Editing Beta distribution (section)

{{short description|Probability distribution}}
{{Distinguish|Beta function}}
{{Probability distribution
| name       = Beta
| type       = density
| pdf_image  = [[File:Beta distribution pdf.svg|325px|Probability density function for the beta distribution]]
| cdf_image  = [[File:Beta distribution cdf.svg|325px|Cumulative distribution function for the beta distribution]]
| notation   = Beta(''α'', ''β'')
| parameters = ''α'' > 0 [[shape parameter|shape]] ([[real number|real]])<br />''β'' > 0 [[shape parameter|shape]] ([[real number|real]])
| support    = <math>x \in [0, 1]\!</math> or <math>x \in (0, 1)\!</math>
| pdf        = <math>\frac{x^{\alpha-1}(1-x)^{\beta-1}} {\Beta(\alpha,\beta)}\!</math><br />where <math>\Beta(\alpha,\beta) = \frac{\Gamma(\alpha)\Gamma(\beta)}{\Gamma(\alpha + \beta)}</math> and <math>\Gamma</math> is the [[Gamma function]].
| cdf    = <math>I_x(\alpha,\beta)\!</math>
(the [[Beta function#Incomplete beta function|regularized incomplete beta function]])
| mean   = <math>\operatorname{E}[X] = \frac{\alpha}{\alpha+\beta}\!</math><br /><math>\operatorname{E}[\ln X] = \psi(\alpha) - \psi(\alpha + \beta)\!</math><br /><br /><math>\operatorname{E}[X \, \ln X] = \frac{\alpha}{\alpha+\beta}\,\left[\psi(\alpha+1)-\psi(\alpha+\beta+1)\right]\!</math><br /> (see section: [[#Geometric mean|Geometric mean]]) <br/>
where <math>\psi</math> is the [[digamma function]]
| median = <math>\begin{matrix}I_{\frac{1}{2}}^{[-1]}(\alpha,\beta)\text{ (in general) }\\[0.5em]
\approx \frac{ \alpha - \tfrac{1}{3} }{ \alpha + \beta - \tfrac{2}{3} }\text{ for }\alpha, \beta >1\end{matrix}</math>
| mode     = 
<math>\frac{\alpha-1}{\alpha+\beta-2}\!</math> for ''α'', ''β'' > 1

Any value in the domain for ''α'' = ''β'' = 1

No mode if ''α''<1 or ''β''<1. Density diverges

at 0 for ''α'' ≤ 1, and at 1 if ''β'' ≤ 1

| variance = <math>\operatorname{var}[X] = \frac{\alpha\beta}{(\alpha+\beta)^2(\alpha+\beta+1)}\!</math><br /><math>\operatorname{var}[\ln X] = \psi_1(\alpha) - \psi_1(\alpha + \beta)\!</math><br />(see [[trigamma function]] and see section: [[#Geometric variance and covariance|Geometric variance]])
| skewness = <math>\frac{2\,(\beta-\alpha)\sqrt{\alpha+\beta+1}}{(\alpha+\beta+2)\sqrt{\alpha\beta}}</math>
| kurtosis = <math>\frac{6[(\alpha - \beta)^2 (\alpha +\beta + 1) - \alpha \beta (\alpha + \beta + 2)]}{\alpha \beta (\alpha + \beta + 2) (\alpha + \beta + 3)}</math>
| entropy  = <math>\begin{matrix}\ln\Beta(\alpha,\beta)-(\alpha-1)\psi(\alpha)-(\beta-1)\psi(\beta)\\[0.5em]
{}+(\alpha+\beta-2)\psi(\alpha+\beta)\end{matrix}</math>
| mgf    = <math>1  +\sum_{k=1}^\infty \left( \prod_{r=0}^{k-1} \frac{\alpha+r}{\alpha+\beta+r} \right) \frac{t^k}{k!}</math>
| char   = <math>{}_1F_1(\alpha; \alpha+\beta; i\,t)\!</math> (see [[Confluent hypergeometric function]])
| fisher = <math>\begin{bmatrix} \operatorname{var}[\ln X] &\operatorname{cov}[\ln X, \ln(1-X)] \\ \operatorname{cov}[\ln X, \ln(1-X)] & \operatorname{var}[\ln (1-X)]\end{bmatrix}</math> <br />see section: [[#Fisher information matrix|Fisher information matrix]]
| moments = <math> \alpha = \left(\frac{E[X](1 - E[X])}{V[X]} - 1 \right)E[X] </math><br /><math>\beta = \left(\frac{E[X](1 - E[X])}{V[X]} - 1 \right)(1 - E[X])</math>
}}

In [[probability theory]] and [[statistics]], the '''beta distribution''' is a family of continuous [[probability distribution]]s defined on the interval [0, 1] or (0, 1) in terms of two positive [[Statistical parameter|parameters]], denoted by ''alpha'' (''α'') and ''beta'' (''β''), that appear as exponents of the variable and its complement to 1, respectively, and control the [[shape parameter|shape]] of the distribution.

The beta distribution has been applied to model the behavior of [[random variables]] limited to intervals of finite length in a wide variety of disciplines. The beta distribution is a suitable model for the random behavior of percentages and proportions.

In [[Bayesian inference]], the beta distribution is the [[conjugate prior distribution|conjugate prior probability distribution]] for the [[Bernoulli distribution|Bernoulli]], [[binomial distribution|binomial]], [[negative binomial distribution|negative binomial]], and [[geometric distribution|geometric]] distributions.

The formulation of the beta distribution discussed here is also known as the '''beta distribution of the first kind''', whereas ''beta distribution of the second kind'' is an alternative name for the [[beta prime distribution]]. The generalization to multiple variables is called a [[Dirichlet distribution]].