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Calculated Shard: 76 (from laksa110)

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curl -X POST \
  'http://laksa076.int.ahrefs:8124/' \
  -H 'Content-Type: text/plain' \
  -H 'X-ClickHouse-Database: crawler3' \
  -H 'Authorization: Basic YXBpOg==' \
  -d 'SELECT getAhrefsURLFromUnparsed(src_unparsed) AS found_url, ifNull(toUnixTimestamp(download_stamp), 0) AS crawl_time, ifNull(toUnixTimestamp(props_url_first_seen), 0) AS first_indexed_time, download_http_code AS http_code, src_unparsed AS src_unparsed, src_root_hash AS src_root_hash, history_drop_reason AS history_drop_reason, meta_title AS meta_title, meta_descriptions AS meta_descriptions, attrs_boilerpipe_text AS attrs_boilerpipe_text, attrs_markdown AS attrs_markdown, attrs_readable_markdown AS attrs_readable_markdown, meta_canonical AS meta_canonical FROM crawler3.page_info_local FINAL PREWHERE (src_root_hash, src_unparsed) IN ((getAhrefsRootHashFromUnparsed(getAhrefsUnparsedNoserviceFromURL(\'https://www.boost.org/doc/libs/1_63_0/libs/math/doc/html/math_toolkit/dist_ref/dists/beta_dist.html\')), getAhrefsUnparsedNoserviceFromURL(\'https://www.boost.org/doc/libs/1_63_0/libs/math/doc/html/math_toolkit/dist_ref/dists/beta_dist.html\'))) FORMAT JSONEachRow'

Response:

{"found_url":"https:\/\/www.boost.org\/doc\/libs\/1_63_0\/libs\/math\/doc\/html\/math_toolkit\/dist_ref\/dists\/beta_dist.html","crawl_time":1774639332,"first_indexed_time":1526473765,"http_code":200,"src_unparsed":"org,boost!www,\/doc\/libs\/1_63_0\/libs\/math\/doc\/html\/math_toolkit\/dist_ref\/dists\/beta_dist.html s443","src_root_hash":"2177253573749532476","history_drop_reason":null,"meta_title":"Beta Distribution","meta_descriptions":[],"attrs_boilerpipe_text":"#include\n<\nboost\n\/\nmath\n\/\ndistributions\n\/\nbeta\n.\nhpp\n>\nnamespace\nboost\n{\nnamespace\nmath\n{\ntemplate\n<\nclass\nRealType\n=\ndouble\n,\nclass\nPolicy\n=\npolicies::policy<>\n>\nclass\nbeta_distribution\n;\ntemplate\n<\nclass\nRealType\n,\nclass\nPolicy\n>\nclass\nbeta_distribution\n{\npublic\n:\ntypedef\nRealType\nvalue_type\n;\ntypedef\nPolicy\npolicy_type\n;\nbeta_distribution\n(\nRealType\na\n,\nRealType\nb\n);\nRealType\nalpha\n()\nconst\n;\nRealType\nbeta\n()\nconst\n;\nstatic\nRealType\nfind_alpha\n(\nRealType\nmean\n,\nRealType\nvariance\n);\nstatic\nRealType\nfind_beta\n(\nRealType\nmean\n,\nRealType\nvariance\n);\nstatic\nRealType\nfind_alpha\n(\nRealType\nbeta\n,\nRealType\nx\n,\nRealType\nprobability\n);\nstatic\nRealType\nfind_beta\n(\nRealType\nalpha\n,\nRealType\nx\n,\nRealType\nprobability\n);\n};\n}}\nThe class type\nbeta_distribution\nrepresents a\nbeta\nprobability\n          distribution function\n.\nThe\nbeta\n          distribution\nis used as a\nprior\n          distribution\nfor binomial proportions in\nBayesian\n          analysis\n.\nSee also:\nbeta\n          distribution\nand\nBayesian\n          statistics\n.\nHow the beta distribution is used for\nBayesian\n          analysis of one parameter models\nis discussed by Jeff Grynaviski.\nThe\nprobability\n          density function PDF\nfor the\nbeta\n          distribution\ndefined on the interval [0,1] is given by:\nf(x;α,β) = x\nα - 1\n(1 - x)\nβ -1\n\/ B(α, β)\nwhere B(α, β) is the\nbeta\n          function\n, implemented in this library as\nbeta\n.\n          Division by the beta function ensures that the pdf is normalized to the\n          range zero to unity.\nThe following graph illustrates examples of the pdf for various values\n          of the shape parameters. Note the α = β = 2 (blue line) is dome-shaped, and\n          might be approximated by a symmetrical triangular distribution.\nIf α = β = 1, then it is a __space\nuniform\n          distribution\n, equal to unity in the entire interval x = 0 to 1.\n          If α __space and β __space are < 1, then the pdf is U-shaped. If α != β, then\n          the shape is asymmetric and could be approximated by a triangle whose apex\n          is away from the centre (where x = half).\nMember\n          Functions\nConstructor\nbeta_distribution\n(\nRealType\nalpha\n,\nRealType\nbeta\n);\nConstructs a beta distribution with shape parameters\nalpha\nand\nbeta\n.\nRequires alpha,beta > 0,otherwise\ndomain_error\nis called. Note that technically the beta distribution is defined for alpha,beta\n          >= 0, but it's not clear whether any program can actually make use of\n          that latitude or how many of the non-member functions can be usefully defined\n          in that case. Therefore for now, we regard it as an error if alpha or beta\n          is zero.\nFor example:\nbeta_distribution\n<>\nmybeta\n(\n2\n,\n5\n);\nConstructs a the beta distribution with alpha=2 and beta=5 (shown in yellow\n          in the graph above).\nParameter\n          Accessors\nRealType\nalpha\n()\nconst\n;\nReturns the parameter\nalpha\nfrom which this distribution\n          was constructed.\nRealType\nbeta\n()\nconst\n;\nReturns the parameter\nbeta\nfrom which this distribution\n          was constructed.\nSo for example:\nbeta_distribution\n<>\nmybeta\n(\n2\n,\n5\n);\nassert\n(\nmybeta\n.\nalpha\n()\n==\n2.\n);\nassert\n(\nmybeta\n.\nbeta\n()\n==\n5.\n);\nParameter\n          Estimators\nTwo pairs of parameter estimators are provided.\nOne estimates either α __space or β __space from presumed-known mean and variance.\nThe other pair estimates either α __space or β __space from the cdf and x.\nIt is also possible to estimate α __space and β __space from 'known' mode &\n          quantile. For example, calculators are provided by the\nPooled\n          Prevalence Calculator\nand\nBeta\n          Buster\nbut this is not yet implemented here.\nstatic\nRealType\nfind_alpha\n(\nRealType\nmean\n,\nRealType\nvariance\n);\nReturns the unique value of α   that corresponds to a beta distribution with\n          mean\nmean\nand variance\nvariance\n.\nstatic\nRealType\nfind_beta\n(\nRealType\nmean\n,\nRealType\nvariance\n);\nReturns the unique value of β   that corresponds to a beta distribution with\n          mean\nmean\nand variance\nvariance\n.\nstatic\nRealType\nfind_alpha\n(\nRealType\nbeta\n,\nRealType\nx\n,\nRealType\nprobability\n);\nReturns the value of α   that gives:\ncdf\n(\nbeta_distribution\n<\nRealType\n>(\nalpha\n,\nbeta\n),\nx\n)\n==\nprobability\n.\nstatic\nRealType\nfind_beta\n(\nRealType\nalpha\n,\nRealType\nx\n,\nRealType\nprobability\n);\nReturns the value of β   that gives:\ncdf\n(\nbeta_distribution\n<\nRealType\n>(\nalpha\n,\nbeta\n),\nx\n)\n==\nprobability\n.\nNon-member\n          Accessor Functions\nAll the\nusual non-member accessor\n          functions\nthat are generic to all distributions are supported:\nCumulative Distribution Function\n,\nProbability Density Function\n,\nQuantile\n,\nHazard Function\n,\nCumulative Hazard Function\n,\nmean\n,\nmedian\n,\nmode\n,\nvariance\n,\nstandard deviation\n,\nskewness\n,\nkurtosis\n,\nkurtosis_excess\n,\nrange\nand\nsupport\n.\nThe formulae for calculating these are shown in the table below, and at\nWolfram\n          Mathworld\n.\nApplications\nThe beta distribution can be used to model events constrained to take place\n          within an interval defined by a minimum and maximum value: so it is used\n          in project management systems.\nIt is also widely used in\nBayesian\n          statistical inference\n.\nRelated\n          distributions\nThe beta distribution with both α  __space and β = 1 follows a\nuniform\n          distribution\n.\nThe\ntriangular\nis used when less precise information is available.\nThe\nbinomial\n          distribution\nis closely related when α  __space and β  __space are integers.\nWith integer values of α  __space and β  __space the distribution B(i, j) is\n          that of the j-th highest of a sample of i + j + 1 independent random variables\n          uniformly distributed between 0 and 1. The cumulative probability from\n          0 to x is thus the probability that the j-th highest value is less than\n          x. Or it is the probability that at least i of the random variables are\n          less than x, a probability given by summing over the\nBinomial\n          Distribution\nwith its p parameter set to x.\nAccuracy\nThis distribution is implemented using the\nbeta\n          functions\nbeta\nand\nincomplete beta\n          functions\nibeta\nand\nibetac\n;\n          please refer to these functions for information on accuracy.\nImplementation\nIn the following table\na\nand\nb\nare the parameters α   and β,\nx\nis the random variable,\np\nis the probability and\nq = 1-p\n.\nFunction\nImplementation Notes\npdf\nf(x;α,β) = x\nα - 1\n(1 - x)\nβ -1\n\/ B(α, β)\nImplemented using\nibeta_derivative\n(a,\n                    b, x).\ncdf\nUsing the incomplete beta function\nibeta\n(a,\n                    b, x)\ncdf complement\nibetac\n(a,\n                    b, x)\nquantile\nUsing the inverse incomplete beta function\nibeta_inv\n(a,\n                    b, p)\nquantile from the complement\nibetac_inv\n(a,\n                    b, q)\nmean\na\n\/(\na\n+\nb\n)\nvariance\na\n*\nb\n\/\n(\na\n+\nb\n)^\n2\n*\n(\na\n+\nb\n+\n1\n)\nmode\n(\na\n-\n1\n)\n\/\n(\na\n+\nb\n-\n2\n)\nskewness\n2\n(\nb\n-\na\n)\nsqrt\n(\na\n+\nb\n+\n1\n)\/(\na\n+\nb\n+\n2\n)\n*\nsqrt\n(\na\n*\nb\n)\nkurtosis excess\nkurtosis\nkurtosis\n+\n3\nparameter estimation\nalpha\nfrom mean and variance\nmean\n*\n((\n(\nmean\n*\n(\n1\n-\nmean\n))\n\/\nvariance\n)-\n1\n)\nbeta\nfrom mean and variance\n(\n1\n-\nmean\n)\n*\n(((\nmean\n*\n(\n1\n-\nmean\n))\n\/\nvariance\n)-\n1\n)\nThe member functions\nfind_alpha\nand\nfind_beta\nfrom cdf and probability x\nand\neither\nalpha\nor\nbeta\nImplemented in terms of the inverse incomplete beta functions\nibeta_inva\n,\n                    and\nibeta_invb\nrespectively.\nfind_alpha\nibeta_inva\n(\nbeta\n,\nx\n,\nprobability\n)\nfind_beta\nibeta_invb\n(\nalpha\n,\nx\n,\nprobability\n)\nReferences\nWikipedia Beta\n          distribution\nNIST\n          Exploratory Data Analysis\nWolfram\n          MathWorld","attrs_markdown":"# [![Boost C++ Libraries](https:\/\/www.boost.org\/static\/img\/original_docs\/space.png) Boost C++ Libraries](https:\/\/www.boost.org\/)\n\"...one of the most highly regarded and expertly designed C++ library projects in the world.\" — [Herb Sutter](https:\/\/herbsutter.com\/) and [Andrei Alexandrescu](http:\/\/en.wikipedia.org\/wiki\/Andrei_Alexandrescu), [C++ Coding Standards](https:\/\/books.google.com\/books\/about\/C++_Coding_Standards.html?id=mmjVIC6WolgC)\n\nSearch...\n\nThis is an older version of Boost and was released in 2016. The [current version](https:\/\/www.boost.org\/doc\/libs\/latest\/libs\/math\/doc\/html\/math_toolkit\/dist_ref\/dists\/beta_dist.html) is 1.90.0.\n\n[![Prev](https:\/\/www.boost.org\/doc\/libs\/1_63_0\/doc\/src\/images\/prev.png)](https:\/\/www.boost.org\/doc\/libs\/1_63_0\/libs\/math\/doc\/html\/math_toolkit\/dist_ref\/dists\/bernoulli_dist.html)[![Up](https:\/\/www.boost.org\/doc\/libs\/1_63_0\/doc\/src\/images\/up.png)](https:\/\/www.boost.org\/doc\/libs\/1_63_0\/libs\/math\/doc\/html\/math_toolkit\/dist_ref\/dists.html)[![Home](https:\/\/www.boost.org\/doc\/libs\/1_63_0\/doc\/src\/images\/home.png)](https:\/\/www.boost.org\/doc\/libs\/1_63_0\/libs\/math\/doc\/html\/index.html)[![Next](https:\/\/www.boost.org\/doc\/libs\/1_63_0\/doc\/src\/images\/next.png)](https:\/\/www.boost.org\/doc\/libs\/1_63_0\/libs\/math\/doc\/html\/math_toolkit\/dist_ref\/dists\/binomial_dist.html)\n\n#### [Beta Distribution](https:\/\/www.boost.org\/doc\/libs\/1_63_0\/libs\/math\/doc\/html\/math_toolkit\/dist_ref\/dists\/beta_dist.html \"Beta Distribution\")\n```\n#include <boost\/math\/distributions\/beta.hpp>\n```\n```\nnamespace boost{ namespace math{\n\n template <class RealType = double,\n           class Policy   = policies::policy<> >\nclass beta_distribution;\n\n\/\/ typedef beta_distribution<double> beta;\n\/\/ Note that this is deliberately NOT provided,\n\/\/ to avoid a clash with the function name beta.\n\ntemplate <class RealType, class Policy>\nclass beta_distribution\n{\npublic:\n   typedef RealType  value_type;\n   typedef Policy    policy_type;\n   \/\/ Constructor from two shape parameters, alpha & beta:\n   beta_distribution(RealType a, RealType b);\n\n   \/\/ Parameter accessors:\n   RealType alpha() const;\n   RealType beta() const;\n\n   \/\/ Parameter estimators of alpha or beta from mean and variance.\n   static RealType find_alpha(\n     RealType mean, \/\/ Expected value of mean.\n     RealType variance); \/\/ Expected value of variance.\n\n   static RealType find_beta(\n     RealType mean, \/\/ Expected value of mean.\n     RealType variance); \/\/ Expected value of variance.\n\n   \/\/ Parameter estimators from\n   \/\/ either alpha or beta, and x and probability.\n\n   static RealType find_alpha(\n     RealType beta, \/\/ from beta.\n     RealType x, \/\/  x.\n     RealType probability); \/\/ cdf\n\n   static RealType find_beta(\n     RealType alpha, \/\/ alpha.\n     RealType x, \/\/ probability x.\n     RealType probability); \/\/ probability cdf.\n};\n\n}} \/\/ namespaces\n```\nThe class type `beta_distribution` represents a [beta](http:\/\/en.wikipedia.org\/wiki\/Beta_distribution)  [probability distribution function](http:\/\/en.wikipedia.org\/wiki\/Probability_distribution).\n\nThe [beta distribution](http:\/\/mathworld.wolfram.com\/BetaDistribution.htm)  is used as a [prior distribution](http:\/\/en.wikipedia.org\/wiki\/Prior_distribution) for binomial proportions in [Bayesian analysis](http:\/\/mathworld.wolfram.com\/BayesianAnalysis.html).\n\nSee also: [beta distribution](http:\/\/documents.wolfram.com\/calculationcenter\/v2\/Functions\/ListsMatrices\/Statistics\/BetaDistribution.html) and [Bayesian statistics](http:\/\/en.wikipedia.org\/wiki\/Bayesian_statistics).\n\nHow the beta distribution is used for [Bayesian analysis of one parameter models](http:\/\/home.uchicago.edu\/~grynav\/bayes\/ABSLec5.ppt) is discussed by Jeff Grynaviski.\n\nThe [probability density function PDF](http:\/\/en.wikipedia.org\/wiki\/Probability_density_function) for the [beta distribution](http:\/\/en.wikipedia.org\/wiki\/Beta_distribution) defined on the interval \\[0,1\\] is given by:\n\nf(x;α,β) = xα - 1 (1 - x)β -1 \/ B(α, β)\n\nwhere B(α, β) is the [beta function](http:\/\/en.wikipedia.org\/wiki\/Beta_function), implemented in this library as [beta](https:\/\/www.boost.org\/doc\/libs\/1_63_0\/libs\/math\/doc\/html\/math_toolkit\/sf_beta\/beta_function.html \"Beta\"). Division by the beta function ensures that the pdf is normalized to the range zero to unity.\n\nThe following graph illustrates examples of the pdf for various values of the shape parameters. Note the α = β = 2 (blue line) is dome-shaped, and might be approximated by a symmetrical triangular distribution.\n\n![](https:\/\/www.boost.org\/doc\/libs\/1_63_0\/libs\/math\/doc\/graphs\/beta_pdf.svg)\n\nIf α = β = 1, then it is a \\_\\_space [uniform distribution](http:\/\/en.wikipedia.org\/wiki\/Uniform_distribution_%28continuous%29), equal to unity in the entire interval x = 0 to 1. If α \\_\\_space and β \\_\\_space are \\< 1, then the pdf is U-shaped. If α != β, then the shape is asymmetric and could be approximated by a triangle whose apex is away from the centre (where x = half).\n\n##### [Member Functions](https:\/\/www.boost.org\/doc\/libs\/1_63_0\/libs\/math\/doc\/html\/math_toolkit\/dist_ref\/dists\/beta_dist.html#math_toolkit.dist_ref.dists.beta_dist.member_functions)\n###### [Constructor](https:\/\/www.boost.org\/doc\/libs\/1_63_0\/libs\/math\/doc\/html\/math_toolkit\/dist_ref\/dists\/beta_dist.html#math_toolkit.dist_ref.dists.beta_dist.constructor)\n```\nbeta_distribution(RealType alpha, RealType beta);\n```\nConstructs a beta distribution with shape parameters *alpha* and *beta*.\n\nRequires alpha,beta \\> 0,otherwise [domain\\_error](https:\/\/www.boost.org\/doc\/libs\/1_63_0\/libs\/math\/doc\/html\/math_toolkit\/error_handling.html#math_toolkit.error_handling.domain_error) is called. Note that technically the beta distribution is defined for alpha,beta \\>= 0, but it's not clear whether any program can actually make use of that latitude or how many of the non-member functions can be usefully defined in that case. Therefore for now, we regard it as an error if alpha or beta is zero.\n\nFor example:\n```\nbeta_distribution<> mybeta(2, 5);\n```\nConstructs a the beta distribution with alpha=2 and beta=5 (shown in yellow in the graph above).\n\n###### [Parameter Accessors](https:\/\/www.boost.org\/doc\/libs\/1_63_0\/libs\/math\/doc\/html\/math_toolkit\/dist_ref\/dists\/beta_dist.html#math_toolkit.dist_ref.dists.beta_dist.parameter_accessors)\n```\nRealType alpha() const;\n```\nReturns the parameter *alpha* from which this distribution was constructed.\n```\nRealType beta() const;\n```\nReturns the parameter *beta* from which this distribution was constructed.\n\nSo for example:\n```\nbeta_distribution<> mybeta(2, 5);\nassert(mybeta.alpha() == 2.);  \/\/ mybeta.alpha() returns 2\nassert(mybeta.beta() == 5.);   \/\/ mybeta.beta()  returns 5\n```\n##### [Parameter Estimators](https:\/\/www.boost.org\/doc\/libs\/1_63_0\/libs\/math\/doc\/html\/math_toolkit\/dist_ref\/dists\/beta_dist.html#math_toolkit.dist_ref.dists.beta_dist.parameter_estimators)\nTwo pairs of parameter estimators are provided.\n\nOne estimates either α \\_\\_space or β \\_\\_space from presumed-known mean and variance.\n\nThe other pair estimates either α \\_\\_space or β \\_\\_space from the cdf and x.\n\nIt is also possible to estimate α \\_\\_space and β \\_\\_space from 'known' mode & quantile. For example, calculators are provided by the [Pooled Prevalence Calculator](http:\/\/www.ausvet.com.au\/pprev\/content.php?page=PPscript) and [Beta Buster](http:\/\/www.epi.ucdavis.edu\/diagnostictests\/betabuster.html) but this is not yet implemented here.\n```\nstatic RealType find_alpha(\n  RealType mean, \/\/ Expected value of mean.\n  RealType variance); \/\/ Expected value of variance.\n```\nReturns the unique value of α that corresponds to a beta distribution with mean *mean* and variance *variance*.\n```\nstatic RealType find_beta(\n  RealType mean, \/\/ Expected value of mean.\n  RealType variance); \/\/ Expected value of variance.\n```\nReturns the unique value of β that corresponds to a beta distribution with mean *mean* and variance *variance*.\n```\nstatic RealType find_alpha(\n  RealType beta, \/\/ from beta.\n  RealType x, \/\/  x.\n  RealType probability); \/\/ probability cdf\n```\nReturns the value of α that gives: `cdf(beta_distribution<RealType>(alpha, beta), x) == probability`.\n```\nstatic RealType find_beta(\n  RealType alpha, \/\/ alpha.\n  RealType x, \/\/ probability x.\n  RealType probability); \/\/ probability cdf.\n```\nReturns the value of β that gives: `cdf(beta_distribution<RealType>(alpha, beta), x) == probability`.\n\n##### [Non-member Accessor Functions](https:\/\/www.boost.org\/doc\/libs\/1_63_0\/libs\/math\/doc\/html\/math_toolkit\/dist_ref\/dists\/beta_dist.html#math_toolkit.dist_ref.dists.beta_dist.non_member_accessor_functions)\nAll the [usual non-member accessor functions](https:\/\/www.boost.org\/doc\/libs\/1_63_0\/libs\/math\/doc\/html\/math_toolkit\/dist_ref\/nmp.html \"Non-Member Properties\") that are generic to all distributions are supported: [Cumulative Distribution Function](https:\/\/www.boost.org\/doc\/libs\/1_63_0\/libs\/math\/doc\/html\/math_toolkit\/dist_ref\/nmp.html#math_toolkit.dist_ref.nmp.cdf), [Probability Density Function](https:\/\/www.boost.org\/doc\/libs\/1_63_0\/libs\/math\/doc\/html\/math_toolkit\/dist_ref\/nmp.html#math_toolkit.dist_ref.nmp.pdf), [Quantile](https:\/\/www.boost.org\/doc\/libs\/1_63_0\/libs\/math\/doc\/html\/math_toolkit\/dist_ref\/nmp.html#math_toolkit.dist_ref.nmp.quantile), [Hazard Function](https:\/\/www.boost.org\/doc\/libs\/1_63_0\/libs\/math\/doc\/html\/math_toolkit\/dist_ref\/nmp.html#math_toolkit.dist_ref.nmp.hazard), [Cumulative Hazard Function](https:\/\/www.boost.org\/doc\/libs\/1_63_0\/libs\/math\/doc\/html\/math_toolkit\/dist_ref\/nmp.html#math_toolkit.dist_ref.nmp.chf), [mean](https:\/\/www.boost.org\/doc\/libs\/1_63_0\/libs\/math\/doc\/html\/math_toolkit\/dist_ref\/nmp.html#math_toolkit.dist_ref.nmp.mean), [median](https:\/\/www.boost.org\/doc\/libs\/1_63_0\/libs\/math\/doc\/html\/math_toolkit\/dist_ref\/nmp.html#math_toolkit.dist_ref.nmp.median), [mode](https:\/\/www.boost.org\/doc\/libs\/1_63_0\/libs\/math\/doc\/html\/math_toolkit\/dist_ref\/nmp.html#math_toolkit.dist_ref.nmp.mode), [variance](https:\/\/www.boost.org\/doc\/libs\/1_63_0\/libs\/math\/doc\/html\/math_toolkit\/dist_ref\/nmp.html#math_toolkit.dist_ref.nmp.variance), [standard deviation](https:\/\/www.boost.org\/doc\/libs\/1_63_0\/libs\/math\/doc\/html\/math_toolkit\/dist_ref\/nmp.html#math_toolkit.dist_ref.nmp.sd), [skewness](https:\/\/www.boost.org\/doc\/libs\/1_63_0\/libs\/math\/doc\/html\/math_toolkit\/dist_ref\/nmp.html#math_toolkit.dist_ref.nmp.skewness), [kurtosis](https:\/\/www.boost.org\/doc\/libs\/1_63_0\/libs\/math\/doc\/html\/math_toolkit\/dist_ref\/nmp.html#math_toolkit.dist_ref.nmp.kurtosis), [kurtosis\\_excess](https:\/\/www.boost.org\/doc\/libs\/1_63_0\/libs\/math\/doc\/html\/math_toolkit\/dist_ref\/nmp.html#math_toolkit.dist_ref.nmp.kurtosis_excess), [range](https:\/\/www.boost.org\/doc\/libs\/1_63_0\/libs\/math\/doc\/html\/math_toolkit\/dist_ref\/nmp.html#math_toolkit.dist_ref.nmp.range) and [support](https:\/\/www.boost.org\/doc\/libs\/1_63_0\/libs\/math\/doc\/html\/math_toolkit\/dist_ref\/nmp.html#math_toolkit.dist_ref.nmp.support).\n\nThe formulae for calculating these are shown in the table below, and at [Wolfram Mathworld](http:\/\/mathworld.wolfram.com\/BetaDistribution.html).\n\n##### [Applications](https:\/\/www.boost.org\/doc\/libs\/1_63_0\/libs\/math\/doc\/html\/math_toolkit\/dist_ref\/dists\/beta_dist.html#math_toolkit.dist_ref.dists.beta_dist.applications)\nThe beta distribution can be used to model events constrained to take place within an interval defined by a minimum and maximum value: so it is used in project management systems.\n\nIt is also widely used in [Bayesian statistical inference](http:\/\/en.wikipedia.org\/wiki\/Bayesian_inference).\n\n##### [Related distributions](https:\/\/www.boost.org\/doc\/libs\/1_63_0\/libs\/math\/doc\/html\/math_toolkit\/dist_ref\/dists\/beta_dist.html#math_toolkit.dist_ref.dists.beta_dist.related_distributions)\nThe beta distribution with both α \\_\\_space and β = 1 follows a [uniform distribution](http:\/\/en.wikipedia.org\/wiki\/Uniform_distribution_%28continuous%29).\n\nThe [triangular](http:\/\/en.wikipedia.org\/wiki\/Triangular_distribution) is used when less precise information is available.\n\nThe [binomial distribution](http:\/\/en.wikipedia.org\/wiki\/Binomial_distribution) is closely related when α \\_\\_space and β \\_\\_space are integers.\n\nWith integer values of α \\_\\_space and β \\_\\_space the distribution B(i, j) is that of the j-th highest of a sample of i + j + 1 independent random variables uniformly distributed between 0 and 1. The cumulative probability from 0 to x is thus the probability that the j-th highest value is less than x. Or it is the probability that at least i of the random variables are less than x, a probability given by summing over the [Binomial Distribution](https:\/\/www.boost.org\/doc\/libs\/1_63_0\/libs\/math\/doc\/html\/math_toolkit\/dist_ref\/dists\/binomial_dist.html \"Binomial Distribution\") with its p parameter set to x.\n\n##### [Accuracy](https:\/\/www.boost.org\/doc\/libs\/1_63_0\/libs\/math\/doc\/html\/math_toolkit\/dist_ref\/dists\/beta_dist.html#math_toolkit.dist_ref.dists.beta_dist.accuracy)\nThis distribution is implemented using the [beta functions](https:\/\/www.boost.org\/doc\/libs\/1_63_0\/libs\/math\/doc\/html\/math_toolkit\/sf_beta\/beta_function.html \"Beta\") [beta](https:\/\/www.boost.org\/doc\/libs\/1_63_0\/libs\/math\/doc\/html\/math_toolkit\/sf_beta\/beta_function.html \"Beta\") and [incomplete beta functions](https:\/\/www.boost.org\/doc\/libs\/1_63_0\/libs\/math\/doc\/html\/math_toolkit\/sf_beta\/ibeta_function.html \"Incomplete Beta Functions\") [ibeta](https:\/\/www.boost.org\/doc\/libs\/1_63_0\/libs\/math\/doc\/html\/math_toolkit\/sf_beta\/ibeta_function.html \"Incomplete Beta Functions\") and [ibetac](https:\/\/www.boost.org\/doc\/libs\/1_63_0\/libs\/math\/doc\/html\/math_toolkit\/sf_beta\/ibeta_function.html \"Incomplete Beta Functions\"); please refer to these functions for information on accuracy.\n\n##### [Implementation](https:\/\/www.boost.org\/doc\/libs\/1_63_0\/libs\/math\/doc\/html\/math_toolkit\/dist_ref\/dists\/beta_dist.html#math_toolkit.dist_ref.dists.beta_dist.implementation)\nIn the following table *a* and *b* are the parameters α and β, *x* is the random variable, *p* is the probability and *q = 1-p*.\n\n| Function | Implementation Notes |\n|---|---|\n| pdf | f(x;α,β) = xα - 1 (1 - x)β -1 \/ B(α, β) Implemented using [ibeta\\_derivative](https:\/\/www.boost.org\/doc\/libs\/1_63_0\/libs\/math\/doc\/html\/math_toolkit\/sf_beta\/beta_derivative.html \"Derivative of the Incomplete Beta Function\")(a, b, x). |\n| cdf | Using the incomplete beta function [ibeta](https:\/\/www.boost.org\/doc\/libs\/1_63_0\/libs\/math\/doc\/html\/math_toolkit\/sf_beta\/ibeta_function.html \"Incomplete Beta Functions\")(a, b, x) |\n| cdf complement | [ibetac](https:\/\/www.boost.org\/doc\/libs\/1_63_0\/libs\/math\/doc\/html\/math_toolkit\/sf_beta\/ibeta_function.html \"Incomplete Beta Functions\")(a, b, x) |\n| quantile | Using the inverse incomplete beta function [ibeta\\_inv](https:\/\/www.boost.org\/doc\/libs\/1_63_0\/libs\/math\/doc\/html\/math_toolkit\/sf_beta\/ibeta_inv_function.html \"The Incomplete Beta Function Inverses\")(a, b, p) |\n| quantile from the complement | [ibetac\\_inv](https:\/\/www.boost.org\/doc\/libs\/1_63_0\/libs\/math\/doc\/html\/math_toolkit\/sf_beta\/ibeta_inv_function.html \"The Incomplete Beta Function Inverses\")(a, b, q) |\n| mean | `a\/(a+b)` |\n| variance |   |\n| mode |   |\n| skewness |   |\n| kurtosis excess | ![](https:\/\/www.boost.org\/doc\/libs\/1_63_0\/libs\/math\/doc\/equations\/beta_dist_kurtosis.svg) |\n| kurtosis |   |\n| parameter estimation |   |\n| alpha from mean and variance |   |\n| beta from mean and variance |   |\n| The member functions `find_alpha` and `find_beta` from cdf and probability x and **either** `alpha` or `beta` | Implemented in terms of the inverse incomplete beta functions [ibeta\\_inva](https:\/\/www.boost.org\/doc\/libs\/1_63_0\/libs\/math\/doc\/html\/math_toolkit\/sf_beta\/ibeta_inv_function.html \"The Incomplete Beta Function Inverses\"), and [ibeta\\_invb](https:\/\/www.boost.org\/doc\/libs\/1_63_0\/libs\/math\/doc\/html\/math_toolkit\/sf_beta\/ibeta_inv_function.html \"The Incomplete Beta Function Inverses\") respectively. |\n| `find_alpha` |   |\n| `find_beta` |   |\n##### [References](https:\/\/www.boost.org\/doc\/libs\/1_63_0\/libs\/math\/doc\/html\/math_toolkit\/dist_ref\/dists\/beta_dist.html#math_toolkit.dist_ref.dists.beta_dist.references)\n[Wikipedia Beta distribution](http:\/\/en.wikipedia.org\/wiki\/Beta_distribution)\n\n[NIST Exploratory Data Analysis](http:\/\/www.itl.nist.gov\/div898\/handbook\/eda\/section3\/eda366h.htm)\n\n[Wolfram MathWorld](http:\/\/mathworld.wolfram.com\/BetaDistribution.html)\n\n|   |   |\n|---|---|\n|   | Copyright © 2006-2010, 2012-2014 Nikhar Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan Råde, Gautam Sewani, Benjamin Sobotta, Thijs van den Berg, Daryle Walker and Xiaogang Zhang Distributed under the Boost Software License, Version 1.0. (See accompanying file LICENSE\\_1\\_0.txt or copy at <http:\/\/www.boost.org\/LICENSE_1_0.txt>) |\n***\n[![Prev](https:\/\/www.boost.org\/doc\/libs\/1_63_0\/doc\/src\/images\/prev.png)](https:\/\/www.boost.org\/doc\/libs\/1_63_0\/libs\/math\/doc\/html\/math_toolkit\/dist_ref\/dists\/bernoulli_dist.html)[![Up](https:\/\/www.boost.org\/doc\/libs\/1_63_0\/doc\/src\/images\/up.png)](https:\/\/www.boost.org\/doc\/libs\/1_63_0\/libs\/math\/doc\/html\/math_toolkit\/dist_ref\/dists.html)[![Home](https:\/\/www.boost.org\/doc\/libs\/1_63_0\/doc\/src\/images\/home.png)](https:\/\/www.boost.org\/doc\/libs\/1_63_0\/libs\/math\/doc\/html\/index.html)[![Next](https:\/\/www.boost.org\/doc\/libs\/1_63_0\/doc\/src\/images\/next.png)](https:\/\/www.boost.org\/doc\/libs\/1_63_0\/libs\/math\/doc\/html\/math_toolkit\/dist_ref\/dists\/binomial_dist.html)","attrs_readable_markdown":"```\n#include <boost\/math\/distributions\/beta.hpp>\n```\n```\nnamespace boost{ namespace math{\n\n template <class RealType = double,\n           class Policy   = policies::policy<> >\nclass beta_distribution;\n\n\n\n\n\ntemplate <class RealType, class Policy>\nclass beta_distribution\n{\npublic:\n   typedef RealType  value_type;\n   typedef Policy    policy_type;\n   \n   beta_distribution(RealType a, RealType b);\n\n   \n   RealType alpha() const;\n   RealType beta() const;\n\n   \n   static RealType find_alpha(\n     RealType mean, \n     RealType variance); \n\n   static RealType find_beta(\n     RealType mean, \n     RealType variance); \n\n   \n   \n\n   static RealType find_alpha(\n     RealType beta, \n     RealType x, \n     RealType probability); \n\n   static RealType find_beta(\n     RealType alpha, \n     RealType x, \n     RealType probability); \n};\n\n}}\n```\nThe class type `beta_distribution` represents a [beta](http:\/\/en.wikipedia.org\/wiki\/Beta_distribution)  [probability distribution function](http:\/\/en.wikipedia.org\/wiki\/Probability_distribution).\n\nThe [beta distribution](http:\/\/mathworld.wolfram.com\/BetaDistribution.htm)  is used as a [prior distribution](http:\/\/en.wikipedia.org\/wiki\/Prior_distribution) for binomial proportions in [Bayesian analysis](http:\/\/mathworld.wolfram.com\/BayesianAnalysis.html).\n\nSee also: [beta distribution](http:\/\/documents.wolfram.com\/calculationcenter\/v2\/Functions\/ListsMatrices\/Statistics\/BetaDistribution.html) and [Bayesian statistics](http:\/\/en.wikipedia.org\/wiki\/Bayesian_statistics).\n\nHow the beta distribution is used for [Bayesian analysis of one parameter models](http:\/\/home.uchicago.edu\/~grynav\/bayes\/ABSLec5.ppt) is discussed by Jeff Grynaviski.\n\nThe [probability density function PDF](http:\/\/en.wikipedia.org\/wiki\/Probability_density_function) for the [beta distribution](http:\/\/en.wikipedia.org\/wiki\/Beta_distribution) defined on the interval \\[0,1\\] is given by:\n\nf(x;α,β) = xα - 1 (1 - x)β -1 \/ B(α, β)\n\nwhere B(α, β) is the [beta function](http:\/\/en.wikipedia.org\/wiki\/Beta_function), implemented in this library as [beta](https:\/\/www.boost.org\/doc\/libs\/1_63_0\/libs\/math\/doc\/html\/math_toolkit\/sf_beta\/beta_function.html \"Beta\"). Division by the beta function ensures that the pdf is normalized to the range zero to unity.\n\nThe following graph illustrates examples of the pdf for various values of the shape parameters. Note the α = β = 2 (blue line) is dome-shaped, and might be approximated by a symmetrical triangular distribution.\n\n![](https:\/\/www.boost.org\/doc\/libs\/1_63_0\/libs\/math\/doc\/graphs\/beta_pdf.svg)\n\nIf α = β = 1, then it is a \\_\\_space [uniform distribution](http:\/\/en.wikipedia.org\/wiki\/Uniform_distribution_%28continuous%29), equal to unity in the entire interval x = 0 to 1. If α \\_\\_space and β \\_\\_space are \\< 1, then the pdf is U-shaped. If α != β, then the shape is asymmetric and could be approximated by a triangle whose apex is away from the centre (where x = half).\n\n##### [Member Functions](https:\/\/www.boost.org\/doc\/libs\/1_63_0\/libs\/math\/doc\/html\/math_toolkit\/dist_ref\/dists\/beta_dist.html#math_toolkit.dist_ref.dists.beta_dist.member_functions)\n###### [Constructor](https:\/\/www.boost.org\/doc\/libs\/1_63_0\/libs\/math\/doc\/html\/math_toolkit\/dist_ref\/dists\/beta_dist.html#math_toolkit.dist_ref.dists.beta_dist.constructor)\n```\nbeta_distribution(RealType alpha, RealType beta);\n```\nConstructs a beta distribution with shape parameters *alpha* and *beta*.\n\nRequires alpha,beta \\> 0,otherwise [domain\\_error](https:\/\/www.boost.org\/doc\/libs\/1_63_0\/libs\/math\/doc\/html\/math_toolkit\/error_handling.html#math_toolkit.error_handling.domain_error) is called. Note that technically the beta distribution is defined for alpha,beta \\>= 0, but it's not clear whether any program can actually make use of that latitude or how many of the non-member functions can be usefully defined in that case. Therefore for now, we regard it as an error if alpha or beta is zero.\n\nFor example:\n```\nbeta_distribution<> mybeta(2, 5);\n```\nConstructs a the beta distribution with alpha=2 and beta=5 (shown in yellow in the graph above).\n\n###### [Parameter Accessors](https:\/\/www.boost.org\/doc\/libs\/1_63_0\/libs\/math\/doc\/html\/math_toolkit\/dist_ref\/dists\/beta_dist.html#math_toolkit.dist_ref.dists.beta_dist.parameter_accessors)\n```\nRealType alpha() const;\n```\nReturns the parameter *alpha* from which this distribution was constructed.\n```\nRealType beta() const;\n```\nReturns the parameter *beta* from which this distribution was constructed.\n\nSo for example:\n```\nbeta_distribution<> mybeta(2, 5);\nassert(mybeta.alpha() == 2.);  \nassert(mybeta.beta() == 5.);\n```\n##### [Parameter Estimators](https:\/\/www.boost.org\/doc\/libs\/1_63_0\/libs\/math\/doc\/html\/math_toolkit\/dist_ref\/dists\/beta_dist.html#math_toolkit.dist_ref.dists.beta_dist.parameter_estimators)\nTwo pairs of parameter estimators are provided.\n\nOne estimates either α \\_\\_space or β \\_\\_space from presumed-known mean and variance.\n\nThe other pair estimates either α \\_\\_space or β \\_\\_space from the cdf and x.\n\nIt is also possible to estimate α \\_\\_space and β \\_\\_space from 'known' mode & quantile. For example, calculators are provided by the [Pooled Prevalence Calculator](http:\/\/www.ausvet.com.au\/pprev\/content.php?page=PPscript) and [Beta Buster](http:\/\/www.epi.ucdavis.edu\/diagnostictests\/betabuster.html) but this is not yet implemented here.\n```\nstatic RealType find_alpha(\n  RealType mean, \n  RealType variance);\n```\nReturns the unique value of α that corresponds to a beta distribution with mean *mean* and variance *variance*.\n```\nstatic RealType find_beta(\n  RealType mean, \n  RealType variance);\n```\nReturns the unique value of β that corresponds to a beta distribution with mean *mean* and variance *variance*.\n```\nstatic RealType find_alpha(\n  RealType beta, \n  RealType x, \n  RealType probability);\n```\nReturns the value of α that gives: `cdf(beta_distribution<RealType>(alpha, beta), x) == probability`.\n```\nstatic RealType find_beta(\n  RealType alpha, \n  RealType x, \n  RealType probability);\n```\nReturns the value of β that gives: `cdf(beta_distribution<RealType>(alpha, beta), x) == probability`.\n\n##### [Non-member Accessor Functions](https:\/\/www.boost.org\/doc\/libs\/1_63_0\/libs\/math\/doc\/html\/math_toolkit\/dist_ref\/dists\/beta_dist.html#math_toolkit.dist_ref.dists.beta_dist.non_member_accessor_functions)\nAll the [usual non-member accessor functions](https:\/\/www.boost.org\/doc\/libs\/1_63_0\/libs\/math\/doc\/html\/math_toolkit\/dist_ref\/nmp.html \"Non-Member Properties\") that are generic to all distributions are supported: [Cumulative Distribution Function](https:\/\/www.boost.org\/doc\/libs\/1_63_0\/libs\/math\/doc\/html\/math_toolkit\/dist_ref\/nmp.html#math_toolkit.dist_ref.nmp.cdf), [Probability Density Function](https:\/\/www.boost.org\/doc\/libs\/1_63_0\/libs\/math\/doc\/html\/math_toolkit\/dist_ref\/nmp.html#math_toolkit.dist_ref.nmp.pdf), [Quantile](https:\/\/www.boost.org\/doc\/libs\/1_63_0\/libs\/math\/doc\/html\/math_toolkit\/dist_ref\/nmp.html#math_toolkit.dist_ref.nmp.quantile), [Hazard Function](https:\/\/www.boost.org\/doc\/libs\/1_63_0\/libs\/math\/doc\/html\/math_toolkit\/dist_ref\/nmp.html#math_toolkit.dist_ref.nmp.hazard), [Cumulative Hazard Function](https:\/\/www.boost.org\/doc\/libs\/1_63_0\/libs\/math\/doc\/html\/math_toolkit\/dist_ref\/nmp.html#math_toolkit.dist_ref.nmp.chf), [mean](https:\/\/www.boost.org\/doc\/libs\/1_63_0\/libs\/math\/doc\/html\/math_toolkit\/dist_ref\/nmp.html#math_toolkit.dist_ref.nmp.mean), [median](https:\/\/www.boost.org\/doc\/libs\/1_63_0\/libs\/math\/doc\/html\/math_toolkit\/dist_ref\/nmp.html#math_toolkit.dist_ref.nmp.median), [mode](https:\/\/www.boost.org\/doc\/libs\/1_63_0\/libs\/math\/doc\/html\/math_toolkit\/dist_ref\/nmp.html#math_toolkit.dist_ref.nmp.mode), [variance](https:\/\/www.boost.org\/doc\/libs\/1_63_0\/libs\/math\/doc\/html\/math_toolkit\/dist_ref\/nmp.html#math_toolkit.dist_ref.nmp.variance), [standard deviation](https:\/\/www.boost.org\/doc\/libs\/1_63_0\/libs\/math\/doc\/html\/math_toolkit\/dist_ref\/nmp.html#math_toolkit.dist_ref.nmp.sd), [skewness](https:\/\/www.boost.org\/doc\/libs\/1_63_0\/libs\/math\/doc\/html\/math_toolkit\/dist_ref\/nmp.html#math_toolkit.dist_ref.nmp.skewness), [kurtosis](https:\/\/www.boost.org\/doc\/libs\/1_63_0\/libs\/math\/doc\/html\/math_toolkit\/dist_ref\/nmp.html#math_toolkit.dist_ref.nmp.kurtosis), [kurtosis\\_excess](https:\/\/www.boost.org\/doc\/libs\/1_63_0\/libs\/math\/doc\/html\/math_toolkit\/dist_ref\/nmp.html#math_toolkit.dist_ref.nmp.kurtosis_excess), [range](https:\/\/www.boost.org\/doc\/libs\/1_63_0\/libs\/math\/doc\/html\/math_toolkit\/dist_ref\/nmp.html#math_toolkit.dist_ref.nmp.range) and [support](https:\/\/www.boost.org\/doc\/libs\/1_63_0\/libs\/math\/doc\/html\/math_toolkit\/dist_ref\/nmp.html#math_toolkit.dist_ref.nmp.support).\n\nThe formulae for calculating these are shown in the table below, and at [Wolfram Mathworld](http:\/\/mathworld.wolfram.com\/BetaDistribution.html).\n\n##### [Applications](https:\/\/www.boost.org\/doc\/libs\/1_63_0\/libs\/math\/doc\/html\/math_toolkit\/dist_ref\/dists\/beta_dist.html#math_toolkit.dist_ref.dists.beta_dist.applications)\nThe beta distribution can be used to model events constrained to take place within an interval defined by a minimum and maximum value: so it is used in project management systems.\n\nIt is also widely used in [Bayesian statistical inference](http:\/\/en.wikipedia.org\/wiki\/Bayesian_inference).\n\n##### [Related distributions](https:\/\/www.boost.org\/doc\/libs\/1_63_0\/libs\/math\/doc\/html\/math_toolkit\/dist_ref\/dists\/beta_dist.html#math_toolkit.dist_ref.dists.beta_dist.related_distributions)\nThe beta distribution with both α \\_\\_space and β = 1 follows a [uniform distribution](http:\/\/en.wikipedia.org\/wiki\/Uniform_distribution_%28continuous%29).\n\nThe [triangular](http:\/\/en.wikipedia.org\/wiki\/Triangular_distribution) is used when less precise information is available.\n\nThe [binomial distribution](http:\/\/en.wikipedia.org\/wiki\/Binomial_distribution) is closely related when α \\_\\_space and β \\_\\_space are integers.\n\nWith integer values of α \\_\\_space and β \\_\\_space the distribution B(i, j) is that of the j-th highest of a sample of i + j + 1 independent random variables uniformly distributed between 0 and 1. The cumulative probability from 0 to x is thus the probability that the j-th highest value is less than x. Or it is the probability that at least i of the random variables are less than x, a probability given by summing over the [Binomial Distribution](https:\/\/www.boost.org\/doc\/libs\/1_63_0\/libs\/math\/doc\/html\/math_toolkit\/dist_ref\/dists\/binomial_dist.html \"Binomial Distribution\") with its p parameter set to x.\n\n##### [Accuracy](https:\/\/www.boost.org\/doc\/libs\/1_63_0\/libs\/math\/doc\/html\/math_toolkit\/dist_ref\/dists\/beta_dist.html#math_toolkit.dist_ref.dists.beta_dist.accuracy)\nThis distribution is implemented using the [beta functions](https:\/\/www.boost.org\/doc\/libs\/1_63_0\/libs\/math\/doc\/html\/math_toolkit\/sf_beta\/beta_function.html \"Beta\") [beta](https:\/\/www.boost.org\/doc\/libs\/1_63_0\/libs\/math\/doc\/html\/math_toolkit\/sf_beta\/beta_function.html \"Beta\") and [incomplete beta functions](https:\/\/www.boost.org\/doc\/libs\/1_63_0\/libs\/math\/doc\/html\/math_toolkit\/sf_beta\/ibeta_function.html \"Incomplete Beta Functions\") [ibeta](https:\/\/www.boost.org\/doc\/libs\/1_63_0\/libs\/math\/doc\/html\/math_toolkit\/sf_beta\/ibeta_function.html \"Incomplete Beta Functions\") and [ibetac](https:\/\/www.boost.org\/doc\/libs\/1_63_0\/libs\/math\/doc\/html\/math_toolkit\/sf_beta\/ibeta_function.html \"Incomplete Beta Functions\"); please refer to these functions for information on accuracy.\n\n##### [Implementation](https:\/\/www.boost.org\/doc\/libs\/1_63_0\/libs\/math\/doc\/html\/math_toolkit\/dist_ref\/dists\/beta_dist.html#math_toolkit.dist_ref.dists.beta_dist.implementation)\nIn the following table *a* and *b* are the parameters α and β, *x* is the random variable, *p* is the probability and *q = 1-p*.\n\n| Function | Implementation Notes |\n|---|---|\n| pdf | f(x;α,β) = xα - 1 (1 - x)β -1 \/ B(α, β) Implemented using [ibeta\\_derivative](https:\/\/www.boost.org\/doc\/libs\/1_63_0\/libs\/math\/doc\/html\/math_toolkit\/sf_beta\/beta_derivative.html \"Derivative of the Incomplete Beta Function\")(a, b, x). |\n| cdf | Using the incomplete beta function [ibeta](https:\/\/www.boost.org\/doc\/libs\/1_63_0\/libs\/math\/doc\/html\/math_toolkit\/sf_beta\/ibeta_function.html \"Incomplete Beta Functions\")(a, b, x) |\n| cdf complement | [ibetac](https:\/\/www.boost.org\/doc\/libs\/1_63_0\/libs\/math\/doc\/html\/math_toolkit\/sf_beta\/ibeta_function.html \"Incomplete Beta Functions\")(a, b, x) |\n| quantile | Using the inverse incomplete beta function [ibeta\\_inv](https:\/\/www.boost.org\/doc\/libs\/1_63_0\/libs\/math\/doc\/html\/math_toolkit\/sf_beta\/ibeta_inv_function.html \"The Incomplete Beta Function Inverses\")(a, b, p) |\n| quantile from the complement | [ibetac\\_inv](https:\/\/www.boost.org\/doc\/libs\/1_63_0\/libs\/math\/doc\/html\/math_toolkit\/sf_beta\/ibeta_inv_function.html \"The Incomplete Beta Function Inverses\")(a, b, q) |\n| mean | `a\/(a+b)` |\n| variance |   |\n| mode |   |\n| skewness |   |\n| kurtosis excess | ![](https:\/\/www.boost.org\/doc\/libs\/1_63_0\/libs\/math\/doc\/equations\/beta_dist_kurtosis.svg) |\n| kurtosis |   |\n| parameter estimation |   |\n| alpha from mean and variance |   |\n| beta from mean and variance |   |\n| The member functions `find_alpha` and `find_beta` from cdf and probability x and **either** `alpha` or `beta` | Implemented in terms of the inverse incomplete beta functions [ibeta\\_inva](https:\/\/www.boost.org\/doc\/libs\/1_63_0\/libs\/math\/doc\/html\/math_toolkit\/sf_beta\/ibeta_inv_function.html \"The Incomplete Beta Function Inverses\"), and [ibeta\\_invb](https:\/\/www.boost.org\/doc\/libs\/1_63_0\/libs\/math\/doc\/html\/math_toolkit\/sf_beta\/ibeta_inv_function.html \"The Incomplete Beta Function Inverses\") respectively. |\n| `find_alpha` |   |\n| `find_beta` |   |\n##### [References](https:\/\/www.boost.org\/doc\/libs\/1_63_0\/libs\/math\/doc\/html\/math_toolkit\/dist_ref\/dists\/beta_dist.html#math_toolkit.dist_ref.dists.beta_dist.references)\n[Wikipedia Beta distribution](http:\/\/en.wikipedia.org\/wiki\/Beta_distribution)\n\n[NIST Exploratory Data Analysis](http:\/\/www.itl.nist.gov\/div898\/handbook\/eda\/section3\/eda366h.htm)\n\n[Wolfram MathWorld](http:\/\/mathworld.wolfram.com\/BetaDistribution.html)","meta_canonical":"org,boost!www,\/doc\/libs\/latest\/libs\/math\/doc\/html\/math_toolkit\/dist_ref\/dists\/beta_dist.html s443"}

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Boilerpipe Text	#include < boost / math / distributions / beta . hpp > namespace boost { namespace math { template < class RealType = double , class Policy = policies::policy<> > class beta_distribution ; template < class RealType , class Policy > class beta_distribution { public : typedef RealType value_type ; typedef Policy policy_type ; beta_distribution ( RealType a , RealType b ); RealType alpha () const ; RealType beta () const ; static RealType find_alpha ( RealType mean , RealType variance ); static RealType find_beta ( RealType mean , RealType variance ); static RealType find_alpha ( RealType beta , RealType x , RealType probability ); static RealType find_beta ( RealType alpha , RealType x , RealType probability ); }; }} The class type beta_distribution represents a beta probability distribution function . The beta distribution is used as a prior distribution for binomial proportions in Bayesian analysis . See also: beta distribution and Bayesian statistics . How the beta distribution is used for Bayesian analysis of one parameter models is discussed by Jeff Grynaviski. The probability density function PDF for the beta distribution defined on the interval [0,1] is given by: f(x;α,β) = x α - 1 (1 - x) β -1 / B(α, β) where B(α, β) is the beta function , implemented in this library as beta . Division by the beta function ensures that the pdf is normalized to the range zero to unity. The following graph illustrates examples of the pdf for various values of the shape parameters. Note the α = β = 2 (blue line) is dome-shaped, and might be approximated by a symmetrical triangular distribution. If α = β = 1, then it is a __space uniform distribution , equal to unity in the entire interval x = 0 to 1. If α __space and β __space are < 1, then the pdf is U-shaped. If α != β, then the shape is asymmetric and could be approximated by a triangle whose apex is away from the centre (where x = half). Member Functions Constructor beta_distribution ( RealType alpha , RealType beta ); Constructs a beta distribution with shape parameters alpha and beta . Requires alpha,beta > 0,otherwise domain_error is called. Note that technically the beta distribution is defined for alpha,beta >= 0, but it's not clear whether any program can actually make use of that latitude or how many of the non-member functions can be usefully defined in that case. Therefore for now, we regard it as an error if alpha or beta is zero. For example: beta_distribution <> mybeta ( 2 , 5 ); Constructs a the beta distribution with alpha=2 and beta=5 (shown in yellow in the graph above). Parameter Accessors RealType alpha () const ; Returns the parameter alpha from which this distribution was constructed. RealType beta () const ; Returns the parameter beta from which this distribution was constructed. So for example: beta_distribution <> mybeta ( 2 , 5 ); assert ( mybeta . alpha () == 2. ); assert ( mybeta . beta () == 5. ); Parameter Estimators Two pairs of parameter estimators are provided. One estimates either α __space or β __space from presumed-known mean and variance. The other pair estimates either α __space or β __space from the cdf and x. It is also possible to estimate α __space and β __space from 'known' mode & quantile. For example, calculators are provided by the Pooled Prevalence Calculator and Beta Buster but this is not yet implemented here. static RealType find_alpha ( RealType mean , RealType variance ); Returns the unique value of α that corresponds to a beta distribution with mean mean and variance variance . static RealType find_beta ( RealType mean , RealType variance ); Returns the unique value of β that corresponds to a beta distribution with mean mean and variance variance . static RealType find_alpha ( RealType beta , RealType x , RealType probability ); Returns the value of α that gives: cdf ( beta_distribution < RealType >( alpha , beta ), x ) == probability . static RealType find_beta ( RealType alpha , RealType x , RealType probability ); Returns the value of β that gives: cdf ( beta_distribution < RealType >( alpha , beta ), x ) == probability . Non-member Accessor Functions All the usual non-member accessor functions that are generic to all distributions are supported: Cumulative Distribution Function , Probability Density Function , Quantile , Hazard Function , Cumulative Hazard Function , mean , median , mode , variance , standard deviation , skewness , kurtosis , kurtosis_excess , range and support . The formulae for calculating these are shown in the table below, and at Wolfram Mathworld . Applications The beta distribution can be used to model events constrained to take place within an interval defined by a minimum and maximum value: so it is used in project management systems. It is also widely used in Bayesian statistical inference . Related distributions The beta distribution with both α __space and β = 1 follows a uniform distribution . The triangular is used when less precise information is available. The binomial distribution is closely related when α __space and β __space are integers. With integer values of α __space and β __space the distribution B(i, j) is that of the j-th highest of a sample of i + j + 1 independent random variables uniformly distributed between 0 and 1. The cumulative probability from 0 to x is thus the probability that the j-th highest value is less than x. Or it is the probability that at least i of the random variables are less than x, a probability given by summing over the Binomial Distribution with its p parameter set to x. Accuracy This distribution is implemented using the beta functions beta and incomplete beta functions ibeta and ibetac ; please refer to these functions for information on accuracy. Implementation In the following table a and b are the parameters α and β, x is the random variable, p is the probability and q = 1-p . Function Implementation Notes pdf f(x;α,β) = x α - 1 (1 - x) β -1 / B(α, β) Implemented using ibeta_derivative (a, b, x). cdf Using the incomplete beta function ibeta (a, b, x) cdf complement ibetac (a, b, x) quantile Using the inverse incomplete beta function ibeta_inv (a, b, p) quantile from the complement ibetac_inv (a, b, q) mean a /( a + b ) variance a * b / ( a + b )^ 2 * ( a + b + 1 ) mode ( a - 1 ) / ( a + b - 2 ) skewness 2 ( b - a ) sqrt ( a + b + 1 )/( a + b + 2 ) * sqrt ( a * b ) kurtosis excess kurtosis kurtosis + 3 parameter estimation alpha from mean and variance mean * (( ( mean * ( 1 - mean )) / variance )- 1 ) beta from mean and variance ( 1 - mean ) * ((( mean * ( 1 - mean )) / variance )- 1 ) The member functions find_alpha and find_beta from cdf and probability x and either alpha or beta Implemented in terms of the inverse incomplete beta functions ibeta_inva , and ibeta_invb respectively. find_alpha ibeta_inva ( beta , x , probability ) find_beta ibeta_invb ( alpha , x , probability ) References Wikipedia Beta distribution NIST Exploratory Data Analysis Wolfram MathWorld
Markdown	# [![Boost C++ Libraries](https://www.boost.org/static/img/original_docs/space.png) Boost C++ Libraries](https://www.boost.org/) "...one of the most highly regarded and expertly designed C++ library projects in the world." — [Herb Sutter](https://herbsutter.com/) and [Andrei Alexandrescu](http://en.wikipedia.org/wiki/Andrei_Alexandrescu), [C++ Coding Standards](https://books.google.com/books/about/C++_Coding_Standards.html?id=mmjVIC6WolgC) Search... This is an older version of Boost and was released in 2016. The [current version](https://www.boost.org/doc/libs/latest/libs/math/doc/html/math_toolkit/dist_ref/dists/beta_dist.html) is 1.90.0. [![Prev](https://www.boost.org/doc/libs/1_63_0/doc/src/images/prev.png)](https://www.boost.org/doc/libs/1_63_0/libs/math/doc/html/math_toolkit/dist_ref/dists/bernoulli_dist.html)[![Up](https://www.boost.org/doc/libs/1_63_0/doc/src/images/up.png)](https://www.boost.org/doc/libs/1_63_0/libs/math/doc/html/math_toolkit/dist_ref/dists.html)[![Home](https://www.boost.org/doc/libs/1_63_0/doc/src/images/home.png)](https://www.boost.org/doc/libs/1_63_0/libs/math/doc/html/index.html)[![Next](https://www.boost.org/doc/libs/1_63_0/doc/src/images/next.png)](https://www.boost.org/doc/libs/1_63_0/libs/math/doc/html/math_toolkit/dist_ref/dists/binomial_dist.html) #### [Beta Distribution](https://www.boost.org/doc/libs/1_63_0/libs/math/doc/html/math_toolkit/dist_ref/dists/beta_dist.html "Beta Distribution") ``` #include <boost/math/distributions/beta.hpp> ``` ``` namespace boost{ namespace math{ template <class RealType = double, class Policy = policies::policy<> > class beta_distribution; // typedef beta_distribution<double> beta; // Note that this is deliberately NOT provided, // to avoid a clash with the function name beta. template <class RealType, class Policy> class beta_distribution { public: typedef RealType value_type; typedef Policy policy_type; // Constructor from two shape parameters, alpha & beta: beta_distribution(RealType a, RealType b); // Parameter accessors: RealType alpha() const; RealType beta() const; // Parameter estimators of alpha or beta from mean and variance. static RealType find_alpha( RealType mean, // Expected value of mean. RealType variance); // Expected value of variance. static RealType find_beta( RealType mean, // Expected value of mean. RealType variance); // Expected value of variance. // Parameter estimators from // either alpha or beta, and x and probability. static RealType find_alpha( RealType beta, // from beta. RealType x, // x. RealType probability); // cdf static RealType find_beta( RealType alpha, // alpha. RealType x, // probability x. RealType probability); // probability cdf. }; }} // namespaces ``` The class type `beta_distribution` represents a [beta](http://en.wikipedia.org/wiki/Beta_distribution) [probability distribution function](http://en.wikipedia.org/wiki/Probability_distribution). The [beta distribution](http://mathworld.wolfram.com/BetaDistribution.htm) is used as a [prior distribution](http://en.wikipedia.org/wiki/Prior_distribution) for binomial proportions in [Bayesian analysis](http://mathworld.wolfram.com/BayesianAnalysis.html). See also: [beta distribution](http://documents.wolfram.com/calculationcenter/v2/Functions/ListsMatrices/Statistics/BetaDistribution.html) and [Bayesian statistics](http://en.wikipedia.org/wiki/Bayesian_statistics). How the beta distribution is used for [Bayesian analysis of one parameter models](http://home.uchicago.edu/~grynav/bayes/ABSLec5.ppt) is discussed by Jeff Grynaviski. The [probability density function PDF](http://en.wikipedia.org/wiki/Probability_density_function) for the [beta distribution](http://en.wikipedia.org/wiki/Beta_distribution) defined on the interval \[0,1\] is given by: f(x;α,β) = xα - 1 (1 - x)β -1 / B(α, β) where B(α, β) is the [beta function](http://en.wikipedia.org/wiki/Beta_function), implemented in this library as [beta](https://www.boost.org/doc/libs/1_63_0/libs/math/doc/html/math_toolkit/sf_beta/beta_function.html "Beta"). Division by the beta function ensures that the pdf is normalized to the range zero to unity. The following graph illustrates examples of the pdf for various values of the shape parameters. Note the α = β = 2 (blue line) is dome-shaped, and might be approximated by a symmetrical triangular distribution. ![](https://www.boost.org/doc/libs/1_63_0/libs/math/doc/graphs/beta_pdf.svg) If α = β = 1, then it is a \_\_space [uniform distribution](http://en.wikipedia.org/wiki/Uniform_distribution_%28continuous%29), equal to unity in the entire interval x = 0 to 1. If α \_\_space and β \_\_space are \< 1, then the pdf is U-shaped. If α != β, then the shape is asymmetric and could be approximated by a triangle whose apex is away from the centre (where x = half). ##### [Member Functions](https://www.boost.org/doc/libs/1_63_0/libs/math/doc/html/math_toolkit/dist_ref/dists/beta_dist.html#math_toolkit.dist_ref.dists.beta_dist.member_functions) ###### [Constructor](https://www.boost.org/doc/libs/1_63_0/libs/math/doc/html/math_toolkit/dist_ref/dists/beta_dist.html#math_toolkit.dist_ref.dists.beta_dist.constructor) ``` beta_distribution(RealType alpha, RealType beta); ``` Constructs a beta distribution with shape parameters alpha and beta. Requires alpha,beta \> 0,otherwise [domain\_error](https://www.boost.org/doc/libs/1_63_0/libs/math/doc/html/math_toolkit/error_handling.html#math_toolkit.error_handling.domain_error) is called. Note that technically the beta distribution is defined for alpha,beta \>= 0, but it's not clear whether any program can actually make use of that latitude or how many of the non-member functions can be usefully defined in that case. Therefore for now, we regard it as an error if alpha or beta is zero. For example: ``` beta_distribution<> mybeta(2, 5); ``` Constructs a the beta distribution with alpha=2 and beta=5 (shown in yellow in the graph above). ###### [Parameter Accessors](https://www.boost.org/doc/libs/1_63_0/libs/math/doc/html/math_toolkit/dist_ref/dists/beta_dist.html#math_toolkit.dist_ref.dists.beta_dist.parameter_accessors) ``` RealType alpha() const; ``` Returns the parameter alpha from which this distribution was constructed. ``` RealType beta() const; ``` Returns the parameter beta from which this distribution was constructed. So for example: ``` beta_distribution<> mybeta(2, 5); assert(mybeta.alpha() == 2.); // mybeta.alpha() returns 2 assert(mybeta.beta() == 5.); // mybeta.beta() returns 5 ``` ##### [Parameter Estimators](https://www.boost.org/doc/libs/1_63_0/libs/math/doc/html/math_toolkit/dist_ref/dists/beta_dist.html#math_toolkit.dist_ref.dists.beta_dist.parameter_estimators) Two pairs of parameter estimators are provided. One estimates either α \_\_space or β \_\_space from presumed-known mean and variance. The other pair estimates either α \_\_space or β \_\_space from the cdf and x. It is also possible to estimate α \_\_space and β \_\_space from 'known' mode & quantile. For example, calculators are provided by the [Pooled Prevalence Calculator](http://www.ausvet.com.au/pprev/content.php?page=PPscript) and [Beta Buster](http://www.epi.ucdavis.edu/diagnostictests/betabuster.html) but this is not yet implemented here. ``` static RealType find_alpha( RealType mean, // Expected value of mean. RealType variance); // Expected value of variance. ``` Returns the unique value of α that corresponds to a beta distribution with mean mean and variance variance. ``` static RealType find_beta( RealType mean, // Expected value of mean. RealType variance); // Expected value of variance. ``` Returns the unique value of β that corresponds to a beta distribution with mean mean and variance variance. ``` static RealType find_alpha( RealType beta, // from beta. RealType x, // x. RealType probability); // probability cdf ``` Returns the value of α that gives: `cdf(beta_distribution<RealType>(alpha, beta), x) == probability`. ``` static RealType find_beta( RealType alpha, // alpha. RealType x, // probability x. RealType probability); // probability cdf. ``` Returns the value of β that gives: `cdf(beta_distribution<RealType>(alpha, beta), x) == probability`. ##### [Non-member Accessor Functions](https://www.boost.org/doc/libs/1_63_0/libs/math/doc/html/math_toolkit/dist_ref/dists/beta_dist.html#math_toolkit.dist_ref.dists.beta_dist.non_member_accessor_functions) All the [usual non-member accessor functions](https://www.boost.org/doc/libs/1_63_0/libs/math/doc/html/math_toolkit/dist_ref/nmp.html "Non-Member Properties") that are generic to all distributions are supported: [Cumulative Distribution Function](https://www.boost.org/doc/libs/1_63_0/libs/math/doc/html/math_toolkit/dist_ref/nmp.html#math_toolkit.dist_ref.nmp.cdf), [Probability Density Function](https://www.boost.org/doc/libs/1_63_0/libs/math/doc/html/math_toolkit/dist_ref/nmp.html#math_toolkit.dist_ref.nmp.pdf), [Quantile](https://www.boost.org/doc/libs/1_63_0/libs/math/doc/html/math_toolkit/dist_ref/nmp.html#math_toolkit.dist_ref.nmp.quantile), [Hazard Function](https://www.boost.org/doc/libs/1_63_0/libs/math/doc/html/math_toolkit/dist_ref/nmp.html#math_toolkit.dist_ref.nmp.hazard), [Cumulative Hazard Function](https://www.boost.org/doc/libs/1_63_0/libs/math/doc/html/math_toolkit/dist_ref/nmp.html#math_toolkit.dist_ref.nmp.chf), [mean](https://www.boost.org/doc/libs/1_63_0/libs/math/doc/html/math_toolkit/dist_ref/nmp.html#math_toolkit.dist_ref.nmp.mean), [median](https://www.boost.org/doc/libs/1_63_0/libs/math/doc/html/math_toolkit/dist_ref/nmp.html#math_toolkit.dist_ref.nmp.median), [mode](https://www.boost.org/doc/libs/1_63_0/libs/math/doc/html/math_toolkit/dist_ref/nmp.html#math_toolkit.dist_ref.nmp.mode), [variance](https://www.boost.org/doc/libs/1_63_0/libs/math/doc/html/math_toolkit/dist_ref/nmp.html#math_toolkit.dist_ref.nmp.variance), [standard deviation](https://www.boost.org/doc/libs/1_63_0/libs/math/doc/html/math_toolkit/dist_ref/nmp.html#math_toolkit.dist_ref.nmp.sd), [skewness](https://www.boost.org/doc/libs/1_63_0/libs/math/doc/html/math_toolkit/dist_ref/nmp.html#math_toolkit.dist_ref.nmp.skewness), [kurtosis](https://www.boost.org/doc/libs/1_63_0/libs/math/doc/html/math_toolkit/dist_ref/nmp.html#math_toolkit.dist_ref.nmp.kurtosis), [kurtosis\_excess](https://www.boost.org/doc/libs/1_63_0/libs/math/doc/html/math_toolkit/dist_ref/nmp.html#math_toolkit.dist_ref.nmp.kurtosis_excess), [range](https://www.boost.org/doc/libs/1_63_0/libs/math/doc/html/math_toolkit/dist_ref/nmp.html#math_toolkit.dist_ref.nmp.range) and [support](https://www.boost.org/doc/libs/1_63_0/libs/math/doc/html/math_toolkit/dist_ref/nmp.html#math_toolkit.dist_ref.nmp.support). The formulae for calculating these are shown in the table below, and at [Wolfram Mathworld](http://mathworld.wolfram.com/BetaDistribution.html). ##### [Applications](https://www.boost.org/doc/libs/1_63_0/libs/math/doc/html/math_toolkit/dist_ref/dists/beta_dist.html#math_toolkit.dist_ref.dists.beta_dist.applications) The beta distribution can be used to model events constrained to take place within an interval defined by a minimum and maximum value: so it is used in project management systems. It is also widely used in [Bayesian statistical inference](http://en.wikipedia.org/wiki/Bayesian_inference). ##### [Related distributions](https://www.boost.org/doc/libs/1_63_0/libs/math/doc/html/math_toolkit/dist_ref/dists/beta_dist.html#math_toolkit.dist_ref.dists.beta_dist.related_distributions) The beta distribution with both α \_\_space and β = 1 follows a [uniform distribution](http://en.wikipedia.org/wiki/Uniform_distribution_%28continuous%29). The [triangular](http://en.wikipedia.org/wiki/Triangular_distribution) is used when less precise information is available. The [binomial distribution](http://en.wikipedia.org/wiki/Binomial_distribution) is closely related when α \_\_space and β \_\_space are integers. With integer values of α \_\_space and β \_\_space the distribution B(i, j) is that of the j-th highest of a sample of i + j + 1 independent random variables uniformly distributed between 0 and 1. The cumulative probability from 0 to x is thus the probability that the j-th highest value is less than x. Or it is the probability that at least i of the random variables are less than x, a probability given by summing over the [Binomial Distribution](https://www.boost.org/doc/libs/1_63_0/libs/math/doc/html/math_toolkit/dist_ref/dists/binomial_dist.html "Binomial Distribution") with its p parameter set to x. ##### [Accuracy](https://www.boost.org/doc/libs/1_63_0/libs/math/doc/html/math_toolkit/dist_ref/dists/beta_dist.html#math_toolkit.dist_ref.dists.beta_dist.accuracy) This distribution is implemented using the [beta functions](https://www.boost.org/doc/libs/1_63_0/libs/math/doc/html/math_toolkit/sf_beta/beta_function.html "Beta") [beta](https://www.boost.org/doc/libs/1_63_0/libs/math/doc/html/math_toolkit/sf_beta/beta_function.html "Beta") and [incomplete beta functions](https://www.boost.org/doc/libs/1_63_0/libs/math/doc/html/math_toolkit/sf_beta/ibeta_function.html "Incomplete Beta Functions") [ibeta](https://www.boost.org/doc/libs/1_63_0/libs/math/doc/html/math_toolkit/sf_beta/ibeta_function.html "Incomplete Beta Functions") and [ibetac](https://www.boost.org/doc/libs/1_63_0/libs/math/doc/html/math_toolkit/sf_beta/ibeta_function.html "Incomplete Beta Functions"); please refer to these functions for information on accuracy. ##### [Implementation](https://www.boost.org/doc/libs/1_63_0/libs/math/doc/html/math_toolkit/dist_ref/dists/beta_dist.html#math_toolkit.dist_ref.dists.beta_dist.implementation) In the following table a and b are the parameters α and β, x is the random variable, p is the probability and q = 1-p. \| Function \| Implementation Notes \| \|---\|---\| \| pdf \| f(x;α,β) = xα - 1 (1 - x)β -1 / B(α, β) Implemented using [ibeta\_derivative](https://www.boost.org/doc/libs/1_63_0/libs/math/doc/html/math_toolkit/sf_beta/beta_derivative.html "Derivative of the Incomplete Beta Function")(a, b, x). \| \| cdf \| Using the incomplete beta function [ibeta](https://www.boost.org/doc/libs/1_63_0/libs/math/doc/html/math_toolkit/sf_beta/ibeta_function.html "Incomplete Beta Functions")(a, b, x) \| \| cdf complement \| [ibetac](https://www.boost.org/doc/libs/1_63_0/libs/math/doc/html/math_toolkit/sf_beta/ibeta_function.html "Incomplete Beta Functions")(a, b, x) \| \| quantile \| Using the inverse incomplete beta function [ibeta\_inv](https://www.boost.org/doc/libs/1_63_0/libs/math/doc/html/math_toolkit/sf_beta/ibeta_inv_function.html "The Incomplete Beta Function Inverses")(a, b, p) \| \| quantile from the complement \| [ibetac\_inv](https://www.boost.org/doc/libs/1_63_0/libs/math/doc/html/math_toolkit/sf_beta/ibeta_inv_function.html "The Incomplete Beta Function Inverses")(a, b, q) \| \| mean \| `a/(a+b)` \| \| variance \| \| \| mode \| \| \| skewness \| \| \| kurtosis excess \| ![](https://www.boost.org/doc/libs/1_63_0/libs/math/doc/equations/beta_dist_kurtosis.svg) \| \| kurtosis \| \| \| parameter estimation \| \| \| alpha from mean and variance \| \| \| beta from mean and variance \| \| \| The member functions `find_alpha` and `find_beta` from cdf and probability x and either `alpha` or `beta` \| Implemented in terms of the inverse incomplete beta functions [ibeta\_inva](https://www.boost.org/doc/libs/1_63_0/libs/math/doc/html/math_toolkit/sf_beta/ibeta_inv_function.html "The Incomplete Beta Function Inverses"), and [ibeta\_invb](https://www.boost.org/doc/libs/1_63_0/libs/math/doc/html/math_toolkit/sf_beta/ibeta_inv_function.html "The Incomplete Beta Function Inverses") respectively. \| \| `find_alpha` \| \| \| `find_beta` \| \| ##### [References](https://www.boost.org/doc/libs/1_63_0/libs/math/doc/html/math_toolkit/dist_ref/dists/beta_dist.html#math_toolkit.dist_ref.dists.beta_dist.references) [Wikipedia Beta distribution](http://en.wikipedia.org/wiki/Beta_distribution) [NIST Exploratory Data Analysis](http://www.itl.nist.gov/div898/handbook/eda/section3/eda366h.htm) [Wolfram MathWorld](http://mathworld.wolfram.com/BetaDistribution.html) \| \| \| \|---\|---\| \| \| Copyright © 2006-2010, 2012-2014 Nikhar Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan Råde, Gautam Sewani, Benjamin Sobotta, Thijs van den Berg, Daryle Walker and Xiaogang Zhang Distributed under the Boost Software License, Version 1.0. 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Readable Markdown	``` #include <boost/math/distributions/beta.hpp> ``` ``` namespace boost{ namespace math{ template <class RealType = double, class Policy = policies::policy<> > class beta_distribution; template <class RealType, class Policy> class beta_distribution { public: typedef RealType value_type; typedef Policy policy_type; beta_distribution(RealType a, RealType b); RealType alpha() const; RealType beta() const; static RealType find_alpha( RealType mean, RealType variance); static RealType find_beta( RealType mean, RealType variance); static RealType find_alpha( RealType beta, RealType x, RealType probability); static RealType find_beta( RealType alpha, RealType x, RealType probability); }; }} ``` The class type `beta_distribution` represents a [beta](http://en.wikipedia.org/wiki/Beta_distribution) [probability distribution function](http://en.wikipedia.org/wiki/Probability_distribution). The [beta distribution](http://mathworld.wolfram.com/BetaDistribution.htm) is used as a [prior distribution](http://en.wikipedia.org/wiki/Prior_distribution) for binomial proportions in [Bayesian analysis](http://mathworld.wolfram.com/BayesianAnalysis.html). See also: [beta distribution](http://documents.wolfram.com/calculationcenter/v2/Functions/ListsMatrices/Statistics/BetaDistribution.html) and [Bayesian statistics](http://en.wikipedia.org/wiki/Bayesian_statistics). How the beta distribution is used for [Bayesian analysis of one parameter models](http://home.uchicago.edu/~grynav/bayes/ABSLec5.ppt) is discussed by Jeff Grynaviski. The [probability density function PDF](http://en.wikipedia.org/wiki/Probability_density_function) for the [beta distribution](http://en.wikipedia.org/wiki/Beta_distribution) defined on the interval \[0,1\] is given by: f(x;α,β) = xα - 1 (1 - x)β -1 / B(α, β) where B(α, β) is the [beta function](http://en.wikipedia.org/wiki/Beta_function), implemented in this library as [beta](https://www.boost.org/doc/libs/1_63_0/libs/math/doc/html/math_toolkit/sf_beta/beta_function.html "Beta"). Division by the beta function ensures that the pdf is normalized to the range zero to unity. The following graph illustrates examples of the pdf for various values of the shape parameters. Note the α = β = 2 (blue line) is dome-shaped, and might be approximated by a symmetrical triangular distribution. ![](https://www.boost.org/doc/libs/1_63_0/libs/math/doc/graphs/beta_pdf.svg) If α = β = 1, then it is a \_\_space [uniform distribution](http://en.wikipedia.org/wiki/Uniform_distribution_%28continuous%29), equal to unity in the entire interval x = 0 to 1. If α \_\_space and β \_\_space are \< 1, then the pdf is U-shaped. If α != β, then the shape is asymmetric and could be approximated by a triangle whose apex is away from the centre (where x = half). ##### [Member Functions](https://www.boost.org/doc/libs/1_63_0/libs/math/doc/html/math_toolkit/dist_ref/dists/beta_dist.html#math_toolkit.dist_ref.dists.beta_dist.member_functions) ###### [Constructor](https://www.boost.org/doc/libs/1_63_0/libs/math/doc/html/math_toolkit/dist_ref/dists/beta_dist.html#math_toolkit.dist_ref.dists.beta_dist.constructor) ``` beta_distribution(RealType alpha, RealType beta); ``` Constructs a beta distribution with shape parameters alpha and beta. Requires alpha,beta \> 0,otherwise [domain\_error](https://www.boost.org/doc/libs/1_63_0/libs/math/doc/html/math_toolkit/error_handling.html#math_toolkit.error_handling.domain_error) is called. Note that technically the beta distribution is defined for alpha,beta \>= 0, but it's not clear whether any program can actually make use of that latitude or how many of the non-member functions can be usefully defined in that case. Therefore for now, we regard it as an error if alpha or beta is zero. For example: ``` beta_distribution<> mybeta(2, 5); ``` Constructs a the beta distribution with alpha=2 and beta=5 (shown in yellow in the graph above). ###### [Parameter Accessors](https://www.boost.org/doc/libs/1_63_0/libs/math/doc/html/math_toolkit/dist_ref/dists/beta_dist.html#math_toolkit.dist_ref.dists.beta_dist.parameter_accessors) ``` RealType alpha() const; ``` Returns the parameter alpha from which this distribution was constructed. ``` RealType beta() const; ``` Returns the parameter beta from which this distribution was constructed. So for example: ``` beta_distribution<> mybeta(2, 5); assert(mybeta.alpha() == 2.); assert(mybeta.beta() == 5.); ``` ##### [Parameter Estimators](https://www.boost.org/doc/libs/1_63_0/libs/math/doc/html/math_toolkit/dist_ref/dists/beta_dist.html#math_toolkit.dist_ref.dists.beta_dist.parameter_estimators) Two pairs of parameter estimators are provided. One estimates either α \_\_space or β \_\_space from presumed-known mean and variance. The other pair estimates either α \_\_space or β \_\_space from the cdf and x. It is also possible to estimate α \_\_space and β \_\_space from 'known' mode & quantile. For example, calculators are provided by the [Pooled Prevalence Calculator](http://www.ausvet.com.au/pprev/content.php?page=PPscript) and [Beta Buster](http://www.epi.ucdavis.edu/diagnostictests/betabuster.html) but this is not yet implemented here. ``` static RealType find_alpha( RealType mean, RealType variance); ``` Returns the unique value of α that corresponds to a beta distribution with mean mean and variance variance. ``` static RealType find_beta( RealType mean, RealType variance); ``` Returns the unique value of β that corresponds to a beta distribution with mean mean and variance variance. ``` static RealType find_alpha( RealType beta, RealType x, RealType probability); ``` Returns the value of α that gives: `cdf(beta_distribution<RealType>(alpha, beta), x) == probability`. ``` static RealType find_beta( RealType alpha, RealType x, RealType probability); ``` Returns the value of β that gives: `cdf(beta_distribution<RealType>(alpha, beta), x) == probability`. ##### [Non-member Accessor Functions](https://www.boost.org/doc/libs/1_63_0/libs/math/doc/html/math_toolkit/dist_ref/dists/beta_dist.html#math_toolkit.dist_ref.dists.beta_dist.non_member_accessor_functions) All the [usual non-member accessor functions](https://www.boost.org/doc/libs/1_63_0/libs/math/doc/html/math_toolkit/dist_ref/nmp.html "Non-Member Properties") that are generic to all distributions are supported: [Cumulative Distribution Function](https://www.boost.org/doc/libs/1_63_0/libs/math/doc/html/math_toolkit/dist_ref/nmp.html#math_toolkit.dist_ref.nmp.cdf), [Probability Density Function](https://www.boost.org/doc/libs/1_63_0/libs/math/doc/html/math_toolkit/dist_ref/nmp.html#math_toolkit.dist_ref.nmp.pdf), [Quantile](https://www.boost.org/doc/libs/1_63_0/libs/math/doc/html/math_toolkit/dist_ref/nmp.html#math_toolkit.dist_ref.nmp.quantile), [Hazard Function](https://www.boost.org/doc/libs/1_63_0/libs/math/doc/html/math_toolkit/dist_ref/nmp.html#math_toolkit.dist_ref.nmp.hazard), [Cumulative Hazard Function](https://www.boost.org/doc/libs/1_63_0/libs/math/doc/html/math_toolkit/dist_ref/nmp.html#math_toolkit.dist_ref.nmp.chf), [mean](https://www.boost.org/doc/libs/1_63_0/libs/math/doc/html/math_toolkit/dist_ref/nmp.html#math_toolkit.dist_ref.nmp.mean), [median](https://www.boost.org/doc/libs/1_63_0/libs/math/doc/html/math_toolkit/dist_ref/nmp.html#math_toolkit.dist_ref.nmp.median), [mode](https://www.boost.org/doc/libs/1_63_0/libs/math/doc/html/math_toolkit/dist_ref/nmp.html#math_toolkit.dist_ref.nmp.mode), [variance](https://www.boost.org/doc/libs/1_63_0/libs/math/doc/html/math_toolkit/dist_ref/nmp.html#math_toolkit.dist_ref.nmp.variance), [standard deviation](https://www.boost.org/doc/libs/1_63_0/libs/math/doc/html/math_toolkit/dist_ref/nmp.html#math_toolkit.dist_ref.nmp.sd), [skewness](https://www.boost.org/doc/libs/1_63_0/libs/math/doc/html/math_toolkit/dist_ref/nmp.html#math_toolkit.dist_ref.nmp.skewness), [kurtosis](https://www.boost.org/doc/libs/1_63_0/libs/math/doc/html/math_toolkit/dist_ref/nmp.html#math_toolkit.dist_ref.nmp.kurtosis), [kurtosis\_excess](https://www.boost.org/doc/libs/1_63_0/libs/math/doc/html/math_toolkit/dist_ref/nmp.html#math_toolkit.dist_ref.nmp.kurtosis_excess), [range](https://www.boost.org/doc/libs/1_63_0/libs/math/doc/html/math_toolkit/dist_ref/nmp.html#math_toolkit.dist_ref.nmp.range) and [support](https://www.boost.org/doc/libs/1_63_0/libs/math/doc/html/math_toolkit/dist_ref/nmp.html#math_toolkit.dist_ref.nmp.support). The formulae for calculating these are shown in the table below, and at [Wolfram Mathworld](http://mathworld.wolfram.com/BetaDistribution.html). ##### [Applications](https://www.boost.org/doc/libs/1_63_0/libs/math/doc/html/math_toolkit/dist_ref/dists/beta_dist.html#math_toolkit.dist_ref.dists.beta_dist.applications) The beta distribution can be used to model events constrained to take place within an interval defined by a minimum and maximum value: so it is used in project management systems. It is also widely used in [Bayesian statistical inference](http://en.wikipedia.org/wiki/Bayesian_inference). ##### [Related distributions](https://www.boost.org/doc/libs/1_63_0/libs/math/doc/html/math_toolkit/dist_ref/dists/beta_dist.html#math_toolkit.dist_ref.dists.beta_dist.related_distributions) The beta distribution with both α \_\_space and β = 1 follows a [uniform distribution](http://en.wikipedia.org/wiki/Uniform_distribution_%28continuous%29). The [triangular](http://en.wikipedia.org/wiki/Triangular_distribution) is used when less precise information is available. The [binomial distribution](http://en.wikipedia.org/wiki/Binomial_distribution) is closely related when α \_\_space and β \_\_space are integers. With integer values of α \_\_space and β \_\_space the distribution B(i, j) is that of the j-th highest of a sample of i + j + 1 independent random variables uniformly distributed between 0 and 1. The cumulative probability from 0 to x is thus the probability that the j-th highest value is less than x. Or it is the probability that at least i of the random variables are less than x, a probability given by summing over the [Binomial Distribution](https://www.boost.org/doc/libs/1_63_0/libs/math/doc/html/math_toolkit/dist_ref/dists/binomial_dist.html "Binomial Distribution") with its p parameter set to x. ##### [Accuracy](https://www.boost.org/doc/libs/1_63_0/libs/math/doc/html/math_toolkit/dist_ref/dists/beta_dist.html#math_toolkit.dist_ref.dists.beta_dist.accuracy) This distribution is implemented using the [beta functions](https://www.boost.org/doc/libs/1_63_0/libs/math/doc/html/math_toolkit/sf_beta/beta_function.html "Beta") [beta](https://www.boost.org/doc/libs/1_63_0/libs/math/doc/html/math_toolkit/sf_beta/beta_function.html "Beta") and [incomplete beta functions](https://www.boost.org/doc/libs/1_63_0/libs/math/doc/html/math_toolkit/sf_beta/ibeta_function.html "Incomplete Beta Functions") [ibeta](https://www.boost.org/doc/libs/1_63_0/libs/math/doc/html/math_toolkit/sf_beta/ibeta_function.html "Incomplete Beta Functions") and [ibetac](https://www.boost.org/doc/libs/1_63_0/libs/math/doc/html/math_toolkit/sf_beta/ibeta_function.html "Incomplete Beta Functions"); please refer to these functions for information on accuracy. ##### [Implementation](https://www.boost.org/doc/libs/1_63_0/libs/math/doc/html/math_toolkit/dist_ref/dists/beta_dist.html#math_toolkit.dist_ref.dists.beta_dist.implementation) In the following table a and b are the parameters α and β, x is the random variable, p is the probability and q = 1-p. \| Function \| Implementation Notes \| \|---\|---\| \| pdf \| f(x;α,β) = xα - 1 (1 - x)β -1 / B(α, β) Implemented using [ibeta\_derivative](https://www.boost.org/doc/libs/1_63_0/libs/math/doc/html/math_toolkit/sf_beta/beta_derivative.html "Derivative of the Incomplete Beta Function")(a, b, x). \| \| cdf \| Using the incomplete beta function [ibeta](https://www.boost.org/doc/libs/1_63_0/libs/math/doc/html/math_toolkit/sf_beta/ibeta_function.html "Incomplete Beta Functions")(a, b, x) \| \| cdf complement \| [ibetac](https://www.boost.org/doc/libs/1_63_0/libs/math/doc/html/math_toolkit/sf_beta/ibeta_function.html "Incomplete Beta Functions")(a, b, x) \| \| quantile \| Using the inverse incomplete beta function [ibeta\_inv](https://www.boost.org/doc/libs/1_63_0/libs/math/doc/html/math_toolkit/sf_beta/ibeta_inv_function.html "The Incomplete Beta Function Inverses")(a, b, p) \| \| quantile from the complement \| [ibetac\_inv](https://www.boost.org/doc/libs/1_63_0/libs/math/doc/html/math_toolkit/sf_beta/ibeta_inv_function.html "The Incomplete Beta Function Inverses")(a, b, q) \| \| mean \| `a/(a+b)` \| \| variance \| \| \| mode \| \| \| skewness \| \| \| kurtosis excess \| ![](https://www.boost.org/doc/libs/1_63_0/libs/math/doc/equations/beta_dist_kurtosis.svg) \| \| kurtosis \| \| \| parameter estimation \| \| \| alpha from mean and variance \| \| \| beta from mean and variance \| \| \| The member functions `find_alpha` and `find_beta` from cdf and probability x and either `alpha` or `beta` \| Implemented in terms of the inverse incomplete beta functions [ibeta\_inva](https://www.boost.org/doc/libs/1_63_0/libs/math/doc/html/math_toolkit/sf_beta/ibeta_inv_function.html "The Incomplete Beta Function Inverses"), and [ibeta\_invb](https://www.boost.org/doc/libs/1_63_0/libs/math/doc/html/math_toolkit/sf_beta/ibeta_inv_function.html "The Incomplete Beta Function Inverses") respectively. \| \| `find_alpha` \| \| \| `find_beta` \| \| ##### [References](https://www.boost.org/doc/libs/1_63_0/libs/math/doc/html/math_toolkit/dist_ref/dists/beta_dist.html#math_toolkit.dist_ref.dists.beta_dist.references) [Wikipedia Beta distribution](http://en.wikipedia.org/wiki/Beta_distribution) [NIST Exploratory Data Analysis](http://www.itl.nist.gov/div898/handbook/eda/section3/eda366h.htm) [Wolfram MathWorld](http://mathworld.wolfram.com/BetaDistribution.html)
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