re-scaling that can also be accomplished with the four parameter parametrization of the beta distribution).
* The beta distribution is the special case of the [[noncentral beta distribution]] where <math>\lambda = 0</math>: <math>\operatorname{Beta}(\alpha, \beta) = \operatorname{NonCentralBeta}(\alpha,\beta,0)</math>.
* The [[generalized beta distribution]] is a five-parameter distribution family which has the beta distribution as a special case.
* The [[matrix variate beta distribution]] is a distribution for [[positive-definite matrices]].