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Raw Queries and Responses

1. Shard Calculation

Query:

Response:

Calculated Shard: 184 (from laksa007)

2. Crawled Status Check

Query:

curl -X POST \
  'http://laksa184.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://reference.wolfram.com/language/ref/BetaDistribution.html\')), getAhrefsUnparsedNoserviceFromURL(\'https://reference.wolfram.com/language/ref/BetaDistribution.html\'))) FORMAT JSONEachRow'

Response:

{"found_url":"https:\/\/reference.wolfram.com\/language\/ref\/BetaDistribution.html","crawl_time":1775919086,"first_indexed_time":1411424807,"http_code":200,"src_unparsed":"com,wolfram!reference,\/language\/ref\/BetaDistribution.html s443","src_root_hash":"3744487911316863784","history_drop_reason":null,"meta_title":"BetaDistribution—Wolfram Documentation","meta_descriptions":["BetaDistribution[\\[Alpha], \\[Beta]] represents a continuous beta distribution with shape parameters \\[Alpha] and \\[Beta]."],"attrs_boilerpipe_text":"See Also\nGammaDistribution\nBinomialDistribution\nBeta\nBetaRegularized\nInverseBetaRegularized\nFRatioDistribution\nDirichletDistribution\nRelated Guides\nBounded Domain Distributions\nParametric Statistical Distributions\nFunctions Used in Statistics\nTech Notes\nContinuous Distributions\nSee Also\nGammaDistribution\nBinomialDistribution\nBeta\nBetaRegularized\nInverseBetaRegularized\nFRatioDistribution\nDirichletDistribution\nRelated Guides\nBounded Domain Distributions\nParametric Statistical Distributions\nFunctions Used in Statistics\nTech Notes\nContinuous Distributions\nExamples\n \n \nBasic Examples\n \n \nScope\n \n \nApplications\n \n \nProperties & Relations\n \n \nPossible Issues\n \n \nBetaDistribution\n[\nα\n,\nβ\n]\nCopy to clipboard.\nBetaDistribution[α,β]\nrepresents a continuous beta distribution with shape parameters\nα\nand\nβ\n.\nDetails\nBackground & Context\nBetaDistribution\n[\nα\n,\nβ\n]\nrepresents a statistical distribution defined over the interval\nand parametrized by two positive values\nα\n,\nβ\nknown as \"shape parameters\", which, roughly speaking, determine the \"fatness\" of the left and right tails in the probability density function (PDF). Depending on the values of\nα\nand\nβ\n, the PDF of the beta distribution may be monotonic increasing, monotonic decreasing, or unimodal with potential singularities approaching the boundaries of its domain.\nThe beta distribution arises as a prior distribution for binomial proportions in Bayesian analysis. It is also commonly used to model random variables limited to a finite interval. For example, the distribution of the\nsmallest element in a continuous, independent, and uniformly distributed sample of size of\ncan be computed using\nOrderDistribution\n[\n{\nUniformDistribution\n[\n]\n,\nn\n}\n,\nk\n]\nand is precisely equal to\nBetaDistribution\n[\nk\n,\nn\n-\nk\n+1\n]\n. In addition to its statistical significance, the beta distribution also plays a fundamental role in a number of scientific fields, including phenomena related to allele frequency distribution, soil property variability, geological mineral-to-rock ratios, and HIV transmission behavior.\nRandomVariate\ncan be used to give one or more machine- or arbitrary-precision (the latter via the\nWorkingPrecision\noption) pseudorandom variates from a beta distribution.\nDistributed\n[\nx\n,\nBetaDistribution\n[\nα\n,\nβ\n]\n]\n, written more concisely as\nx\n\nBetaDistribution\n[\nα\n,\nβ\n]\n, can be used to assert that a random variable\nx\nis distributed according to a beta distribution. Such an assertion can then be used in functions such as\nProbability\n,\nNProbability\n,\nExpectation\n, and\nNExpectation\n.\nThe probability density and cumulative distribution functions may be given using\nPDF\n[\nBetaDistribution\n[\nα\n,\nβ\n]\n,\nx\n]\nand\nCDF\n[\nBetaDistribution\n[\nα\n,\nβ\n]\n,\nx\n]\n. The mean, median, variance, raw moments, and central moments may be computed using\nMean\n,\nMedian\n,\nVariance\n,\nMoment\n, and\nCentralMoment\n, respectively.\nDistributionFitTest\ncan be used to test if a given dataset is consistent with a beta distribution,\nEstimatedDistribution\nto estimate a beta parametric distribution from given data, and\nFindDistributionParameters\nto fit data to a beta distribution.\nProbabilityPlot\ncan be used to generate a plot of the CDF of given data against the CDF of a symbolic beta distribution and\nQuantilePlot\nto generate a plot of the quantiles of given data against the quantiles of a symbolic beta distribution.\nTransformedDistribution\ncan be used to represent a transformed beta distribution,\nCensoredDistribution\nto represent the distribution of values censored between upper and lower values, and\nTruncatedDistribution\nto represent the distribution of values truncated between upper and lower values.\nCopulaDistribution\ncan be used to build higher-dimensional distributions that contain a beta distribution, and\nProductDistribution\ncan be used to compute a joint distribution with independent component distributions involving beta distributions.\nThe beta distribution is related to a number of other distributions. For example,\nBetaDistribution\nis the so-called \"conjugate prior\" for the parameters of a number of other distributions, including\nBernoulliDistribution\n,\nBinomialDistribution\n,\nNegativeBinomialDistribution\n, and\nGeometricDistribution\n. Moreover,\nBetaDistribution\ngeneralizes both\nUniformDistribution\nand\nPowerDistribution\nin the sense that (modulo inclusion of the endpoints\nand\n),\nPDF\n[\nBetaDistribution\n[\n1\n,\n1\n]\n,\nx\n]\nis equal to both\nPDF\n[\nUniformDistribution\n[\n]\n,\nx\n]\nand\nPDF\n[\nPowerDistribution\n[\n1\n,\n1\n]\n,\nx\n]\n.\nBetaDistribution\ncan also be obtained as transformations of\nKumaraswamyDistribution\nand\nNoncentralBetaDistribution\nand is closely related to\nPERTDistribution\n,\nPearsonDistribution\n,\nChiSquareDistribution\n,\nGammaDistribution\n,\nFRatioDistribution\n, and\nBetaPrimeDistribution\n.\nExamples\nopen all\nclose all\nBasic Examples\n \n \n(4)\nSummary of the most common use cases\nScope\n \n \n(8)\nSurvey of the scope of standard use cases\nApplications\n \n \n(3)\nSample problems that can be solved with this function\nProperties & Relations\n \n \n(21)\nProperties of the function, and connections to other functions\nPossible Issues\n \n \n(2)\nCommon pitfalls and unexpected behavior\nWolfram Research (2007), BetaDistribution, Wolfram Language function, https:\/\/reference.wolfram.com\/language\/ref\/BetaDistribution.html (updated 2016).\nCopy to clipboard.\nWolfram Research (2007), BetaDistribution, Wolfram Language function, https:\/\/reference.wolfram.com\/language\/ref\/BetaDistribution.html (updated 2016).\nText\nWolfram Research (2007), BetaDistribution, Wolfram Language function, https:\/\/reference.wolfram.com\/language\/ref\/BetaDistribution.html (updated 2016).\nCopy to clipboard.\nWolfram Research (2007), BetaDistribution, Wolfram Language function, https:\/\/reference.wolfram.com\/language\/ref\/BetaDistribution.html (updated 2016).\nCMS\nWolfram Language. 2007. \"BetaDistribution.\" Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2016. https:\/\/reference.wolfram.com\/language\/ref\/BetaDistribution.html.\nCopy to clipboard.\nWolfram Language. 2007. \"BetaDistribution.\" Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2016. https:\/\/reference.wolfram.com\/language\/ref\/BetaDistribution.html.\nAPA\nWolfram Language. (2007). BetaDistribution. Wolfram Language & System Documentation Center. Retrieved from https:\/\/reference.wolfram.com\/language\/ref\/BetaDistribution.html\nCopy to clipboard.\nWolfram Language. (2007). BetaDistribution. Wolfram Language & System Documentation Center. Retrieved from https:\/\/reference.wolfram.com\/language\/ref\/BetaDistribution.html\nBibTeX\n@misc{reference.wolfram_2025_betadistribution, author=\"Wolfram Research\", title=\"{BetaDistribution}\", year=\"2016\", howpublished=\"\\url{https:\/\/reference.wolfram.com\/language\/ref\/BetaDistribution.html}\", note=[Accessed: 11-April-2026]}\nCopy to clipboard.\n@misc{reference.wolfram_2025_betadistribution, author=\"Wolfram Research\", title=\"{BetaDistribution}\", year=\"2016\", howpublished=\"\\url{https:\/\/reference.wolfram.com\/language\/ref\/BetaDistribution.html}\", note=[Accessed: 11-April-2026]}\nBibLaTeX\n@online{reference.wolfram_2025_betadistribution, organization={Wolfram Research}, title={BetaDistribution}, year={2016}, url={https:\/\/reference.wolfram.com\/language\/ref\/BetaDistribution.html}, note=[Accessed: 11-April-2026]}\nCopy to clipboard.\n@online{reference.wolfram_2025_betadistribution, organization={Wolfram Research}, title={BetaDistribution}, year={2016}, url={https:\/\/reference.wolfram.com\/language\/ref\/BetaDistribution.html}, note=[Accessed: 11-April-2026]}","attrs_markdown":"#\n[Products](https:\/\/reference.wolfram.com\/language\/ref\/BetaDistribution.html)\n\n- [Wolfram\\|One The definitive Wolfram Language and notebook experience](https:\/\/www.wolfram.com\/wolfram-one\/?source=nav)\n- [Mathematica The original technical computing environment](https:\/\/www.wolfram.com\/mathematica\/?source=nav)\n- [Notebook Assistant + LLM Kit All-in-one AI assistance for your Wolfram experience](https:\/\/www.wolfram.com\/notebook-assistant-llm-kit\/?source=nav)\n\n- [Compute Services](https:\/\/www.wolfram.com\/compute-services\/?source=nav)\n- [System Modeler](https:\/\/www.wolfram.com\/system-modeler\/?source=nav)\n- [Finance Platform](https:\/\/www.wolfram.com\/finance-platform\/?source=nav)\n- [Wolfram\\|Alpha Notebook Edition](https:\/\/www.wolfram.com\/wolfram-alpha-notebook-edition\/?source=nav)\n- [Application 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[BetaRegularized](https:\/\/reference.wolfram.com\/language\/ref\/BetaRegularized.html)\n  - [InverseBetaRegularized](https:\/\/reference.wolfram.com\/language\/ref\/InverseBetaRegularized.html)\n  - [FRatioDistribution](https:\/\/reference.wolfram.com\/language\/ref\/FRatioDistribution.html)\n  - [DirichletDistribution](https:\/\/reference.wolfram.com\/language\/ref\/DirichletDistribution.html)\n- [Related Guides](https:\/\/reference.wolfram.com\/language\/ref\/BetaDistribution.html)\n  - [Bounded Domain Distributions](https:\/\/reference.wolfram.com\/language\/guide\/BoundedDomainDistributions.html)\n  - [Parametric Statistical Distributions](https:\/\/reference.wolfram.com\/language\/guide\/ParametricStatisticalDistributions.html)\n  - [Functions Used in Statistics](https:\/\/reference.wolfram.com\/language\/guide\/FunctionsUsedInStatistics.html)\n- [Tech Notes](https:\/\/reference.wolfram.com\/language\/ref\/BetaDistribution.html)\n  - [Continuous Distributions](https:\/\/reference.wolfram.com\/language\/tutorial\/NumericalOperationsOnData.html#11002)\n- - [See Also](https:\/\/reference.wolfram.com\/language\/ref\/BetaDistribution.html)\n    - [GammaDistribution](https:\/\/reference.wolfram.com\/language\/ref\/GammaDistribution.html)\n    - [BinomialDistribution](https:\/\/reference.wolfram.com\/language\/ref\/BinomialDistribution.html)\n    - [Beta](https:\/\/reference.wolfram.com\/language\/ref\/Beta.html)\n    - [BetaRegularized](https:\/\/reference.wolfram.com\/language\/ref\/BetaRegularized.html)\n    - [InverseBetaRegularized](https:\/\/reference.wolfram.com\/language\/ref\/InverseBetaRegularized.html)\n    - [FRatioDistribution](https:\/\/reference.wolfram.com\/language\/ref\/FRatioDistribution.html)\n    - [DirichletDistribution](https:\/\/reference.wolfram.com\/language\/ref\/DirichletDistribution.html)\n  - [Related Guides](https:\/\/reference.wolfram.com\/language\/ref\/BetaDistribution.html)\n    - [Bounded Domain Distributions](https:\/\/reference.wolfram.com\/language\/guide\/BoundedDomainDistributions.html)\n    - [Parametric Statistical Distributions](https:\/\/reference.wolfram.com\/language\/guide\/ParametricStatisticalDistributions.html)\n    - [Functions Used in Statistics](https:\/\/reference.wolfram.com\/language\/guide\/FunctionsUsedInStatistics.html)\n  - [Tech Notes](https:\/\/reference.wolfram.com\/language\/ref\/BetaDistribution.html)\n    - [Continuous Distributions](https:\/\/reference.wolfram.com\/language\/tutorial\/NumericalOperationsOnData.html#11002)\n\n[BetaDistribution](https:\/\/reference.wolfram.com\/language\/ref\/BetaDistribution.html)\\[α,β\\]\n\nrepresents a continuous beta distribution with shape parameters α and β.\n\nDetails\n\n![Details and Options](https:\/\/reference.wolfram.com\/language-assets-v2\/img\/DetailsOptions-gray.png) ![Details and Options](https:\/\/reference.wolfram.com\/language-assets-v2\/img\/DetailsOptions-hover.png)\n\nBackground & Context\n\nExamples\n\nBasic Examples\n\nScope\n\nApplications\n\nProperties & Relations\n\nPossible Issues\n\nSee Also\n\nTech Notes\n\nRelated Guides\n\nHistory\n\nCite this Page\n\nBUILT-IN SYMBOL\n\n- [See Also](https:\/\/reference.wolfram.com\/language\/ref\/BetaDistribution.html)\n  - [GammaDistribution](https:\/\/reference.wolfram.com\/language\/ref\/GammaDistribution.html)\n  - [BinomialDistribution](https:\/\/reference.wolfram.com\/language\/ref\/BinomialDistribution.html)\n  - [Beta](https:\/\/reference.wolfram.com\/language\/ref\/Beta.html)\n  - [BetaRegularized](https:\/\/reference.wolfram.com\/language\/ref\/BetaRegularized.html)\n  - [InverseBetaRegularized](https:\/\/reference.wolfram.com\/language\/ref\/InverseBetaRegularized.html)\n  - [FRatioDistribution](https:\/\/reference.wolfram.com\/language\/ref\/FRatioDistribution.html)\n  - [DirichletDistribution](https:\/\/reference.wolfram.com\/language\/ref\/DirichletDistribution.html)\n- [Related Guides](https:\/\/reference.wolfram.com\/language\/ref\/BetaDistribution.html)\n  - [Bounded Domain Distributions](https:\/\/reference.wolfram.com\/language\/guide\/BoundedDomainDistributions.html)\n  - [Parametric Statistical Distributions](https:\/\/reference.wolfram.com\/language\/guide\/ParametricStatisticalDistributions.html)\n  - [Functions Used in Statistics](https:\/\/reference.wolfram.com\/language\/guide\/FunctionsUsedInStatistics.html)\n- [Tech Notes](https:\/\/reference.wolfram.com\/language\/ref\/BetaDistribution.html)\n  - [Continuous Distributions](https:\/\/reference.wolfram.com\/language\/tutorial\/NumericalOperationsOnData.html#11002)\n- - [See Also](https:\/\/reference.wolfram.com\/language\/ref\/BetaDistribution.html)\n    - [GammaDistribution](https:\/\/reference.wolfram.com\/language\/ref\/GammaDistribution.html)\n    - [BinomialDistribution](https:\/\/reference.wolfram.com\/language\/ref\/BinomialDistribution.html)\n    - [Beta](https:\/\/reference.wolfram.com\/language\/ref\/Beta.html)\n    - [BetaRegularized](https:\/\/reference.wolfram.com\/language\/ref\/BetaRegularized.html)\n    - [InverseBetaRegularized](https:\/\/reference.wolfram.com\/language\/ref\/InverseBetaRegularized.html)\n    - [FRatioDistribution](https:\/\/reference.wolfram.com\/language\/ref\/FRatioDistribution.html)\n    - [DirichletDistribution](https:\/\/reference.wolfram.com\/language\/ref\/DirichletDistribution.html)\n  - [Related Guides](https:\/\/reference.wolfram.com\/language\/ref\/BetaDistribution.html)\n    - [Bounded Domain Distributions](https:\/\/reference.wolfram.com\/language\/guide\/BoundedDomainDistributions.html)\n    - [Parametric Statistical Distributions](https:\/\/reference.wolfram.com\/language\/guide\/ParametricStatisticalDistributions.html)\n    - [Functions Used in Statistics](https:\/\/reference.wolfram.com\/language\/guide\/FunctionsUsedInStatistics.html)\n  - [Tech Notes](https:\/\/reference.wolfram.com\/language\/ref\/BetaDistribution.html)\n    - [Continuous Distributions](https:\/\/reference.wolfram.com\/language\/tutorial\/NumericalOperationsOnData.html#11002)\n# BetaDistributionCopy to clipboard. ✖ `BetaDistribution`\n\n[BetaDistribution](https:\/\/reference.wolfram.com\/language\/ref\/BetaDistribution.html)\\[α,β\\]\n\nCopy to clipboard.\n\n✖\n\n`BetaDistribution[α,β]`\n\nrepresents a continuous beta distribution with shape parameters α and β.\n\n# Details\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/details_1.png)\n\n- The probability density for value ![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/1.png) in a beta distribution is proportional to ![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/2.png) for ![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/3.png), and is zero for ![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/4.png) or ![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/5.png). [»](https:\/\/reference.wolfram.com\/language\/ref\/BetaDistribution.html#2046786362)\n- [BetaDistribution](https:\/\/reference.wolfram.com\/language\/ref\/BetaDistribution.html) allows α and β to be any positive real numbers.\n- [BetaDistribution](https:\/\/reference.wolfram.com\/language\/ref\/BetaDistribution.html) allows α and β to be dimensionless quantities. [»](https:\/\/reference.wolfram.com\/language\/ref\/BetaDistribution.html#504092574)\n- [BetaDistribution](https:\/\/reference.wolfram.com\/language\/ref\/BetaDistribution.html) can be used with such functions as [Mean](https:\/\/reference.wolfram.com\/language\/ref\/Mean.html), [CDF](https:\/\/reference.wolfram.com\/language\/ref\/CDF.html), and [RandomVariate](https:\/\/reference.wolfram.com\/language\/ref\/RandomVariate.html). [»](https:\/\/reference.wolfram.com\/language\/ref\/BetaDistribution.html#10542)\n\n# Background & Context\n- [BetaDistribution](https:\/\/reference.wolfram.com\/language\/ref\/BetaDistribution.html)\\[α,β\\] represents a statistical distribution defined over the interval ![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/6.png) and parametrized by two positive values α, β known as \"shape parameters\", which, roughly speaking, determine the \"fatness\" of the left and right tails in the probability density function (PDF). Depending on the values of α and β, the PDF of the beta distribution may be monotonic increasing, monotonic decreasing, or unimodal with potential singularities approaching the boundaries of its domain.\n- The beta distribution arises as a prior distribution for binomial proportions in Bayesian analysis. It is also commonly used to model random variables limited to a finite interval. For example, the distribution of the ![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/7.png)![^(th)](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/8.png) smallest element in a continuous, independent, and uniformly distributed sample of size of ![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/9.png) can be computed using [OrderDistribution](https:\/\/reference.wolfram.com\/language\/ref\/OrderDistribution.html)\\[{[UniformDistribution](https:\/\/reference.wolfram.com\/language\/ref\/UniformDistribution.html)\\[\\],n},k\\] and is precisely equal to [BetaDistribution](https:\/\/reference.wolfram.com\/language\/ref\/BetaDistribution.html)\\[k,n\\-k\\+1\\]. In addition to its statistical significance, the beta distribution also plays a fundamental role in a number of scientific fields, including phenomena related to allele frequency distribution, soil property variability, geological mineral-to-rock ratios, and HIV transmission behavior.\n- [RandomVariate](https:\/\/reference.wolfram.com\/language\/ref\/RandomVariate.html) can be used to give one or more machine- or arbitrary-precision (the latter via the [WorkingPrecision](https:\/\/reference.wolfram.com\/language\/ref\/WorkingPrecision.html) option) pseudorandom variates from a beta distribution. [Distributed](https:\/\/reference.wolfram.com\/language\/ref\/Distributed.html)\\[x,BetaDistribution\\[α,β\\]\\], written more concisely as xBetaDistribution\\[α,β\\], can be used to assert that a random variable x is distributed according to a beta distribution. Such an assertion can then be used in functions such as [Probability](https:\/\/reference.wolfram.com\/language\/ref\/Probability.html), [NProbability](https:\/\/reference.wolfram.com\/language\/ref\/NProbability.html), [Expectation](https:\/\/reference.wolfram.com\/language\/ref\/Expectation.html), and [NExpectation](https:\/\/reference.wolfram.com\/language\/ref\/NExpectation.html).\n- The probability density and cumulative distribution functions may be given using [PDF](https:\/\/reference.wolfram.com\/language\/ref\/PDF.html)\\[BetaDistribution\\[α,β\\],x\\] and [CDF](https:\/\/reference.wolfram.com\/language\/ref\/CDF.html)\\[BetaDistribution\\[α,β\\],x\\]. The mean, median, variance, raw moments, and central moments may be computed using [Mean](https:\/\/reference.wolfram.com\/language\/ref\/Mean.html), [Median](https:\/\/reference.wolfram.com\/language\/ref\/Median.html), [Variance](https:\/\/reference.wolfram.com\/language\/ref\/Variance.html), [Moment](https:\/\/reference.wolfram.com\/language\/ref\/Moment.html), and [CentralMoment](https:\/\/reference.wolfram.com\/language\/ref\/CentralMoment.html), respectively.\n- [DistributionFitTest](https:\/\/reference.wolfram.com\/language\/ref\/DistributionFitTest.html) can be used to test if a given dataset is consistent with a beta distribution, [EstimatedDistribution](https:\/\/reference.wolfram.com\/language\/ref\/EstimatedDistribution.html) to estimate a beta parametric distribution from given data, and [FindDistributionParameters](https:\/\/reference.wolfram.com\/language\/ref\/FindDistributionParameters.html) to fit data to a beta distribution. [ProbabilityPlot](https:\/\/reference.wolfram.com\/language\/ref\/ProbabilityPlot.html) can be used to generate a plot of the CDF of given data against the CDF of a symbolic beta distribution and [QuantilePlot](https:\/\/reference.wolfram.com\/language\/ref\/QuantilePlot.html) to generate a plot of the quantiles of given data against the quantiles of a symbolic beta distribution.\n- [TransformedDistribution](https:\/\/reference.wolfram.com\/language\/ref\/TransformedDistribution.html) can be used to represent a transformed beta distribution, [CensoredDistribution](https:\/\/reference.wolfram.com\/language\/ref\/CensoredDistribution.html) to represent the distribution of values censored between upper and lower values, and [TruncatedDistribution](https:\/\/reference.wolfram.com\/language\/ref\/TruncatedDistribution.html) to represent the distribution of values truncated between upper and lower values. [CopulaDistribution](https:\/\/reference.wolfram.com\/language\/ref\/CopulaDistribution.html) can be used to build higher-dimensional distributions that contain a beta distribution, and [ProductDistribution](https:\/\/reference.wolfram.com\/language\/ref\/ProductDistribution.html) can be used to compute a joint distribution with independent component distributions involving beta distributions.\n- The beta distribution is related to a number of other distributions. For example, [BetaDistribution](https:\/\/reference.wolfram.com\/language\/ref\/BetaDistribution.html) is the so-called \"conjugate prior\" for the parameters of a number of other distributions, including [BernoulliDistribution](https:\/\/reference.wolfram.com\/language\/ref\/BernoulliDistribution.html), [BinomialDistribution](https:\/\/reference.wolfram.com\/language\/ref\/BinomialDistribution.html), [NegativeBinomialDistribution](https:\/\/reference.wolfram.com\/language\/ref\/NegativeBinomialDistribution.html), and [GeometricDistribution](https:\/\/reference.wolfram.com\/language\/ref\/GeometricDistribution.html). Moreover, [BetaDistribution](https:\/\/reference.wolfram.com\/language\/ref\/BetaDistribution.html) generalizes both [UniformDistribution](https:\/\/reference.wolfram.com\/language\/ref\/UniformDistribution.html) and [PowerDistribution](https:\/\/reference.wolfram.com\/language\/ref\/PowerDistribution.html) in the sense that (modulo inclusion of the endpoints ![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/10.png) and ![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/11.png)), [PDF](https:\/\/reference.wolfram.com\/language\/ref\/PDF.html)\\[BetaDistribution\\[1,1\\],x\\] is equal to both [PDF](https:\/\/reference.wolfram.com\/language\/ref\/PDF.html)\\[[UniformDistribution](https:\/\/reference.wolfram.com\/language\/ref\/UniformDistribution.html)\\[\\],x\\] and [PDF](https:\/\/reference.wolfram.com\/language\/ref\/PDF.html)\\[[PowerDistribution](https:\/\/reference.wolfram.com\/language\/ref\/PowerDistribution.html)\\[1,1\\],x\\]. [BetaDistribution](https:\/\/reference.wolfram.com\/language\/ref\/BetaDistribution.html) can also be obtained as transformations of [KumaraswamyDistribution](https:\/\/reference.wolfram.com\/language\/ref\/KumaraswamyDistribution.html) and [NoncentralBetaDistribution](https:\/\/reference.wolfram.com\/language\/ref\/NoncentralBetaDistribution.html) and is closely related to [PERTDistribution](https:\/\/reference.wolfram.com\/language\/ref\/PERTDistribution.html), [PearsonDistribution](https:\/\/reference.wolfram.com\/language\/ref\/PearsonDistribution.html), [ChiSquareDistribution](https:\/\/reference.wolfram.com\/language\/ref\/ChiSquareDistribution.html), [GammaDistribution](https:\/\/reference.wolfram.com\/language\/ref\/GammaDistribution.html), [FRatioDistribution](https:\/\/reference.wolfram.com\/language\/ref\/FRatioDistribution.html), and [BetaPrimeDistribution](https:\/\/reference.wolfram.com\/language\/ref\/BetaPrimeDistribution.html).\n\n# Examples\nopen all close all\n\n## Basic Examples  (4)Summary of the most common use cases\nProbability density function:\n\nCopy to clipboard.\n\nIn\\[1\\]:=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/I_1.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0g7kenjf9sq-6ghfcq`\n\nDirect link to example\n\nOut\\[1\\]=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/O_1.png)\n\nCopy to clipboard.\n\nIn\\[2\\]:=2\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/I_2.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0g7kenjf9sq-xi7qe0`\n\nDirect link to example\n\nOut\\[2\\]=2\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/O_2.png)\n\nCopy to clipboard.\n\nIn\\[3\\]:=3\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/I_3.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0g7kenjf9sq-crqw28`\n\nDirect link to example\n\nOut\\[3\\]=3\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/O_3.png)\n\nCumulative distribution function:\n\nCopy to clipboard.\n\nIn\\[1\\]:=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/I_4.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0g7kenjf9sq-o4atla`\n\nDirect link to example\n\nOut\\[1\\]=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/O_4.png)\n\nCopy to clipboard.\n\nIn\\[2\\]:=2\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/I_5.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0g7kenjf9sq-8fypqg`\n\nDirect link to example\n\nOut\\[2\\]=2\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/O_5.png)\n\nCopy to clipboard.\n\nIn\\[3\\]:=3\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/I_6.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0g7kenjf9sq-xzaznn`\n\nDirect link to example\n\nOut\\[3\\]=3\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/O_6.png)\n\nMean and variance:\n\nCopy to clipboard.\n\nIn\\[1\\]:=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/I_7.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0g7kenjf9sq-kc5`\n\nDirect link to example\n\nOut\\[1\\]=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/O_7.png)\n\nCopy to clipboard.\n\nIn\\[2\\]:=2\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/I_8.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0g7kenjf9sq-xtk`\n\nDirect link to example\n\nOut\\[2\\]=2\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/O_8.png)\n\nMedian:\n\nCopy to clipboard.\n\nIn\\[1\\]:=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/I_9.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0g7kenjf9sq-4zlnan`\n\nDirect link to example\n\nOut\\[1\\]=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/O_9.png)\n\n## Scope  (8)Survey of the scope of standard use cases\nGenerate a sample of pseudorandom numbers from a beta distribution:\n\nCopy to clipboard.\n\nIn\\[1\\]:=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/I_10.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0g7kenjf9sq-tscvhm`\n\nDirect link to example\n\nCompare the histogram to the PDF:\n\nCopy to clipboard.\n\nIn\\[2\\]:=2\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/I_11.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0g7kenjf9sq-fw0ala`\n\nDirect link to example\n\nOut\\[2\\]=2\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/O_10.png)\n\nDistribution parameters estimation:\n\nCopy to clipboard.\n\nIn\\[1\\]:=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/I_12.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0g7kenjf9sq-45b7g2`\n\nDirect link to example\n\nEstimate the distribution parameters from sample data:\n\nCopy to clipboard.\n\nIn\\[2\\]:=2\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/I_13.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0g7kenjf9sq-epi747`\n\nDirect link to example\n\nOut\\[2\\]=2\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/O_11.png)\n\nCompare a density histogram of the sample with the PDF of the estimated distribution:\n\nCopy to clipboard.\n\nIn\\[3\\]:=3\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/I_14.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0g7kenjf9sq-f8ui5o`\n\nDirect link to example\n\nOut\\[3\\]=3\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/O_12.png)\n\nSkewness varies with shape parameters:\n\nCopy to clipboard.\n\nIn\\[1\\]:=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/I_15.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0g7kenjf9sq-3n7jeg`\n\nDirect link to example\n\nOut\\[1\\]=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/O_13.png)\n\nCopy to clipboard.\n\nIn\\[2\\]:=2\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/I_16.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0g7kenjf9sq-i6l`\n\nDirect link to example\n\nOut\\[2\\]=2\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/O_14.png)\n\nWhen both parameters go to ![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/12.png), the distribution becomes symmetric:\n\nCopy to clipboard.\n\nIn\\[3\\]:=3\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/I_17.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0g7kenjf9sq-hcr61q`\n\nDirect link to example\n\nOut\\[3\\]=3\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/O_15.png)\n\nKurtosis varies with shape parameters:\n\nCopy to clipboard.\n\nIn\\[1\\]:=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/I_18.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0g7kenjf9sq-6aiaq1`\n\nDirect link to example\n\nOut\\[1\\]=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/O_16.png)\n\nCopy to clipboard.\n\nIn\\[2\\]:=2\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/I_19.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0g7kenjf9sq-gft`\n\nDirect link to example\n\nOut\\[2\\]=2\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/O_17.png)\n\nIn the limit, the kurtosis becomes the same as for [NormalDistribution](https:\/\/reference.wolfram.com\/language\/ref\/NormalDistribution.html):\n\nCopy to clipboard.\n\nIn\\[3\\]:=3\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/I_20.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0g7kenjf9sq-70koo4`\n\nDirect link to example\n\nOut\\[3\\]=3\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/O_18.png)\n\nCopy to clipboard.\n\nIn\\[4\\]:=4\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/I_21.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0g7kenjf9sq-hn3lr`\n\nDirect link to example\n\nOut\\[4\\]=4\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/O_19.png)\n\nDifferent moments with closed forms as functions of parameters:\n\nCopy to clipboard.\n\nIn\\[1\\]:=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/I_22.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0g7kenjf9sq-js043h`\n\nDirect link to example\n\n[Moment](https:\/\/reference.wolfram.com\/language\/ref\/Moment.html):\n\nCopy to clipboard.\n\nIn\\[2\\]:=2\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/I_23.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0g7kenjf9sq-rx074o`\n\nDirect link to example\n\nOut\\[2\\]=2\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/O_20.png)\n\n[CentralMoment](https:\/\/reference.wolfram.com\/language\/ref\/CentralMoment.html):\n\nCopy to clipboard.\n\nIn\\[3\\]:=3\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/I_24.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0g7kenjf9sq-pknsqa`\n\nDirect link to example\n\nOut\\[3\\]=3\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/O_21.png)\n\n[FactorialMoment](https:\/\/reference.wolfram.com\/language\/ref\/FactorialMoment.html):\n\nCopy to clipboard.\n\nIn\\[4\\]:=4\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/I_25.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0g7kenjf9sq-zg9ct4`\n\nDirect link to example\n\nOut\\[4\\]=4\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/O_22.png)\n\n[Cumulant](https:\/\/reference.wolfram.com\/language\/ref\/Cumulant.html):\n\nCopy to clipboard.\n\nIn\\[5\\]:=5\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/I_26.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0g7kenjf9sq-9gzmth`\n\nDirect link to example\n\nOut\\[5\\]=5\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/O_23.png)\n\nHazard function:\n\nCopy to clipboard.\n\nIn\\[1\\]:=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/I_27.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0g7kenjf9sq-cly108`\n\nDirect link to example\n\nOut\\[1\\]=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/O_24.png)\n\nCopy to clipboard.\n\nIn\\[2\\]:=2\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/I_28.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0g7kenjf9sq-cmoefp`\n\nDirect link to example\n\nOut\\[2\\]=2\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/O_25.png)\n\nCopy to clipboard.\n\nIn\\[3\\]:=3\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/I_29.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0g7kenjf9sq-zet7d0`\n\nDirect link to example\n\nOut\\[3\\]=3\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/O_26.png)\n\nQuantile function:\n\nCopy to clipboard.\n\nIn\\[1\\]:=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/I_30.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0g7kenjf9sq-8ljlc3`\n\nDirect link to example\n\nOut\\[1\\]=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/O_27.png)\n\nCopy to clipboard.\n\nIn\\[2\\]:=2\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/I_31.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0g7kenjf9sq-ffrzb1`\n\nDirect link to example\n\nOut\\[2\\]=2\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/O_28.png)\n\nCopy to clipboard.\n\nIn\\[3\\]:=3\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/I_32.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0g7kenjf9sq-m8kdz6`\n\nDirect link to example\n\nOut\\[3\\]=3\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/O_29.png)\n\nConsistent use of [Quantity](https:\/\/reference.wolfram.com\/language\/ref\/Quantity.html) in parameters expands them into their numeric values:\n\nCopy to clipboard.\n\nIn\\[1\\]:=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/I_33.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0g7kenjf9sq-gr2bgd`\n\nDirect link to example\n\nOut\\[1\\]=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/O_30.png)\n\nFind the mean:\n\nCopy to clipboard.\n\nIn\\[2\\]:=2\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/I_34.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0g7kenjf9sq-h594k5`\n\nDirect link to example\n\nOut\\[2\\]=2\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/O_31.png)\n\n## Applications  (3)Sample problems that can be solved with this function\nCloud duration approximately follows a beta distribution with parameters 0.3 and 0.4 for a particular location. Find the probability that cloud duration will be longer than half a day:\n\nCopy to clipboard.\n\nIn\\[1\\]:=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/I_35.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0g7kenjf9sq-1og0gu`\n\nDirect link to example\n\nOut\\[1\\]=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/O_32.png)\n\nSimulate the fraction of the day that is cloudy over a period of one month:\n\nCopy to clipboard.\n\nIn\\[2\\]:=2\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/I_36.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0g7kenjf9sq-ex9npo`\n\nDirect link to example\n\nCopy to clipboard.\n\nIn\\[3\\]:=3\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/I_37.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0g7kenjf9sq-txou6v`\n\nDirect link to example\n\nOut\\[3\\]=3\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/O_33.png)\n\nFind the average cloudiness duration for a day:\n\nCopy to clipboard.\n\nIn\\[4\\]:=4\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/I_38.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0g7kenjf9sq-3hhdl7`\n\nDirect link to example\n\nOut\\[4\\]=4\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/O_34.png)\n\nFind the probability of having exactly 20 days in a month with cloud duration less than 10%:\n\nCopy to clipboard.\n\nIn\\[5\\]:=5\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/I_39.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0g7kenjf9sq-dgsxa0`\n\nDirect link to example\n\nCopy to clipboard.\n\nIn\\[6\\]:=6\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/I_40.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0g7kenjf9sq-cww8e7`\n\nDirect link to example\n\nOut\\[6\\]=6\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/O_35.png)\n\nFind the probability of at least 20 days in a month with cloud duration less than 10%:\n\nCopy to clipboard.\n\nIn\\[7\\]:=7\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/I_41.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0g7kenjf9sq-gdcxid`\n\nDirect link to example\n\nOut\\[7\\]=7\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/O_36.png)\n\nBeta distribution can be used to model the proportion of the stocks that increase in value on a given day. Fit beta distribution to the Dow Jones Industrial stocks data:\n\nCopy to clipboard.\n\nIn\\[1\\]:=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/I_42.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0g7kenjf9sq-hhnw8h`\n\nDirect link to example\n\nOut\\[1\\]=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/O_37.png)\n\nFind daily change:\n\nCopy to clipboard.\n\nIn\\[2\\]:=2\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/I_43.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0g7kenjf9sq-vzl714`\n\nDirect link to example\n\nNumber of days for each financial entity:\n\nCopy to clipboard.\n\nIn\\[3\\]:=3\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/I_44.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0g7kenjf9sq-e15an6`\n\nDirect link to example\n\nOut\\[3\\]=3\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/O_38.png)\n\nExtract values from time series for each entity and normalize numeric quantities:\n\nCopy to clipboard.\n\nIn\\[4\\]:=4\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/I_45.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0g7kenjf9sq-bdcjdx`\n\nDirect link to example\n\nCheck if each entity has the same length of data:\n\nCopy to clipboard.\n\nIn\\[5\\]:=5\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/I_46.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0g7kenjf9sq-ldtuwa`\n\nDirect link to example\n\nOut\\[5\\]=5\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/O_39.png)\n\nCopy to clipboard.\n\nIn\\[6\\]:=6\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/I_47.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0g7kenjf9sq-oibkmk`\n\nDirect link to example\n\nOut\\[6\\]=6\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/O_40.png)\n\nCalculate the daily ratio of companies with an increase in value:\n\nCopy to clipboard.\n\nIn\\[7\\]:=7\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/I_48.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0g7kenjf9sq-b1xazx`\n\nDirect link to example\n\nFind fit, excluding days with no companies having an increase in value:\n\nCopy to clipboard.\n\nIn\\[8\\]:=8\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/I_49.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0g7kenjf9sq-18xqlp`\n\nDirect link to example\n\nOut\\[8\\]=8\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/O_41.png)\n\nCompare the histogram of the data with the PDF of the estimated distribution:\n\nCopy to clipboard.\n\nIn\\[9\\]:=9\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/I_50.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0g7kenjf9sq-z8idun`\n\nDirect link to example\n\nOut\\[9\\]=9\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/O_42.png)\n\nFind the probability that at least 60% of Dow Jones Industrial stocks will increase in value:\n\nCopy to clipboard.\n\nIn\\[10\\]:=10\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/I_51.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0g7kenjf9sq-k3z2y6`\n\nDirect link to example\n\nOut\\[10\\]=10\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/O_43.png)\n\nFind the average percentage of Dow Jones Industrial stocks that will increase in value:\n\nCopy to clipboard.\n\nIn\\[11\\]:=11\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/I_52.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0g7kenjf9sq-l8d1x5`\n\nDirect link to example\n\nOut\\[11\\]=11\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/O_44.png)\n\nSimulate the percentage of Dow Jones Industrial stocks that will increase in value for 30 days:\n\nCopy to clipboard.\n\nIn\\[12\\]:=12\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/I_53.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0g7kenjf9sq-ogrhpe`\n\nDirect link to example\n\nOut\\[12\\]=12\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/O_45.png)\n\nDiscrete-time Markov chain ![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/13.png), where ![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/14.png) is the sequence of independent and identically distributed (iid) standard uniform random variables, and ![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/15.png) is the sequence of iid Bernoulli random variables with success probability of ![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/16.png) converges to stationary distribution [BetaDistribution](https:\/\/reference.wolfram.com\/language\/ref\/BetaDistribution.html)\\[p,1-p\\] for any initial condition ![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/17.png) such that ![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/18.png):\n\nCopy to clipboard.\n\nIn\\[1\\]:=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/I_54.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0g7kenjf9sq-d8x1dq`\n\nDirect link to example\n\nSample a realization of the Markov chain and discard the burn-in portion of the path:\n\nCopy to clipboard.\n\nIn\\[2\\]:=2\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/I_55.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0g7kenjf9sq-wea5d`\n\nDirect link to example\n\nSamples from the Markov chain are not independent and exhibit internal structure:\n\nCopy to clipboard.\n\nIn\\[3\\]:=3\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/I_56.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0g7kenjf9sq-jhh7sk`\n\nDirect link to example\n\nOut\\[3\\]=3\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/O_46.png)\n\nCompare the histogram of path values to the [PDF](https:\/\/reference.wolfram.com\/language\/ref\/PDF.html) of the Markov chain's stationary distribution:\n\nCopy to clipboard.\n\nIn\\[4\\]:=4\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/I_57.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0g7kenjf9sq-cwr1w`\n\nDirect link to example\n\nOut\\[4\\]=4\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/O_47.png)\n\nUse path values to approximate an expectation:\n\nCopy to clipboard.\n\nIn\\[5\\]:=5\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/I_58.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0g7kenjf9sq-dcivz6`\n\nDirect link to example\n\nOut\\[5\\]=5\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/O_48.png)\n\nCompare with the quadrature value:\n\nCopy to clipboard.\n\nIn\\[6\\]:=6\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/I_59.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0g7kenjf9sq-f6nfr1`\n\nDirect link to example\n\nOut\\[6\\]=6\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/O_49.png)\n\n## Properties & Relations  (21)Properties of the function, and connections to other functions\nIf a variate ![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/19.png) follows beta distribution, then ![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/20.png) follows the reflected distribution:\n\nCopy to clipboard.\n\nIn\\[1\\]:=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/I_60.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0g7kenjf9sq-sdo87r`\n\nDirect link to example\n\nOut\\[1\\]=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/O_50.png)\n\nRelationships to other distributions:\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/Image_21.gif)\n\n[BetaDistribution](https:\/\/reference.wolfram.com\/language\/ref\/BetaDistribution.html)\\[1,1\\] is equivalent to [UniformDistribution](https:\/\/reference.wolfram.com\/language\/ref\/UniformDistribution.html)\\[{0,1}\\]:\n\nCopy to clipboard.\n\nIn\\[1\\]:=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/I_61.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0g7kenjf9sq-res`\n\nDirect link to example\n\nOut\\[1\\]=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/O_51.png)\n\nCopy to clipboard.\n\nIn\\[2\\]:=2\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/I_62.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0g7kenjf9sq-r6e`\n\nDirect link to example\n\nOut\\[2\\]=2\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/O_52.png)\n\n[BetaDistribution](https:\/\/reference.wolfram.com\/language\/ref\/BetaDistribution.html) is a transformation of [UniformDistribution](https:\/\/reference.wolfram.com\/language\/ref\/UniformDistribution.html):\n\nCopy to clipboard.\n\nIn\\[1\\]:=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/I_63.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0g7kenjf9sq-r56wcy`\n\nDirect link to example\n\nOut\\[1\\]=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/O_53.png)\n\n[UniformDistribution](https:\/\/reference.wolfram.com\/language\/ref\/UniformDistribution.html) is a transformation of [BetaDistribution](https:\/\/reference.wolfram.com\/language\/ref\/BetaDistribution.html):\n\nCopy to clipboard.\n\nIn\\[1\\]:=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/I_64.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0g7kenjf9sq-oumvvz`\n\nDirect link to example\n\nOut\\[1\\]=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/O_54.png)\n\n[BetaDistribution](https:\/\/reference.wolfram.com\/language\/ref\/BetaDistribution.html) is a limiting case of [NoncentralBetaDistribution](https:\/\/reference.wolfram.com\/language\/ref\/NoncentralBetaDistribution.html):\n\nCopy to clipboard.\n\nIn\\[1\\]:=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/I_65.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0g7kenjf9sq-dlrhy8`\n\nDirect link to example\n\nOut\\[1\\]=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/O_55.png)\n\nCopy to clipboard.\n\nIn\\[2\\]:=2\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/I_66.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0g7kenjf9sq-ewhxcn`\n\nDirect link to example\n\nOut\\[2\\]=2\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/O_56.png)\n\nCopy to clipboard.\n\nIn\\[3\\]:=3\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/I_67.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0g7kenjf9sq-pe9es2`\n\nDirect link to example\n\nOut\\[3\\]=3\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/O_57.png)\n\n[BetaPrimeDistribution](https:\/\/reference.wolfram.com\/language\/ref\/BetaPrimeDistribution.html) can be obtained as a transformation of the beta-distributed variable:\n\nCopy to clipboard.\n\nIn\\[1\\]:=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/I_68.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0g7kenjf9sq-sywssu`\n\nDirect link to example\n\nOut\\[1\\]=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/O_58.png)\n\nCopy to clipboard.\n\nIn\\[2\\]:=2\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/I_69.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0g7kenjf9sq-yd91s5`\n\nDirect link to example\n\nOut\\[2\\]=2\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/O_59.png)\n\nBeta distribution is a special case of [PearsonDistribution](https:\/\/reference.wolfram.com\/language\/ref\/PearsonDistribution.html) of type 1:\n\nCopy to clipboard.\n\nIn\\[1\\]:=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/I_70.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0g7kenjf9sq-x3yp0y`\n\nDirect link to example\n\nOut\\[1\\]=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/O_60.png)\n\nCopy to clipboard.\n\nIn\\[2\\]:=2\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/I_71.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0g7kenjf9sq-negofw`\n\nDirect link to example\n\nOut\\[2\\]=2\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/O_61.png)\n\nCopy to clipboard.\n\nIn\\[3\\]:=3\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/I_72.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0g7kenjf9sq-of2jpp`\n\nDirect link to example\n\nOut\\[3\\]=3\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/O_62.png)\n\nBeta distribution can be obtained as a transformation of [GammaDistribution](https:\/\/reference.wolfram.com\/language\/ref\/GammaDistribution.html):\n\nCopy to clipboard.\n\nIn\\[1\\]:=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/I_73.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0g7kenjf9sq-ub1zwk`\n\nDirect link to example\n\nOut\\[1\\]=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/O_63.png)\n\nBeta distribution can be obtained as a transformation of [ChiSquareDistribution](https:\/\/reference.wolfram.com\/language\/ref\/ChiSquareDistribution.html):\n\nCopy to clipboard.\n\nIn\\[1\\]:=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/I_74.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0g7kenjf9sq-bak0x0`\n\nDirect link to example\n\nOut\\[1\\]=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/O_64.png)\n\n[FRatioDistribution](https:\/\/reference.wolfram.com\/language\/ref\/FRatioDistribution.html) can be obtained from beta distribution:\n\nCopy to clipboard.\n\nIn\\[1\\]:=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/I_75.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0g7kenjf9sq-px8s1z`\n\nDirect link to example\n\nCopy to clipboard.\n\nIn\\[2\\]:=2\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/I_76.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0g7kenjf9sq-opzpze`\n\nDirect link to example\n\nOut\\[2\\]=2\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/O_65.png)\n\nCopy to clipboard.\n\nIn\\[3\\]:=3\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/I_77.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0g7kenjf9sq-yz6lah`\n\nDirect link to example\n\nOut\\[3\\]=3\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/O_66.png)\n\nCopy to clipboard.\n\nIn\\[4\\]:=4\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/I_78.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0g7kenjf9sq-tyc15u`\n\nDirect link to example\n\nOut\\[4\\]=4\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/O_67.png)\n\nBeta distribution is an order distribution of variables from [UniformDistribution](https:\/\/reference.wolfram.com\/language\/ref\/UniformDistribution.html):\n\nCopy to clipboard.\n\nIn\\[1\\]:=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/I_79.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0g7kenjf9sq-xgzy15`\n\nDirect link to example\n\nOut\\[1\\]=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/O_68.png)\n\n[ExponentialDistribution](https:\/\/reference.wolfram.com\/language\/ref\/ExponentialDistribution.html) is a limit of a scaled beta distribution:\n\nCopy to clipboard.\n\nIn\\[1\\]:=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/I_80.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0g7kenjf9sq-d5fhku`\n\nDirect link to example\n\nCopy to clipboard.\n\nIn\\[2\\]:=2\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/I_81.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0g7kenjf9sq-don187`\n\nDirect link to example\n\nOut\\[2\\]=2\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/O_69.png)\n\nCopy to clipboard.\n\nIn\\[3\\]:=3\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/I_82.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0g7kenjf9sq-itl7pb`\n\nDirect link to example\n\nOut\\[3\\]=3\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/O_70.png)\n\nCopy to clipboard.\n\nIn\\[4\\]:=4\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/I_83.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0g7kenjf9sq-8b3vgg`\n\nDirect link to example\n\nOut\\[4\\]=4\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/O_71.png)\n\nCopy to clipboard.\n\nIn\\[5\\]:=5\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/I_84.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0g7kenjf9sq-th75rv`\n\nDirect link to example\n\nOut\\[5\\]=5\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/O_72.png)\n\n[ExponentialDistribution](https:\/\/reference.wolfram.com\/language\/ref\/ExponentialDistribution.html) is a transformation of beta distribution:\n\nCopy to clipboard.\n\nIn\\[1\\]:=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/I_85.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0g7kenjf9sq-e3bu74`\n\nDirect link to example\n\nOut\\[1\\]=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/O_73.png)\n\nCopy to clipboard.\n\nIn\\[2\\]:=2\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/I_86.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0g7kenjf9sq-7zbko5`\n\nDirect link to example\n\nOut\\[2\\]=2\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/O_74.png)\n\n[KumaraswamyDistribution](https:\/\/reference.wolfram.com\/language\/ref\/KumaraswamyDistribution.html) is a transformation of beta distribution:\n\nCopy to clipboard.\n\nIn\\[1\\]:=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/I_87.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0g7kenjf9sq-h1ppe8`\n\nDirect link to example\n\nOut\\[1\\]=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/O_75.png)\n\n[KumaraswamyDistribution](https:\/\/reference.wolfram.com\/language\/ref\/KumaraswamyDistribution.html) simplifies to a special case of beta distribution:\n\nCopy to clipboard.\n\nIn\\[1\\]:=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/I_88.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0g7kenjf9sq-tyjged`\n\nDirect link to example\n\nOut\\[1\\]=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/O_76.png)\n\nCopy to clipboard.\n\nIn\\[2\\]:=2\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/I_89.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0g7kenjf9sq-y0ly7p`\n\nDirect link to example\n\nOut\\[2\\]=2\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/O_77.png)\n\nCopy to clipboard.\n\nIn\\[3\\]:=3\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/I_90.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0g7kenjf9sq-gmfqcx`\n\nDirect link to example\n\nOut\\[3\\]=3\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/O_78.png)\n\n[PERTDistribution](https:\/\/reference.wolfram.com\/language\/ref\/PERTDistribution.html) is a transformation of beta distribution:\n\nCopy to clipboard.\n\nIn\\[1\\]:=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/I_91.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0g7kenjf9sq-pvrrzu`\n\nDirect link to example\n\nCopy to clipboard.\n\nIn\\[2\\]:=2\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/I_92.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0g7kenjf9sq-ojaijt`\n\nDirect link to example\n\nOut\\[2\\]=2\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/O_79.png)\n\nCopy to clipboard.\n\nIn\\[3\\]:=3\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/I_93.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0g7kenjf9sq-xapy0l`\n\nDirect link to example\n\nOut\\[3\\]=3\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/O_80.png)\n\nCopy to clipboard.\n\nIn\\[4\\]:=4\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/I_94.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0g7kenjf9sq-tg7dcq`\n\nDirect link to example\n\nOut\\[4\\]=4\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/O_81.png)\n\n[WignerSemicircleDistribution](https:\/\/reference.wolfram.com\/language\/ref\/WignerSemicircleDistribution.html) is a transformation of special beta distribution:\n\nCopy to clipboard.\n\nIn\\[1\\]:=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/I_95.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0g7kenjf9sq-zo9b0n`\n\nDirect link to example\n\nCopy to clipboard.\n\nIn\\[2\\]:=2\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/I_96.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0g7kenjf9sq-djj5oc`\n\nDirect link to example\n\nOut\\[2\\]=2\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/O_82.png)\n\nCopy to clipboard.\n\nIn\\[3\\]:=3\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/I_97.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0g7kenjf9sq-ng3g1m`\n\nDirect link to example\n\nOut\\[3\\]=3\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/O_83.png)\n\nCopy to clipboard.\n\nIn\\[4\\]:=4\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/I_98.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0g7kenjf9sq-s0lugl`\n\nDirect link to example\n\nOut\\[4\\]=4\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/O_84.png)\n\nUnivariate marginals of [DirichletDistribution](https:\/\/reference.wolfram.com\/language\/ref\/DirichletDistribution.html) have beta distribution:\n\nCopy to clipboard.\n\nIn\\[1\\]:=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/I_99.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0g7kenjf9sq-5he2sh`\n\nDirect link to example\n\nOut\\[1\\]=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/O_85.png)\n\nCopy to clipboard.\n\nIn\\[2\\]:=2\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/I_100.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0g7kenjf9sq-3oha4x`\n\nDirect link to example\n\nOut\\[2\\]=2\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/O_86.png)\n\n[BetaBinomialDistribution](https:\/\/reference.wolfram.com\/language\/ref\/BetaBinomialDistribution.html) is a mixture of [BinomialDistribution](https:\/\/reference.wolfram.com\/language\/ref\/BinomialDistribution.html) and [BetaDistribution](https:\/\/reference.wolfram.com\/language\/ref\/BetaDistribution.html):\n\nCopy to clipboard.\n\nIn\\[1\\]:=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/I_101.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0g7kenjf9sq-6a4ms5`\n\nDirect link to example\n\nOut\\[1\\]=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/O_87.png)\n\n[BetaNegativeBinomialDistribution](https:\/\/reference.wolfram.com\/language\/ref\/BetaNegativeBinomialDistribution.html) is a mixture of [NegativeBinomialDistribution](https:\/\/reference.wolfram.com\/language\/ref\/NegativeBinomialDistribution.html) and [BetaDistribution](https:\/\/reference.wolfram.com\/language\/ref\/BetaDistribution.html):\n\nCopy to clipboard.\n\nIn\\[1\\]:=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/I_102.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0g7kenjf9sq-vysest`\n\nDirect link to example\n\nOut\\[1\\]=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/O_88.png)\n\n## Possible Issues  (2)Common pitfalls and unexpected behavior\n[BetaDistribution](https:\/\/reference.wolfram.com\/language\/ref\/BetaDistribution.html) is not defined when either α or β is not a positive real number:\n\nCopy to clipboard.\n\nIn\\[1\\]:=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/I_103.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0g7kenjf9sq-dny`\n\nDirect link to example\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/22.gif)\n\nOut\\[1\\]=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/O_89.png)\n\nSubstitution of invalid parameters into symbolic outputs gives results that are not meaningful:\n\nCopy to clipboard.\n\nIn\\[1\\]:=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/I_104.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0g7kenjf9sq-srb`\n\nDirect link to example\n\nOut\\[1\\]=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/O_90.png)\n\n# See Also\n[GammaDistribution](https:\/\/reference.wolfram.com\/language\/ref\/GammaDistribution.html)  [BinomialDistribution](https:\/\/reference.wolfram.com\/language\/ref\/BinomialDistribution.html)  [Beta](https:\/\/reference.wolfram.com\/language\/ref\/Beta.html)  [BetaRegularized](https:\/\/reference.wolfram.com\/language\/ref\/BetaRegularized.html)  [InverseBetaRegularized](https:\/\/reference.wolfram.com\/language\/ref\/InverseBetaRegularized.html)  [FRatioDistribution](https:\/\/reference.wolfram.com\/language\/ref\/FRatioDistribution.html)  [DirichletDistribution](https:\/\/reference.wolfram.com\/language\/ref\/DirichletDistribution.html)\n\nFunction Repository:  [MeanSpreadBetaDistribution](https:\/\/resources.wolframcloud.com\/FunctionRepository\/resources\/MeanSpreadBetaDistribution)\n\n# Tech Notes\n- [Continuous Distributions](https:\/\/reference.wolfram.com\/language\/tutorial\/NumericalOperationsOnData.html#11002)\n\n# Related Guides\n- [Bounded Domain Distributions](https:\/\/reference.wolfram.com\/language\/guide\/BoundedDomainDistributions.html)\n- [Parametric Statistical Distributions](https:\/\/reference.wolfram.com\/language\/guide\/ParametricStatisticalDistributions.html)\n- [Functions Used in Statistics](https:\/\/reference.wolfram.com\/language\/guide\/FunctionsUsedInStatistics.html)\n\n# History\n[Introduced in 2007 (6.0)](https:\/\/reference.wolfram.com\/language\/guide\/SummaryOfNewFeaturesIn60)  \\|  [Updated in 2016 (10.4)](https:\/\/reference.wolfram.com\/language\/guide\/SummaryOfNewFeaturesIn104)\n\nCite this as:\n\nWolfram Research (2007), BetaDistribution, Wolfram Language function, https:\/\/reference.wolfram.com\/language\/ref\/BetaDistribution.html (updated 2016).\n\nCopy to clipboard.\n\n✖\n\n`Wolfram Research (2007), BetaDistribution, Wolfram Language function, https:\/\/reference.wolfram.com\/language\/ref\/BetaDistribution.html (updated 2016).`\n\n![](https:\/\/reference.wolfram.com\/language-assets-v2\/img\/CitationCloseButton.svg)\n\n#### Text\nWolfram Research (2007), BetaDistribution, Wolfram Language function, https:\/\/reference.wolfram.com\/language\/ref\/BetaDistribution.html (updated 2016).\n\nCopy to clipboard.\n\n✖\n\n`Wolfram Research (2007), BetaDistribution, Wolfram Language function, https:\/\/reference.wolfram.com\/language\/ref\/BetaDistribution.html (updated 2016).`\n\n#### CMS\nWolfram Language. 2007. \"BetaDistribution.\" Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2016. https:\/\/reference.wolfram.com\/language\/ref\/BetaDistribution.html.\n\nCopy to clipboard.\n\n✖\n\n`Wolfram Language. 2007. \"BetaDistribution.\" Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2016. https:\/\/reference.wolfram.com\/language\/ref\/BetaDistribution.html.`\n\n#### APA\nWolfram Language. (2007). BetaDistribution. Wolfram Language & System Documentation Center. Retrieved from https:\/\/reference.wolfram.com\/language\/ref\/BetaDistribution.html\n\nCopy to clipboard.\n\n✖\n\n`Wolfram Language. (2007). BetaDistribution. Wolfram Language & System Documentation Center. Retrieved from https:\/\/reference.wolfram.com\/language\/ref\/BetaDistribution.html`\n\n#### BibTeX\n@misc{reference.wolfram\\_2025\\_betadistribution, author=\"Wolfram Research\", title=\"{BetaDistribution}\", year=\"2016\", howpublished=\"\\\\url{https:\/\/reference.wolfram.com\/language\/ref\/BetaDistribution.html}\", note=\\[Accessed: 11-April-2026\\]}\n\nCopy to clipboard.\n\n✖\n\n`@misc{reference.wolfram_2025_betadistribution, author=\"Wolfram Research\", title=\"{BetaDistribution}\", year=\"2016\", howpublished=\"\\url{https:\/\/reference.wolfram.com\/language\/ref\/BetaDistribution.html}\", note=[Accessed: 11-April-2026]}`\n\n#### BibLaTeX\n@online{reference.wolfram\\_2025\\_betadistribution, organization={Wolfram Research}, title={BetaDistribution}, year={2016}, url={https:\/\/reference.wolfram.com\/language\/ref\/BetaDistribution.html}, note=\\[Accessed: 11-April-2026\\]}\n\nCopy to clipboard.\n\n✖\n\n`@online{reference.wolfram_2025_betadistribution, organization={Wolfram Research}, title={BetaDistribution}, year={2016}, url={https:\/\/reference.wolfram.com\/language\/ref\/BetaDistribution.html}, note=[Accessed: 11-April-2026]}`\n\nGive Feedback\n\n[Top](https:\/\/reference.wolfram.com\/language\/ref\/BetaDistribution.html#top)\n\n[Introduction for Programmers](https:\/\/www.wolfram.com\/language\/fast-introduction-for-programmers\/en\/)\n\n[Introductory Book](https:\/\/www.wolfram.com\/language\/elementary-introduction\/)\n\n[Wolfram Function Repository](http:\/\/resources.wolframcloud.com\/FunctionRepository\/)  \\|  [Wolfram Data Repository](https:\/\/datarepository.wolframcloud.com\/)  \\|  [Wolfram Data Drop](https:\/\/datadrop.wolframcloud.com\/)  \\|  [Wolfram Language Products](https:\/\/www.wolfram.com\/products\/)\n\n[Top](https:\/\/reference.wolfram.com\/language\/ref\/BetaDistribution.html#top)\n\n- Products\n- [Wolfram\\|One](https:\/\/www.wolfram.com\/wolfram-one\/?source=footer)\n- 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Also](https:\/\/reference.wolfram.com\/language\/ref\/BetaDistribution.html)\n  - [GammaDistribution](https:\/\/reference.wolfram.com\/language\/ref\/GammaDistribution.html)\n  - [BinomialDistribution](https:\/\/reference.wolfram.com\/language\/ref\/BinomialDistribution.html)\n  - [Beta](https:\/\/reference.wolfram.com\/language\/ref\/Beta.html)\n  - [BetaRegularized](https:\/\/reference.wolfram.com\/language\/ref\/BetaRegularized.html)\n  - [InverseBetaRegularized](https:\/\/reference.wolfram.com\/language\/ref\/InverseBetaRegularized.html)\n  - [FRatioDistribution](https:\/\/reference.wolfram.com\/language\/ref\/FRatioDistribution.html)\n  - [DirichletDistribution](https:\/\/reference.wolfram.com\/language\/ref\/DirichletDistribution.html)\n- [Related Guides](https:\/\/reference.wolfram.com\/language\/ref\/BetaDistribution.html)\n  - [Bounded Domain Distributions](https:\/\/reference.wolfram.com\/language\/guide\/BoundedDomainDistributions.html)\n  - [Parametric Statistical Distributions](https:\/\/reference.wolfram.com\/language\/guide\/ParametricStatisticalDistributions.html)\n  - [Functions Used in Statistics](https:\/\/reference.wolfram.com\/language\/guide\/FunctionsUsedInStatistics.html)\n- [Tech Notes](https:\/\/reference.wolfram.com\/language\/ref\/BetaDistribution.html)\n  - [Continuous Distributions](https:\/\/reference.wolfram.com\/language\/tutorial\/NumericalOperationsOnData.html#11002)\n- - [See Also](https:\/\/reference.wolfram.com\/language\/ref\/BetaDistribution.html)\n    - [GammaDistribution](https:\/\/reference.wolfram.com\/language\/ref\/GammaDistribution.html)\n    - [BinomialDistribution](https:\/\/reference.wolfram.com\/language\/ref\/BinomialDistribution.html)\n    - [Beta](https:\/\/reference.wolfram.com\/language\/ref\/Beta.html)\n    - [BetaRegularized](https:\/\/reference.wolfram.com\/language\/ref\/BetaRegularized.html)\n    - [InverseBetaRegularized](https:\/\/reference.wolfram.com\/language\/ref\/InverseBetaRegularized.html)\n    - [FRatioDistribution](https:\/\/reference.wolfram.com\/language\/ref\/FRatioDistribution.html)\n    - [DirichletDistribution](https:\/\/reference.wolfram.com\/language\/ref\/DirichletDistribution.html)\n  - [Related Guides](https:\/\/reference.wolfram.com\/language\/ref\/BetaDistribution.html)\n    - [Bounded Domain Distributions](https:\/\/reference.wolfram.com\/language\/guide\/BoundedDomainDistributions.html)\n    - [Parametric Statistical Distributions](https:\/\/reference.wolfram.com\/language\/guide\/ParametricStatisticalDistributions.html)\n    - [Functions Used in Statistics](https:\/\/reference.wolfram.com\/language\/guide\/FunctionsUsedInStatistics.html)\n  - [Tech Notes](https:\/\/reference.wolfram.com\/language\/ref\/BetaDistribution.html)\n    - [Continuous Distributions](https:\/\/reference.wolfram.com\/language\/tutorial\/NumericalOperationsOnData.html#11002)\n\nExamples\n\nBasic Examples\n\nScope\n\nApplications\n\nProperties & Relations\n\nPossible Issues\n\n[BetaDistribution](https:\/\/reference.wolfram.com\/language\/ref\/BetaDistribution.html)\\[α,β\\]\n\nCopy to clipboard.\n\n`BetaDistribution[α,β]`\n\nrepresents a continuous beta distribution with shape parameters α and β.\n\n## Details\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/details_1.png)\n\n## Background & Context\n- [BetaDistribution](https:\/\/reference.wolfram.com\/language\/ref\/BetaDistribution.html)\\[α,β\\] represents a statistical distribution defined over the interval ![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/6.png) and parametrized by two positive values α, β known as \"shape parameters\", which, roughly speaking, determine the \"fatness\" of the left and right tails in the probability density function (PDF). Depending on the values of α and β, the PDF of the beta distribution may be monotonic increasing, monotonic decreasing, or unimodal with potential singularities approaching the boundaries of its domain.\n- The beta distribution arises as a prior distribution for binomial proportions in Bayesian analysis. It is also commonly used to model random variables limited to a finite interval. For example, the distribution of the ![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/7.png)![^(th)](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/8.png) smallest element in a continuous, independent, and uniformly distributed sample of size of ![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/9.png) can be computed using [OrderDistribution](https:\/\/reference.wolfram.com\/language\/ref\/OrderDistribution.html)\\[{[UniformDistribution](https:\/\/reference.wolfram.com\/language\/ref\/UniformDistribution.html)\\[\\],n},k\\] and is precisely equal to [BetaDistribution](https:\/\/reference.wolfram.com\/language\/ref\/BetaDistribution.html)\\[k,n\\-k\\+1\\]. In addition to its statistical significance, the beta distribution also plays a fundamental role in a number of scientific fields, including phenomena related to allele frequency distribution, soil property variability, geological mineral-to-rock ratios, and HIV transmission behavior.\n- [RandomVariate](https:\/\/reference.wolfram.com\/language\/ref\/RandomVariate.html) can be used to give one or more machine- or arbitrary-precision (the latter via the [WorkingPrecision](https:\/\/reference.wolfram.com\/language\/ref\/WorkingPrecision.html) option) pseudorandom variates from a beta distribution. [Distributed](https:\/\/reference.wolfram.com\/language\/ref\/Distributed.html)\\[x,BetaDistribution\\[α,β\\]\\], written more concisely as xBetaDistribution\\[α,β\\], can be used to assert that a random variable x is distributed according to a beta distribution. Such an assertion can then be used in functions such as [Probability](https:\/\/reference.wolfram.com\/language\/ref\/Probability.html), [NProbability](https:\/\/reference.wolfram.com\/language\/ref\/NProbability.html), [Expectation](https:\/\/reference.wolfram.com\/language\/ref\/Expectation.html), and [NExpectation](https:\/\/reference.wolfram.com\/language\/ref\/NExpectation.html).\n- The probability density and cumulative distribution functions may be given using [PDF](https:\/\/reference.wolfram.com\/language\/ref\/PDF.html)\\[BetaDistribution\\[α,β\\],x\\] and [CDF](https:\/\/reference.wolfram.com\/language\/ref\/CDF.html)\\[BetaDistribution\\[α,β\\],x\\]. The mean, median, variance, raw moments, and central moments may be computed using [Mean](https:\/\/reference.wolfram.com\/language\/ref\/Mean.html), [Median](https:\/\/reference.wolfram.com\/language\/ref\/Median.html), [Variance](https:\/\/reference.wolfram.com\/language\/ref\/Variance.html), [Moment](https:\/\/reference.wolfram.com\/language\/ref\/Moment.html), and [CentralMoment](https:\/\/reference.wolfram.com\/language\/ref\/CentralMoment.html), respectively.\n- [DistributionFitTest](https:\/\/reference.wolfram.com\/language\/ref\/DistributionFitTest.html) can be used to test if a given dataset is consistent with a beta distribution, [EstimatedDistribution](https:\/\/reference.wolfram.com\/language\/ref\/EstimatedDistribution.html) to estimate a beta parametric distribution from given data, and [FindDistributionParameters](https:\/\/reference.wolfram.com\/language\/ref\/FindDistributionParameters.html) to fit data to a beta distribution. [ProbabilityPlot](https:\/\/reference.wolfram.com\/language\/ref\/ProbabilityPlot.html) can be used to generate a plot of the CDF of given data against the CDF of a symbolic beta distribution and [QuantilePlot](https:\/\/reference.wolfram.com\/language\/ref\/QuantilePlot.html) to generate a plot of the quantiles of given data against the quantiles of a symbolic beta distribution.\n- [TransformedDistribution](https:\/\/reference.wolfram.com\/language\/ref\/TransformedDistribution.html) can be used to represent a transformed beta distribution, [CensoredDistribution](https:\/\/reference.wolfram.com\/language\/ref\/CensoredDistribution.html) to represent the distribution of values censored between upper and lower values, and [TruncatedDistribution](https:\/\/reference.wolfram.com\/language\/ref\/TruncatedDistribution.html) to represent the distribution of values truncated between upper and lower values. [CopulaDistribution](https:\/\/reference.wolfram.com\/language\/ref\/CopulaDistribution.html) can be used to build higher-dimensional distributions that contain a beta distribution, and [ProductDistribution](https:\/\/reference.wolfram.com\/language\/ref\/ProductDistribution.html) can be used to compute a joint distribution with independent component distributions involving beta distributions.\n- The beta distribution is related to a number of other distributions. For example, [BetaDistribution](https:\/\/reference.wolfram.com\/language\/ref\/BetaDistribution.html) is the so-called \"conjugate prior\" for the parameters of a number of other distributions, including [BernoulliDistribution](https:\/\/reference.wolfram.com\/language\/ref\/BernoulliDistribution.html), [BinomialDistribution](https:\/\/reference.wolfram.com\/language\/ref\/BinomialDistribution.html), [NegativeBinomialDistribution](https:\/\/reference.wolfram.com\/language\/ref\/NegativeBinomialDistribution.html), and [GeometricDistribution](https:\/\/reference.wolfram.com\/language\/ref\/GeometricDistribution.html). Moreover, [BetaDistribution](https:\/\/reference.wolfram.com\/language\/ref\/BetaDistribution.html) generalizes both [UniformDistribution](https:\/\/reference.wolfram.com\/language\/ref\/UniformDistribution.html) and [PowerDistribution](https:\/\/reference.wolfram.com\/language\/ref\/PowerDistribution.html) in the sense that (modulo inclusion of the endpoints ![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/10.png) and ![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/BetaDistribution.en\/11.png)), [PDF](https:\/\/reference.wolfram.com\/language\/ref\/PDF.html)\\[BetaDistribution\\[1,1\\],x\\] is equal to both [PDF](https:\/\/reference.wolfram.com\/language\/ref\/PDF.html)\\[[UniformDistribution](https:\/\/reference.wolfram.com\/language\/ref\/UniformDistribution.html)\\[\\],x\\] and [PDF](https:\/\/reference.wolfram.com\/language\/ref\/PDF.html)\\[[PowerDistribution](https:\/\/reference.wolfram.com\/language\/ref\/PowerDistribution.html)\\[1,1\\],x\\]. [BetaDistribution](https:\/\/reference.wolfram.com\/language\/ref\/BetaDistribution.html) can also be obtained as transformations of [KumaraswamyDistribution](https:\/\/reference.wolfram.com\/language\/ref\/KumaraswamyDistribution.html) and [NoncentralBetaDistribution](https:\/\/reference.wolfram.com\/language\/ref\/NoncentralBetaDistribution.html) and is closely related to [PERTDistribution](https:\/\/reference.wolfram.com\/language\/ref\/PERTDistribution.html), [PearsonDistribution](https:\/\/reference.wolfram.com\/language\/ref\/PearsonDistribution.html), [ChiSquareDistribution](https:\/\/reference.wolfram.com\/language\/ref\/ChiSquareDistribution.html), [GammaDistribution](https:\/\/reference.wolfram.com\/language\/ref\/GammaDistribution.html), [FRatioDistribution](https:\/\/reference.wolfram.com\/language\/ref\/FRatioDistribution.html), and [BetaPrimeDistribution](https:\/\/reference.wolfram.com\/language\/ref\/BetaPrimeDistribution.html).\n\n## Examples\nopen all close all\n\n## Basic Examples  (4)Summary of the most common use cases\n## Scope  (8)Survey of the scope of standard use cases\n## Applications  (3)Sample problems that can be solved with this function\n## Properties & Relations  (21)Properties of the function, and connections to other functions\n## Possible Issues  (2)Common pitfalls and unexpected behavior\n\nWolfram Research (2007), BetaDistribution, Wolfram Language function, https:\/\/reference.wolfram.com\/language\/ref\/BetaDistribution.html (updated 2016).\n\nCopy to clipboard.\n\n`Wolfram Research (2007), BetaDistribution, Wolfram Language function, https:\/\/reference.wolfram.com\/language\/ref\/BetaDistribution.html (updated 2016).`\n\n#### Text\nWolfram Research (2007), BetaDistribution, Wolfram Language function, https:\/\/reference.wolfram.com\/language\/ref\/BetaDistribution.html (updated 2016).\n\nCopy to clipboard.\n\n`Wolfram Research (2007), BetaDistribution, Wolfram Language function, https:\/\/reference.wolfram.com\/language\/ref\/BetaDistribution.html (updated 2016).`\n\n#### CMS\nWolfram Language. 2007. \"BetaDistribution.\" Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2016. https:\/\/reference.wolfram.com\/language\/ref\/BetaDistribution.html.\n\nCopy to clipboard.\n\n`Wolfram Language. 2007. \"BetaDistribution.\" Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2016. https:\/\/reference.wolfram.com\/language\/ref\/BetaDistribution.html.`\n\n#### APA\nWolfram Language. (2007). BetaDistribution. Wolfram Language & System Documentation Center. Retrieved from https:\/\/reference.wolfram.com\/language\/ref\/BetaDistribution.html\n\nCopy to clipboard.\n\n`Wolfram Language. (2007). BetaDistribution. Wolfram Language & System Documentation Center. Retrieved from https:\/\/reference.wolfram.com\/language\/ref\/BetaDistribution.html`\n\n#### BibTeX\n@misc{reference.wolfram\\_2025\\_betadistribution, author=\"Wolfram Research\", title=\"{BetaDistribution}\", year=\"2016\", howpublished=\"\\\\url{https:\/\/reference.wolfram.com\/language\/ref\/BetaDistribution.html}\", note=\\[Accessed: 11-April-2026\\]}\n\nCopy to clipboard.\n\n`@misc{reference.wolfram_2025_betadistribution, author=\"Wolfram Research\", title=\"{BetaDistribution}\", year=\"2016\", howpublished=\"\\url{https:\/\/reference.wolfram.com\/language\/ref\/BetaDistribution.html}\", note=[Accessed: 11-April-2026]}`\n\n#### BibLaTeX\n@online{reference.wolfram\\_2025\\_betadistribution, organization={Wolfram Research}, title={BetaDistribution}, year={2016}, url={https:\/\/reference.wolfram.com\/language\/ref\/BetaDistribution.html}, note=\\[Accessed: 11-April-2026\\]}\n\nCopy to clipboard.\n\n`@online{reference.wolfram_2025_betadistribution, organization={Wolfram Research}, title={BetaDistribution}, year={2016}, url={https:\/\/reference.wolfram.com\/language\/ref\/BetaDistribution.html}, note=[Accessed: 11-April-2026]}`","meta_canonical":null}

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Boilerpipe Text	See Also GammaDistribution BinomialDistribution Beta BetaRegularized InverseBetaRegularized FRatioDistribution DirichletDistribution Related Guides Bounded Domain Distributions Parametric Statistical Distributions Functions Used in Statistics Tech Notes Continuous Distributions See Also GammaDistribution BinomialDistribution Beta BetaRegularized InverseBetaRegularized FRatioDistribution DirichletDistribution Related Guides Bounded Domain Distributions Parametric Statistical Distributions Functions Used in Statistics Tech Notes Continuous Distributions Examples Basic Examples Scope Applications Properties & Relations Possible Issues BetaDistribution [ α , β ] Copy to clipboard. BetaDistribution[α,β] represents a continuous beta distribution with shape parameters α and β . Details Background & Context BetaDistribution [ α , β ] represents a statistical distribution defined over the interval and parametrized by two positive values α , β known as "shape parameters", which, roughly speaking, determine the "fatness" of the left and right tails in the probability density function (PDF). Depending on the values of α and β , the PDF of the beta distribution may be monotonic increasing, monotonic decreasing, or unimodal with potential singularities approaching the boundaries of its domain. The beta distribution arises as a prior distribution for binomial proportions in Bayesian analysis. It is also commonly used to model random variables limited to a finite interval. For example, the distribution of the smallest element in a continuous, independent, and uniformly distributed sample of size of can be computed using OrderDistribution [ { UniformDistribution [ ] , n } , k ] and is precisely equal to BetaDistribution [ k , n - k +1 ] . In addition to its statistical significance, the beta distribution also plays a fundamental role in a number of scientific fields, including phenomena related to allele frequency distribution, soil property variability, geological mineral-to-rock ratios, and HIV transmission behavior. RandomVariate can be used to give one or more machine- or arbitrary-precision (the latter via the WorkingPrecision option) pseudorandom variates from a beta distribution. Distributed [ x , BetaDistribution [ α , β ] ] , written more concisely as x  BetaDistribution [ α , β ] , can be used to assert that a random variable x is distributed according to a beta distribution. Such an assertion can then be used in functions such as Probability , NProbability , Expectation , and NExpectation . The probability density and cumulative distribution functions may be given using PDF [ BetaDistribution [ α , β ] , x ] and CDF [ BetaDistribution [ α , β ] , x ] . The mean, median, variance, raw moments, and central moments may be computed using Mean , Median , Variance , Moment , and CentralMoment , respectively. DistributionFitTest can be used to test if a given dataset is consistent with a beta distribution, EstimatedDistribution to estimate a beta parametric distribution from given data, and FindDistributionParameters to fit data to a beta distribution. ProbabilityPlot can be used to generate a plot of the CDF of given data against the CDF of a symbolic beta distribution and QuantilePlot to generate a plot of the quantiles of given data against the quantiles of a symbolic beta distribution. TransformedDistribution can be used to represent a transformed beta distribution, CensoredDistribution to represent the distribution of values censored between upper and lower values, and TruncatedDistribution to represent the distribution of values truncated between upper and lower values. CopulaDistribution can be used to build higher-dimensional distributions that contain a beta distribution, and ProductDistribution can be used to compute a joint distribution with independent component distributions involving beta distributions. The beta distribution is related to a number of other distributions. For example, BetaDistribution is the so-called "conjugate prior" for the parameters of a number of other distributions, including BernoulliDistribution , BinomialDistribution , NegativeBinomialDistribution , and GeometricDistribution . Moreover, BetaDistribution generalizes both UniformDistribution and PowerDistribution in the sense that (modulo inclusion of the endpoints and ), PDF [ BetaDistribution [ 1 , 1 ] , x ] is equal to both PDF [ UniformDistribution [ ] , x ] and PDF [ PowerDistribution [ 1 , 1 ] , x ] . BetaDistribution can also be obtained as transformations of KumaraswamyDistribution and NoncentralBetaDistribution and is closely related to PERTDistribution , PearsonDistribution , ChiSquareDistribution , GammaDistribution , FRatioDistribution , and BetaPrimeDistribution . Examples open all close all Basic Examples (4) Summary of the most common use cases Scope (8) Survey of the scope of standard use cases Applications (3) Sample problems that can be solved with this function Properties & Relations (21) Properties of the function, and connections to other functions Possible Issues (2) Common pitfalls and unexpected behavior Wolfram Research (2007), BetaDistribution, Wolfram Language function, https://reference.wolfram.com/language/ref/BetaDistribution.html (updated 2016). Copy to clipboard. Wolfram Research (2007), BetaDistribution, Wolfram Language function, https://reference.wolfram.com/language/ref/BetaDistribution.html (updated 2016). Text Wolfram Research (2007), BetaDistribution, Wolfram Language function, https://reference.wolfram.com/language/ref/BetaDistribution.html (updated 2016). Copy to clipboard. Wolfram Research (2007), BetaDistribution, Wolfram Language function, https://reference.wolfram.com/language/ref/BetaDistribution.html (updated 2016). CMS Wolfram Language. 2007. "BetaDistribution." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2016. https://reference.wolfram.com/language/ref/BetaDistribution.html. Copy to clipboard. Wolfram Language. 2007. "BetaDistribution." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2016. https://reference.wolfram.com/language/ref/BetaDistribution.html. APA Wolfram Language. (2007). BetaDistribution. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/BetaDistribution.html Copy to clipboard. Wolfram Language. (2007). BetaDistribution. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/BetaDistribution.html BibTeX @misc{reference.wolfram_2025_betadistribution, author="Wolfram Research", title="{BetaDistribution}", year="2016", howpublished="\url{https://reference.wolfram.com/language/ref/BetaDistribution.html}", note=[Accessed: 11-April-2026]} Copy to clipboard. @misc{reference.wolfram_2025_betadistribution, author="Wolfram Research", title="{BetaDistribution}", year="2016", howpublished="\url{https://reference.wolfram.com/language/ref/BetaDistribution.html}", note=[Accessed: 11-April-2026]} BibLaTeX @online{reference.wolfram_2025_betadistribution, organization={Wolfram Research}, title={BetaDistribution}, year={2016}, url={https://reference.wolfram.com/language/ref/BetaDistribution.html}, note=[Accessed: 11-April-2026]} Copy to clipboard. @online{reference.wolfram_2025_betadistribution, organization={Wolfram Research}, title={BetaDistribution}, year={2016}, url={https://reference.wolfram.com/language/ref/BetaDistribution.html}, note=[Accessed: 11-April-2026]}
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[DirichletDistribution](https://reference.wolfram.com/language/ref/DirichletDistribution.html) - [Related Guides](https://reference.wolfram.com/language/ref/BetaDistribution.html) - [Bounded Domain Distributions](https://reference.wolfram.com/language/guide/BoundedDomainDistributions.html) - [Parametric Statistical Distributions](https://reference.wolfram.com/language/guide/ParametricStatisticalDistributions.html) - [Functions Used in Statistics](https://reference.wolfram.com/language/guide/FunctionsUsedInStatistics.html) - [Tech Notes](https://reference.wolfram.com/language/ref/BetaDistribution.html) - [Continuous Distributions](https://reference.wolfram.com/language/tutorial/NumericalOperationsOnData.html#11002) - - [See Also](https://reference.wolfram.com/language/ref/BetaDistribution.html) - [GammaDistribution](https://reference.wolfram.com/language/ref/GammaDistribution.html) - [BinomialDistribution](https://reference.wolfram.com/language/ref/BinomialDistribution.html) - [Beta](https://reference.wolfram.com/language/ref/Beta.html) - [BetaRegularized](https://reference.wolfram.com/language/ref/BetaRegularized.html) - [InverseBetaRegularized](https://reference.wolfram.com/language/ref/InverseBetaRegularized.html) - [FRatioDistribution](https://reference.wolfram.com/language/ref/FRatioDistribution.html) - [DirichletDistribution](https://reference.wolfram.com/language/ref/DirichletDistribution.html) - [Related Guides](https://reference.wolfram.com/language/ref/BetaDistribution.html) - [Bounded Domain Distributions](https://reference.wolfram.com/language/guide/BoundedDomainDistributions.html) - [Parametric Statistical Distributions](https://reference.wolfram.com/language/guide/ParametricStatisticalDistributions.html) - [Functions Used in Statistics](https://reference.wolfram.com/language/guide/FunctionsUsedInStatistics.html) - [Tech Notes](https://reference.wolfram.com/language/ref/BetaDistribution.html) - [Continuous Distributions](https://reference.wolfram.com/language/tutorial/NumericalOperationsOnData.html#11002) [BetaDistribution](https://reference.wolfram.com/language/ref/BetaDistribution.html)\[α,β\] represents a continuous beta distribution with shape parameters α and β. Details ![Details and Options](https://reference.wolfram.com/language-assets-v2/img/DetailsOptions-gray.png) ![Details and Options](https://reference.wolfram.com/language-assets-v2/img/DetailsOptions-hover.png) Background & Context Examples Basic Examples Scope Applications Properties & Relations Possible Issues See Also Tech Notes Related Guides History Cite this Page BUILT-IN SYMBOL - [See Also](https://reference.wolfram.com/language/ref/BetaDistribution.html) - [GammaDistribution](https://reference.wolfram.com/language/ref/GammaDistribution.html) - [BinomialDistribution](https://reference.wolfram.com/language/ref/BinomialDistribution.html) - [Beta](https://reference.wolfram.com/language/ref/Beta.html) - [BetaRegularized](https://reference.wolfram.com/language/ref/BetaRegularized.html) - [InverseBetaRegularized](https://reference.wolfram.com/language/ref/InverseBetaRegularized.html) - [FRatioDistribution](https://reference.wolfram.com/language/ref/FRatioDistribution.html) - [DirichletDistribution](https://reference.wolfram.com/language/ref/DirichletDistribution.html) - [Related Guides](https://reference.wolfram.com/language/ref/BetaDistribution.html) - [Bounded Domain Distributions](https://reference.wolfram.com/language/guide/BoundedDomainDistributions.html) - [Parametric Statistical Distributions](https://reference.wolfram.com/language/guide/ParametricStatisticalDistributions.html) - [Functions Used in Statistics](https://reference.wolfram.com/language/guide/FunctionsUsedInStatistics.html) - [Tech Notes](https://reference.wolfram.com/language/ref/BetaDistribution.html) - [Continuous Distributions](https://reference.wolfram.com/language/tutorial/NumericalOperationsOnData.html#11002) - - [See Also](https://reference.wolfram.com/language/ref/BetaDistribution.html) - [GammaDistribution](https://reference.wolfram.com/language/ref/GammaDistribution.html) - [BinomialDistribution](https://reference.wolfram.com/language/ref/BinomialDistribution.html) - [Beta](https://reference.wolfram.com/language/ref/Beta.html) - [BetaRegularized](https://reference.wolfram.com/language/ref/BetaRegularized.html) - [InverseBetaRegularized](https://reference.wolfram.com/language/ref/InverseBetaRegularized.html) - [FRatioDistribution](https://reference.wolfram.com/language/ref/FRatioDistribution.html) - [DirichletDistribution](https://reference.wolfram.com/language/ref/DirichletDistribution.html) - [Related Guides](https://reference.wolfram.com/language/ref/BetaDistribution.html) - [Bounded Domain Distributions](https://reference.wolfram.com/language/guide/BoundedDomainDistributions.html) - [Parametric Statistical Distributions](https://reference.wolfram.com/language/guide/ParametricStatisticalDistributions.html) - [Functions Used in Statistics](https://reference.wolfram.com/language/guide/FunctionsUsedInStatistics.html) - [Tech Notes](https://reference.wolfram.com/language/ref/BetaDistribution.html) - [Continuous Distributions](https://reference.wolfram.com/language/tutorial/NumericalOperationsOnData.html#11002) # BetaDistributionCopy to clipboard. ✖ `BetaDistribution` [BetaDistribution](https://reference.wolfram.com/language/ref/BetaDistribution.html)\[α,β\] Copy to clipboard. ✖ `BetaDistribution[α,β]` represents a continuous beta distribution with shape parameters α and β. # Details ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/details_1.png) - The probability density for value ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/1.png) in a beta distribution is proportional to ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/2.png) for ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/3.png), and is zero for ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/4.png) or ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/5.png). [»](https://reference.wolfram.com/language/ref/BetaDistribution.html#2046786362) - [BetaDistribution](https://reference.wolfram.com/language/ref/BetaDistribution.html) allows α and β to be any positive real numbers. - [BetaDistribution](https://reference.wolfram.com/language/ref/BetaDistribution.html) allows α and β to be dimensionless quantities. [»](https://reference.wolfram.com/language/ref/BetaDistribution.html#504092574) - [BetaDistribution](https://reference.wolfram.com/language/ref/BetaDistribution.html) can be used with such functions as [Mean](https://reference.wolfram.com/language/ref/Mean.html), [CDF](https://reference.wolfram.com/language/ref/CDF.html), and [RandomVariate](https://reference.wolfram.com/language/ref/RandomVariate.html). [»](https://reference.wolfram.com/language/ref/BetaDistribution.html#10542) # Background & Context - [BetaDistribution](https://reference.wolfram.com/language/ref/BetaDistribution.html)\[α,β\] represents a statistical distribution defined over the interval ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/6.png) and parametrized by two positive values α, β known as "shape parameters", which, roughly speaking, determine the "fatness" of the left and right tails in the probability density function (PDF). Depending on the values of α and β, the PDF of the beta distribution may be monotonic increasing, monotonic decreasing, or unimodal with potential singularities approaching the boundaries of its domain. - The beta distribution arises as a prior distribution for binomial proportions in Bayesian analysis. It is also commonly used to model random variables limited to a finite interval. For example, the distribution of the ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/7.png)![^(th)](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/8.png) smallest element in a continuous, independent, and uniformly distributed sample of size of ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/9.png) can be computed using [OrderDistribution](https://reference.wolfram.com/language/ref/OrderDistribution.html)\[{[UniformDistribution](https://reference.wolfram.com/language/ref/UniformDistribution.html)\[\],n},k\] and is precisely equal to [BetaDistribution](https://reference.wolfram.com/language/ref/BetaDistribution.html)\[k,n\-k\+1\]. In addition to its statistical significance, the beta distribution also plays a fundamental role in a number of scientific fields, including phenomena related to allele frequency distribution, soil property variability, geological mineral-to-rock ratios, and HIV transmission behavior. - [RandomVariate](https://reference.wolfram.com/language/ref/RandomVariate.html) can be used to give one or more machine- or arbitrary-precision (the latter via the [WorkingPrecision](https://reference.wolfram.com/language/ref/WorkingPrecision.html) option) pseudorandom variates from a beta distribution. [Distributed](https://reference.wolfram.com/language/ref/Distributed.html)\[x,BetaDistribution\[α,β\]\], written more concisely as xBetaDistribution\[α,β\], can be used to assert that a random variable x is distributed according to a beta distribution. Such an assertion can then be used in functions such as [Probability](https://reference.wolfram.com/language/ref/Probability.html), [NProbability](https://reference.wolfram.com/language/ref/NProbability.html), [Expectation](https://reference.wolfram.com/language/ref/Expectation.html), and [NExpectation](https://reference.wolfram.com/language/ref/NExpectation.html). - The probability density and cumulative distribution functions may be given using [PDF](https://reference.wolfram.com/language/ref/PDF.html)\[BetaDistribution\[α,β\],x\] and [CDF](https://reference.wolfram.com/language/ref/CDF.html)\[BetaDistribution\[α,β\],x\]. The mean, median, variance, raw moments, and central moments may be computed using [Mean](https://reference.wolfram.com/language/ref/Mean.html), [Median](https://reference.wolfram.com/language/ref/Median.html), [Variance](https://reference.wolfram.com/language/ref/Variance.html), [Moment](https://reference.wolfram.com/language/ref/Moment.html), and [CentralMoment](https://reference.wolfram.com/language/ref/CentralMoment.html), respectively. - [DistributionFitTest](https://reference.wolfram.com/language/ref/DistributionFitTest.html) can be used to test if a given dataset is consistent with a beta distribution, [EstimatedDistribution](https://reference.wolfram.com/language/ref/EstimatedDistribution.html) to estimate a beta parametric distribution from given data, and [FindDistributionParameters](https://reference.wolfram.com/language/ref/FindDistributionParameters.html) to fit data to a beta distribution. [ProbabilityPlot](https://reference.wolfram.com/language/ref/ProbabilityPlot.html) can be used to generate a plot of the CDF of given data against the CDF of a symbolic beta distribution and [QuantilePlot](https://reference.wolfram.com/language/ref/QuantilePlot.html) to generate a plot of the quantiles of given data against the quantiles of a symbolic beta distribution. - [TransformedDistribution](https://reference.wolfram.com/language/ref/TransformedDistribution.html) can be used to represent a transformed beta distribution, [CensoredDistribution](https://reference.wolfram.com/language/ref/CensoredDistribution.html) to represent the distribution of values censored between upper and lower values, and [TruncatedDistribution](https://reference.wolfram.com/language/ref/TruncatedDistribution.html) to represent the distribution of values truncated between upper and lower values. [CopulaDistribution](https://reference.wolfram.com/language/ref/CopulaDistribution.html) can be used to build higher-dimensional distributions that contain a beta distribution, and [ProductDistribution](https://reference.wolfram.com/language/ref/ProductDistribution.html) can be used to compute a joint distribution with independent component distributions involving beta distributions. - The beta distribution is related to a number of other distributions. For example, [BetaDistribution](https://reference.wolfram.com/language/ref/BetaDistribution.html) is the so-called "conjugate prior" for the parameters of a number of other distributions, including [BernoulliDistribution](https://reference.wolfram.com/language/ref/BernoulliDistribution.html), [BinomialDistribution](https://reference.wolfram.com/language/ref/BinomialDistribution.html), [NegativeBinomialDistribution](https://reference.wolfram.com/language/ref/NegativeBinomialDistribution.html), and [GeometricDistribution](https://reference.wolfram.com/language/ref/GeometricDistribution.html). Moreover, [BetaDistribution](https://reference.wolfram.com/language/ref/BetaDistribution.html) generalizes both [UniformDistribution](https://reference.wolfram.com/language/ref/UniformDistribution.html) and [PowerDistribution](https://reference.wolfram.com/language/ref/PowerDistribution.html) in the sense that (modulo inclusion of the endpoints ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/10.png) and ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/11.png)), [PDF](https://reference.wolfram.com/language/ref/PDF.html)\[BetaDistribution\[1,1\],x\] is equal to both [PDF](https://reference.wolfram.com/language/ref/PDF.html)\[[UniformDistribution](https://reference.wolfram.com/language/ref/UniformDistribution.html)\[\],x\] and [PDF](https://reference.wolfram.com/language/ref/PDF.html)\[[PowerDistribution](https://reference.wolfram.com/language/ref/PowerDistribution.html)\[1,1\],x\]. [BetaDistribution](https://reference.wolfram.com/language/ref/BetaDistribution.html) can also be obtained as transformations of [KumaraswamyDistribution](https://reference.wolfram.com/language/ref/KumaraswamyDistribution.html) and [NoncentralBetaDistribution](https://reference.wolfram.com/language/ref/NoncentralBetaDistribution.html) and is closely related to [PERTDistribution](https://reference.wolfram.com/language/ref/PERTDistribution.html), [PearsonDistribution](https://reference.wolfram.com/language/ref/PearsonDistribution.html), [ChiSquareDistribution](https://reference.wolfram.com/language/ref/ChiSquareDistribution.html), [GammaDistribution](https://reference.wolfram.com/language/ref/GammaDistribution.html), [FRatioDistribution](https://reference.wolfram.com/language/ref/FRatioDistribution.html), and [BetaPrimeDistribution](https://reference.wolfram.com/language/ref/BetaPrimeDistribution.html). # Examples open all close all ## Basic Examples (4)Summary of the most common use cases Probability density function: Copy to clipboard. In\[1\]:=1 ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/I_1.png) ✖ `https://wolfram.com/xid/0g7kenjf9sq-6ghfcq` Direct link to example Out\[1\]=1 ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/O_1.png) Copy to clipboard. In\[2\]:=2 ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/I_2.png) ✖ `https://wolfram.com/xid/0g7kenjf9sq-xi7qe0` Direct link to example Out\[2\]=2 ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/O_2.png) Copy to clipboard. In\[3\]:=3 ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/I_3.png) ✖ `https://wolfram.com/xid/0g7kenjf9sq-crqw28` Direct link to example Out\[3\]=3 ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/O_3.png) Cumulative distribution function: Copy to clipboard. In\[1\]:=1 ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/I_4.png) ✖ `https://wolfram.com/xid/0g7kenjf9sq-o4atla` Direct link to example Out\[1\]=1 ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/O_4.png) Copy to clipboard. In\[2\]:=2 ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/I_5.png) ✖ `https://wolfram.com/xid/0g7kenjf9sq-8fypqg` Direct link to example Out\[2\]=2 ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/O_5.png) Copy to clipboard. In\[3\]:=3 ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/I_6.png) ✖ `https://wolfram.com/xid/0g7kenjf9sq-xzaznn` Direct link to example Out\[3\]=3 ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/O_6.png) Mean and variance: Copy to clipboard. In\[1\]:=1 ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/I_7.png) ✖ `https://wolfram.com/xid/0g7kenjf9sq-kc5` Direct link to example Out\[1\]=1 ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/O_7.png) Copy to clipboard. In\[2\]:=2 ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/I_8.png) ✖ `https://wolfram.com/xid/0g7kenjf9sq-xtk` Direct link to example Out\[2\]=2 ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/O_8.png) Median: Copy to clipboard. In\[1\]:=1 ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/I_9.png) ✖ `https://wolfram.com/xid/0g7kenjf9sq-4zlnan` Direct link to example Out\[1\]=1 ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/O_9.png) ## Scope (8)Survey of the scope of standard use cases Generate a sample of pseudorandom numbers from a beta distribution: Copy to clipboard. In\[1\]:=1 ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/I_10.png) ✖ `https://wolfram.com/xid/0g7kenjf9sq-tscvhm` Direct link to example Compare the histogram to the PDF: Copy to clipboard. In\[2\]:=2 ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/I_11.png) ✖ `https://wolfram.com/xid/0g7kenjf9sq-fw0ala` Direct link to example Out\[2\]=2 ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/O_10.png) Distribution parameters estimation: Copy to clipboard. In\[1\]:=1 ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/I_12.png) ✖ `https://wolfram.com/xid/0g7kenjf9sq-45b7g2` Direct link to example Estimate the distribution parameters from sample data: Copy to clipboard. In\[2\]:=2 ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/I_13.png) ✖ `https://wolfram.com/xid/0g7kenjf9sq-epi747` Direct link to example Out\[2\]=2 ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/O_11.png) Compare a density histogram of the sample with the PDF of the estimated distribution: Copy to clipboard. In\[3\]:=3 ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/I_14.png) ✖ `https://wolfram.com/xid/0g7kenjf9sq-f8ui5o` Direct link to example Out\[3\]=3 ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/O_12.png) Skewness varies with shape parameters: Copy to clipboard. In\[1\]:=1 ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/I_15.png) ✖ `https://wolfram.com/xid/0g7kenjf9sq-3n7jeg` Direct link to example Out\[1\]=1 ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/O_13.png) Copy to clipboard. In\[2\]:=2 ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/I_16.png) ✖ `https://wolfram.com/xid/0g7kenjf9sq-i6l` Direct link to example Out\[2\]=2 ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/O_14.png) When both parameters go to ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/12.png), the distribution becomes symmetric: Copy to clipboard. In\[3\]:=3 ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/I_17.png) ✖ `https://wolfram.com/xid/0g7kenjf9sq-hcr61q` Direct link to example Out\[3\]=3 ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/O_15.png) Kurtosis varies with shape parameters: Copy to clipboard. In\[1\]:=1 ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/I_18.png) ✖ `https://wolfram.com/xid/0g7kenjf9sq-6aiaq1` Direct link to example Out\[1\]=1 ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/O_16.png) Copy to clipboard. In\[2\]:=2 ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/I_19.png) ✖ `https://wolfram.com/xid/0g7kenjf9sq-gft` Direct link to example Out\[2\]=2 ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/O_17.png) In the limit, the kurtosis becomes the same as for [NormalDistribution](https://reference.wolfram.com/language/ref/NormalDistribution.html): Copy to clipboard. In\[3\]:=3 ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/I_20.png) ✖ `https://wolfram.com/xid/0g7kenjf9sq-70koo4` Direct link to example Out\[3\]=3 ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/O_18.png) Copy to clipboard. In\[4\]:=4 ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/I_21.png) ✖ `https://wolfram.com/xid/0g7kenjf9sq-hn3lr` Direct link to example Out\[4\]=4 ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/O_19.png) Different moments with closed forms as functions of parameters: Copy to clipboard. In\[1\]:=1 ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/I_22.png) ✖ `https://wolfram.com/xid/0g7kenjf9sq-js043h` Direct link to example [Moment](https://reference.wolfram.com/language/ref/Moment.html): Copy to clipboard. In\[2\]:=2 ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/I_23.png) ✖ `https://wolfram.com/xid/0g7kenjf9sq-rx074o` Direct link to example Out\[2\]=2 ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/O_20.png) [CentralMoment](https://reference.wolfram.com/language/ref/CentralMoment.html): Copy to clipboard. In\[3\]:=3 ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/I_24.png) ✖ `https://wolfram.com/xid/0g7kenjf9sq-pknsqa` Direct link to example Out\[3\]=3 ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/O_21.png) [FactorialMoment](https://reference.wolfram.com/language/ref/FactorialMoment.html): Copy to clipboard. In\[4\]:=4 ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/I_25.png) ✖ `https://wolfram.com/xid/0g7kenjf9sq-zg9ct4` Direct link to example Out\[4\]=4 ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/O_22.png) [Cumulant](https://reference.wolfram.com/language/ref/Cumulant.html): Copy to clipboard. In\[5\]:=5 ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/I_26.png) ✖ `https://wolfram.com/xid/0g7kenjf9sq-9gzmth` Direct link to example Out\[5\]=5 ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/O_23.png) Hazard function: Copy to clipboard. In\[1\]:=1 ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/I_27.png) ✖ `https://wolfram.com/xid/0g7kenjf9sq-cly108` Direct link to example Out\[1\]=1 ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/O_24.png) Copy to clipboard. In\[2\]:=2 ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/I_28.png) ✖ `https://wolfram.com/xid/0g7kenjf9sq-cmoefp` Direct link to example Out\[2\]=2 ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/O_25.png) Copy to clipboard. In\[3\]:=3 ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/I_29.png) ✖ `https://wolfram.com/xid/0g7kenjf9sq-zet7d0` Direct link to example Out\[3\]=3 ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/O_26.png) Quantile function: Copy to clipboard. In\[1\]:=1 ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/I_30.png) ✖ `https://wolfram.com/xid/0g7kenjf9sq-8ljlc3` Direct link to example Out\[1\]=1 ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/O_27.png) Copy to clipboard. In\[2\]:=2 ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/I_31.png) ✖ `https://wolfram.com/xid/0g7kenjf9sq-ffrzb1` Direct link to example Out\[2\]=2 ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/O_28.png) Copy to clipboard. In\[3\]:=3 ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/I_32.png) ✖ `https://wolfram.com/xid/0g7kenjf9sq-m8kdz6` Direct link to example Out\[3\]=3 ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/O_29.png) Consistent use of [Quantity](https://reference.wolfram.com/language/ref/Quantity.html) in parameters expands them into their numeric values: Copy to clipboard. In\[1\]:=1 ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/I_33.png) ✖ `https://wolfram.com/xid/0g7kenjf9sq-gr2bgd` Direct link to example Out\[1\]=1 ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/O_30.png) Find the mean: Copy to clipboard. In\[2\]:=2 ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/I_34.png) ✖ `https://wolfram.com/xid/0g7kenjf9sq-h594k5` Direct link to example Out\[2\]=2 ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/O_31.png) ## Applications (3)Sample problems that can be solved with this function Cloud duration approximately follows a beta distribution with parameters 0.3 and 0.4 for a particular location. Find the probability that cloud duration will be longer than half a day: Copy to clipboard. In\[1\]:=1 ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/I_35.png) ✖ `https://wolfram.com/xid/0g7kenjf9sq-1og0gu` Direct link to example Out\[1\]=1 ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/O_32.png) Simulate the fraction of the day that is cloudy over a period of one month: Copy to clipboard. In\[2\]:=2 ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/I_36.png) ✖ `https://wolfram.com/xid/0g7kenjf9sq-ex9npo` Direct link to example Copy to clipboard. In\[3\]:=3 ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/I_37.png) ✖ `https://wolfram.com/xid/0g7kenjf9sq-txou6v` Direct link to example Out\[3\]=3 ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/O_33.png) Find the average cloudiness duration for a day: Copy to clipboard. In\[4\]:=4 ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/I_38.png) ✖ `https://wolfram.com/xid/0g7kenjf9sq-3hhdl7` Direct link to example Out\[4\]=4 ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/O_34.png) Find the probability of having exactly 20 days in a month with cloud duration less than 10%: Copy to clipboard. In\[5\]:=5 ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/I_39.png) ✖ `https://wolfram.com/xid/0g7kenjf9sq-dgsxa0` Direct link to example Copy to clipboard. In\[6\]:=6 ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/I_40.png) ✖ `https://wolfram.com/xid/0g7kenjf9sq-cww8e7` Direct link to example Out\[6\]=6 ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/O_35.png) Find the probability of at least 20 days in a month with cloud duration less than 10%: Copy to clipboard. In\[7\]:=7 ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/I_41.png) ✖ `https://wolfram.com/xid/0g7kenjf9sq-gdcxid` Direct link to example Out\[7\]=7 ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/O_36.png) Beta distribution can be used to model the proportion of the stocks that increase in value on a given day. Fit beta distribution to the Dow Jones Industrial stocks data: Copy to clipboard. In\[1\]:=1 ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/I_42.png) ✖ `https://wolfram.com/xid/0g7kenjf9sq-hhnw8h` Direct link to example Out\[1\]=1 ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/O_37.png) Find daily change: Copy to clipboard. In\[2\]:=2 ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/I_43.png) ✖ `https://wolfram.com/xid/0g7kenjf9sq-vzl714` Direct link to example Number of days for each financial entity: Copy to clipboard. In\[3\]:=3 ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/I_44.png) ✖ `https://wolfram.com/xid/0g7kenjf9sq-e15an6` Direct link to example Out\[3\]=3 ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/O_38.png) Extract values from time series for each entity and normalize numeric quantities: Copy to clipboard. In\[4\]:=4 ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/I_45.png) ✖ `https://wolfram.com/xid/0g7kenjf9sq-bdcjdx` Direct link to example Check if each entity has the same length of data: Copy to clipboard. In\[5\]:=5 ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/I_46.png) ✖ `https://wolfram.com/xid/0g7kenjf9sq-ldtuwa` Direct link to example Out\[5\]=5 ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/O_39.png) Copy to clipboard. In\[6\]:=6 ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/I_47.png) ✖ `https://wolfram.com/xid/0g7kenjf9sq-oibkmk` Direct link to example Out\[6\]=6 ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/O_40.png) Calculate the daily ratio of companies with an increase in value: Copy to clipboard. In\[7\]:=7 ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/I_48.png) ✖ `https://wolfram.com/xid/0g7kenjf9sq-b1xazx` Direct link to example Find fit, excluding days with no companies having an increase in value: Copy to clipboard. In\[8\]:=8 ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/I_49.png) ✖ `https://wolfram.com/xid/0g7kenjf9sq-18xqlp` Direct link to example Out\[8\]=8 ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/O_41.png) Compare the histogram of the data with the PDF of the estimated distribution: Copy to clipboard. In\[9\]:=9 ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/I_50.png) ✖ `https://wolfram.com/xid/0g7kenjf9sq-z8idun` Direct link to example Out\[9\]=9 ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/O_42.png) Find the probability that at least 60% of Dow Jones Industrial stocks will increase in value: Copy to clipboard. In\[10\]:=10 ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/I_51.png) ✖ `https://wolfram.com/xid/0g7kenjf9sq-k3z2y6` Direct link to example Out\[10\]=10 ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/O_43.png) Find the average percentage of Dow Jones Industrial stocks that will increase in value: Copy to clipboard. In\[11\]:=11 ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/I_52.png) ✖ `https://wolfram.com/xid/0g7kenjf9sq-l8d1x5` Direct link to example Out\[11\]=11 ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/O_44.png) Simulate the percentage of Dow Jones Industrial stocks that will increase in value for 30 days: Copy to clipboard. In\[12\]:=12 ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/I_53.png) ✖ `https://wolfram.com/xid/0g7kenjf9sq-ogrhpe` Direct link to example Out\[12\]=12 ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/O_45.png) Discrete-time Markov chain ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/13.png), where ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/14.png) is the sequence of independent and identically distributed (iid) standard uniform random variables, and ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/15.png) is the sequence of iid Bernoulli random variables with success probability of ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/16.png) converges to stationary distribution [BetaDistribution](https://reference.wolfram.com/language/ref/BetaDistribution.html)\[p,1-p\] for any initial condition ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/17.png) such that ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/18.png): Copy to clipboard. In\[1\]:=1 ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/I_54.png) ✖ `https://wolfram.com/xid/0g7kenjf9sq-d8x1dq` Direct link to example Sample a realization of the Markov chain and discard the burn-in portion of the path: Copy to clipboard. In\[2\]:=2 ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/I_55.png) ✖ `https://wolfram.com/xid/0g7kenjf9sq-wea5d` Direct link to example Samples from the Markov chain are not independent and exhibit internal structure: Copy to clipboard. In\[3\]:=3 ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/I_56.png) ✖ `https://wolfram.com/xid/0g7kenjf9sq-jhh7sk` Direct link to example Out\[3\]=3 ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/O_46.png) Compare the histogram of path values to the [PDF](https://reference.wolfram.com/language/ref/PDF.html) of the Markov chain's stationary distribution: Copy to clipboard. In\[4\]:=4 ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/I_57.png) ✖ `https://wolfram.com/xid/0g7kenjf9sq-cwr1w` Direct link to example Out\[4\]=4 ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/O_47.png) Use path values to approximate an expectation: Copy to clipboard. In\[5\]:=5 ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/I_58.png) ✖ `https://wolfram.com/xid/0g7kenjf9sq-dcivz6` Direct link to example Out\[5\]=5 ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/O_48.png) Compare with the quadrature value: Copy to clipboard. In\[6\]:=6 ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/I_59.png) ✖ `https://wolfram.com/xid/0g7kenjf9sq-f6nfr1` Direct link to example Out\[6\]=6 ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/O_49.png) ## Properties & Relations (21)Properties of the function, and connections to other functions If a variate ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/19.png) follows beta distribution, then ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/20.png) follows the reflected distribution: Copy to clipboard. In\[1\]:=1 ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/I_60.png) ✖ `https://wolfram.com/xid/0g7kenjf9sq-sdo87r` Direct link to example Out\[1\]=1 ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/O_50.png) Relationships to other distributions: ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/Image_21.gif) [BetaDistribution](https://reference.wolfram.com/language/ref/BetaDistribution.html)\[1,1\] is equivalent to [UniformDistribution](https://reference.wolfram.com/language/ref/UniformDistribution.html)\[{0,1}\]: Copy to clipboard. In\[1\]:=1 ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/I_61.png) ✖ `https://wolfram.com/xid/0g7kenjf9sq-res` Direct link to example Out\[1\]=1 ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/O_51.png) Copy to clipboard. In\[2\]:=2 ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/I_62.png) ✖ `https://wolfram.com/xid/0g7kenjf9sq-r6e` Direct link to example Out\[2\]=2 ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/O_52.png) [BetaDistribution](https://reference.wolfram.com/language/ref/BetaDistribution.html) is a transformation of [UniformDistribution](https://reference.wolfram.com/language/ref/UniformDistribution.html): Copy to clipboard. In\[1\]:=1 ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/I_63.png) ✖ `https://wolfram.com/xid/0g7kenjf9sq-r56wcy` Direct link to example Out\[1\]=1 ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/O_53.png) [UniformDistribution](https://reference.wolfram.com/language/ref/UniformDistribution.html) is a transformation of [BetaDistribution](https://reference.wolfram.com/language/ref/BetaDistribution.html): Copy to clipboard. In\[1\]:=1 ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/I_64.png) ✖ `https://wolfram.com/xid/0g7kenjf9sq-oumvvz` Direct link to example Out\[1\]=1 ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/O_54.png) [BetaDistribution](https://reference.wolfram.com/language/ref/BetaDistribution.html) is a limiting case of [NoncentralBetaDistribution](https://reference.wolfram.com/language/ref/NoncentralBetaDistribution.html): Copy to clipboard. In\[1\]:=1 ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/I_65.png) ✖ `https://wolfram.com/xid/0g7kenjf9sq-dlrhy8` Direct link to example Out\[1\]=1 ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/O_55.png) Copy to clipboard. In\[2\]:=2 ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/I_66.png) ✖ `https://wolfram.com/xid/0g7kenjf9sq-ewhxcn` Direct link to example Out\[2\]=2 ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/O_56.png) Copy to clipboard. In\[3\]:=3 ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/I_67.png) ✖ `https://wolfram.com/xid/0g7kenjf9sq-pe9es2` Direct link to example Out\[3\]=3 ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/O_57.png) [BetaPrimeDistribution](https://reference.wolfram.com/language/ref/BetaPrimeDistribution.html) can be obtained as a transformation of the beta-distributed variable: Copy to clipboard. In\[1\]:=1 ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/I_68.png) ✖ `https://wolfram.com/xid/0g7kenjf9sq-sywssu` Direct link to example Out\[1\]=1 ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/O_58.png) Copy to clipboard. In\[2\]:=2 ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/I_69.png) ✖ `https://wolfram.com/xid/0g7kenjf9sq-yd91s5` Direct link to example Out\[2\]=2 ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/O_59.png) Beta distribution is a special case of [PearsonDistribution](https://reference.wolfram.com/language/ref/PearsonDistribution.html) of type 1: Copy to clipboard. In\[1\]:=1 ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/I_70.png) ✖ `https://wolfram.com/xid/0g7kenjf9sq-x3yp0y` Direct link to example Out\[1\]=1 ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/O_60.png) Copy to clipboard. In\[2\]:=2 ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/I_71.png) ✖ `https://wolfram.com/xid/0g7kenjf9sq-negofw` Direct link to example Out\[2\]=2 ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/O_61.png) Copy to clipboard. In\[3\]:=3 ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/I_72.png) ✖ `https://wolfram.com/xid/0g7kenjf9sq-of2jpp` Direct link to example Out\[3\]=3 ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/O_62.png) Beta distribution can be obtained as a transformation of [GammaDistribution](https://reference.wolfram.com/language/ref/GammaDistribution.html): Copy to clipboard. In\[1\]:=1 ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/I_73.png) ✖ `https://wolfram.com/xid/0g7kenjf9sq-ub1zwk` Direct link to example Out\[1\]=1 ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/O_63.png) Beta distribution can be obtained as a transformation of [ChiSquareDistribution](https://reference.wolfram.com/language/ref/ChiSquareDistribution.html): Copy to clipboard. In\[1\]:=1 ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/I_74.png) ✖ `https://wolfram.com/xid/0g7kenjf9sq-bak0x0` Direct link to example Out\[1\]=1 ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/O_64.png) [FRatioDistribution](https://reference.wolfram.com/language/ref/FRatioDistribution.html) can be obtained from beta distribution: Copy to clipboard. In\[1\]:=1 ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/I_75.png) ✖ `https://wolfram.com/xid/0g7kenjf9sq-px8s1z` Direct link to example Copy to clipboard. In\[2\]:=2 ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/I_76.png) ✖ `https://wolfram.com/xid/0g7kenjf9sq-opzpze` Direct link to example Out\[2\]=2 ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/O_65.png) Copy to clipboard. In\[3\]:=3 ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/I_77.png) ✖ `https://wolfram.com/xid/0g7kenjf9sq-yz6lah` Direct link to example Out\[3\]=3 ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/O_66.png) Copy to clipboard. In\[4\]:=4 ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/I_78.png) ✖ `https://wolfram.com/xid/0g7kenjf9sq-tyc15u` Direct link to example Out\[4\]=4 ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/O_67.png) Beta distribution is an order distribution of variables from [UniformDistribution](https://reference.wolfram.com/language/ref/UniformDistribution.html): Copy to clipboard. In\[1\]:=1 ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/I_79.png) ✖ `https://wolfram.com/xid/0g7kenjf9sq-xgzy15` Direct link to example Out\[1\]=1 ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/O_68.png) [ExponentialDistribution](https://reference.wolfram.com/language/ref/ExponentialDistribution.html) is a limit of a scaled beta distribution: Copy to clipboard. In\[1\]:=1 ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/I_80.png) ✖ `https://wolfram.com/xid/0g7kenjf9sq-d5fhku` Direct link to example Copy to clipboard. In\[2\]:=2 ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/I_81.png) ✖ `https://wolfram.com/xid/0g7kenjf9sq-don187` Direct link to example Out\[2\]=2 ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/O_69.png) Copy to clipboard. In\[3\]:=3 ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/I_82.png) ✖ `https://wolfram.com/xid/0g7kenjf9sq-itl7pb` Direct link to example Out\[3\]=3 ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/O_70.png) Copy to clipboard. In\[4\]:=4 ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/I_83.png) ✖ `https://wolfram.com/xid/0g7kenjf9sq-8b3vgg` Direct link to example Out\[4\]=4 ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/O_71.png) Copy to clipboard. In\[5\]:=5 ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/I_84.png) ✖ `https://wolfram.com/xid/0g7kenjf9sq-th75rv` Direct link to example Out\[5\]=5 ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/O_72.png) [ExponentialDistribution](https://reference.wolfram.com/language/ref/ExponentialDistribution.html) is a transformation of beta distribution: Copy to clipboard. In\[1\]:=1 ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/I_85.png) ✖ `https://wolfram.com/xid/0g7kenjf9sq-e3bu74` Direct link to example Out\[1\]=1 ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/O_73.png) Copy to clipboard. In\[2\]:=2 ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/I_86.png) ✖ `https://wolfram.com/xid/0g7kenjf9sq-7zbko5` Direct link to example Out\[2\]=2 ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/O_74.png) [KumaraswamyDistribution](https://reference.wolfram.com/language/ref/KumaraswamyDistribution.html) is a transformation of beta distribution: Copy to clipboard. In\[1\]:=1 ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/I_87.png) ✖ `https://wolfram.com/xid/0g7kenjf9sq-h1ppe8` Direct link to example Out\[1\]=1 ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/O_75.png) [KumaraswamyDistribution](https://reference.wolfram.com/language/ref/KumaraswamyDistribution.html) simplifies to a special case of beta distribution: Copy to clipboard. In\[1\]:=1 ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/I_88.png) ✖ `https://wolfram.com/xid/0g7kenjf9sq-tyjged` Direct link to example Out\[1\]=1 ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/O_76.png) Copy to clipboard. In\[2\]:=2 ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/I_89.png) ✖ `https://wolfram.com/xid/0g7kenjf9sq-y0ly7p` Direct link to example Out\[2\]=2 ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/O_77.png) Copy to clipboard. In\[3\]:=3 ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/I_90.png) ✖ `https://wolfram.com/xid/0g7kenjf9sq-gmfqcx` Direct link to example Out\[3\]=3 ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/O_78.png) [PERTDistribution](https://reference.wolfram.com/language/ref/PERTDistribution.html) is a transformation of beta distribution: Copy to clipboard. In\[1\]:=1 ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/I_91.png) ✖ `https://wolfram.com/xid/0g7kenjf9sq-pvrrzu` Direct link to example Copy to clipboard. In\[2\]:=2 ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/I_92.png) ✖ `https://wolfram.com/xid/0g7kenjf9sq-ojaijt` Direct link to example Out\[2\]=2 ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/O_79.png) Copy to clipboard. In\[3\]:=3 ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/I_93.png) ✖ `https://wolfram.com/xid/0g7kenjf9sq-xapy0l` Direct link to example Out\[3\]=3 ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/O_80.png) Copy to clipboard. In\[4\]:=4 ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/I_94.png) ✖ `https://wolfram.com/xid/0g7kenjf9sq-tg7dcq` Direct link to example Out\[4\]=4 ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/O_81.png) [WignerSemicircleDistribution](https://reference.wolfram.com/language/ref/WignerSemicircleDistribution.html) is a transformation of special beta distribution: Copy to clipboard. In\[1\]:=1 ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/I_95.png) ✖ `https://wolfram.com/xid/0g7kenjf9sq-zo9b0n` Direct link to example Copy to clipboard. In\[2\]:=2 ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/I_96.png) ✖ `https://wolfram.com/xid/0g7kenjf9sq-djj5oc` Direct link to example Out\[2\]=2 ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/O_82.png) Copy to clipboard. In\[3\]:=3 ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/I_97.png) ✖ `https://wolfram.com/xid/0g7kenjf9sq-ng3g1m` Direct link to example Out\[3\]=3 ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/O_83.png) Copy to clipboard. In\[4\]:=4 ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/I_98.png) ✖ `https://wolfram.com/xid/0g7kenjf9sq-s0lugl` Direct link to example Out\[4\]=4 ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/O_84.png) Univariate marginals of [DirichletDistribution](https://reference.wolfram.com/language/ref/DirichletDistribution.html) have beta distribution: Copy to clipboard. In\[1\]:=1 ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/I_99.png) ✖ `https://wolfram.com/xid/0g7kenjf9sq-5he2sh` Direct link to example Out\[1\]=1 ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/O_85.png) Copy to clipboard. In\[2\]:=2 ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/I_100.png) ✖ `https://wolfram.com/xid/0g7kenjf9sq-3oha4x` Direct link to example Out\[2\]=2 ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/O_86.png) [BetaBinomialDistribution](https://reference.wolfram.com/language/ref/BetaBinomialDistribution.html) is a mixture of [BinomialDistribution](https://reference.wolfram.com/language/ref/BinomialDistribution.html) and [BetaDistribution](https://reference.wolfram.com/language/ref/BetaDistribution.html): Copy to clipboard. In\[1\]:=1 ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/I_101.png) ✖ `https://wolfram.com/xid/0g7kenjf9sq-6a4ms5` Direct link to example Out\[1\]=1 ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/O_87.png) [BetaNegativeBinomialDistribution](https://reference.wolfram.com/language/ref/BetaNegativeBinomialDistribution.html) is a mixture of [NegativeBinomialDistribution](https://reference.wolfram.com/language/ref/NegativeBinomialDistribution.html) and [BetaDistribution](https://reference.wolfram.com/language/ref/BetaDistribution.html): Copy to clipboard. In\[1\]:=1 ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/I_102.png) ✖ `https://wolfram.com/xid/0g7kenjf9sq-vysest` Direct link to example Out\[1\]=1 ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/O_88.png) ## Possible Issues (2)Common pitfalls and unexpected behavior [BetaDistribution](https://reference.wolfram.com/language/ref/BetaDistribution.html) is not defined when either α or β is not a positive real number: Copy to clipboard. In\[1\]:=1 ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/I_103.png) ✖ `https://wolfram.com/xid/0g7kenjf9sq-dny` Direct link to example ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/22.gif) Out\[1\]=1 ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/O_89.png) Substitution of invalid parameters into symbolic outputs gives results that are not meaningful: Copy to clipboard. In\[1\]:=1 ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/I_104.png) ✖ `https://wolfram.com/xid/0g7kenjf9sq-srb` Direct link to example Out\[1\]=1 ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/O_90.png) # See Also [GammaDistribution](https://reference.wolfram.com/language/ref/GammaDistribution.html) [BinomialDistribution](https://reference.wolfram.com/language/ref/BinomialDistribution.html) [Beta](https://reference.wolfram.com/language/ref/Beta.html) [BetaRegularized](https://reference.wolfram.com/language/ref/BetaRegularized.html) [InverseBetaRegularized](https://reference.wolfram.com/language/ref/InverseBetaRegularized.html) [FRatioDistribution](https://reference.wolfram.com/language/ref/FRatioDistribution.html) [DirichletDistribution](https://reference.wolfram.com/language/ref/DirichletDistribution.html) Function Repository: [MeanSpreadBetaDistribution](https://resources.wolframcloud.com/FunctionRepository/resources/MeanSpreadBetaDistribution) # Tech Notes - [Continuous Distributions](https://reference.wolfram.com/language/tutorial/NumericalOperationsOnData.html#11002) # Related Guides - [Bounded Domain Distributions](https://reference.wolfram.com/language/guide/BoundedDomainDistributions.html) - [Parametric Statistical Distributions](https://reference.wolfram.com/language/guide/ParametricStatisticalDistributions.html) - [Functions Used in Statistics](https://reference.wolfram.com/language/guide/FunctionsUsedInStatistics.html) # History [Introduced in 2007 (6.0)](https://reference.wolfram.com/language/guide/SummaryOfNewFeaturesIn60) \\| [Updated in 2016 (10.4)](https://reference.wolfram.com/language/guide/SummaryOfNewFeaturesIn104) Cite this as: Wolfram Research (2007), BetaDistribution, Wolfram Language function, https://reference.wolfram.com/language/ref/BetaDistribution.html (updated 2016). Copy to clipboard. ✖ `Wolfram Research (2007), BetaDistribution, Wolfram Language function, https://reference.wolfram.com/language/ref/BetaDistribution.html (updated 2016).` ![](https://reference.wolfram.com/language-assets-v2/img/CitationCloseButton.svg) #### Text Wolfram Research (2007), BetaDistribution, Wolfram Language function, https://reference.wolfram.com/language/ref/BetaDistribution.html (updated 2016). Copy to clipboard. ✖ `Wolfram Research (2007), BetaDistribution, Wolfram Language function, https://reference.wolfram.com/language/ref/BetaDistribution.html (updated 2016).` #### CMS Wolfram Language. 2007. "BetaDistribution." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2016. https://reference.wolfram.com/language/ref/BetaDistribution.html. Copy to clipboard. ✖ `Wolfram Language. 2007. "BetaDistribution." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2016. https://reference.wolfram.com/language/ref/BetaDistribution.html.` #### APA Wolfram Language. (2007). BetaDistribution. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/BetaDistribution.html Copy to clipboard. ✖ `Wolfram Language. (2007). BetaDistribution. Wolfram Language & System Documentation Center. 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Readable Markdown	- [See Also](https://reference.wolfram.com/language/ref/BetaDistribution.html) - [GammaDistribution](https://reference.wolfram.com/language/ref/GammaDistribution.html) - [BinomialDistribution](https://reference.wolfram.com/language/ref/BinomialDistribution.html) - [Beta](https://reference.wolfram.com/language/ref/Beta.html) - [BetaRegularized](https://reference.wolfram.com/language/ref/BetaRegularized.html) - [InverseBetaRegularized](https://reference.wolfram.com/language/ref/InverseBetaRegularized.html) - [FRatioDistribution](https://reference.wolfram.com/language/ref/FRatioDistribution.html) - [DirichletDistribution](https://reference.wolfram.com/language/ref/DirichletDistribution.html) - [Related Guides](https://reference.wolfram.com/language/ref/BetaDistribution.html) - [Bounded Domain Distributions](https://reference.wolfram.com/language/guide/BoundedDomainDistributions.html) - [Parametric Statistical Distributions](https://reference.wolfram.com/language/guide/ParametricStatisticalDistributions.html) - [Functions Used in Statistics](https://reference.wolfram.com/language/guide/FunctionsUsedInStatistics.html) - [Tech Notes](https://reference.wolfram.com/language/ref/BetaDistribution.html) - [Continuous Distributions](https://reference.wolfram.com/language/tutorial/NumericalOperationsOnData.html#11002) - - [See Also](https://reference.wolfram.com/language/ref/BetaDistribution.html) - [GammaDistribution](https://reference.wolfram.com/language/ref/GammaDistribution.html) - [BinomialDistribution](https://reference.wolfram.com/language/ref/BinomialDistribution.html) - [Beta](https://reference.wolfram.com/language/ref/Beta.html) - [BetaRegularized](https://reference.wolfram.com/language/ref/BetaRegularized.html) - [InverseBetaRegularized](https://reference.wolfram.com/language/ref/InverseBetaRegularized.html) - [FRatioDistribution](https://reference.wolfram.com/language/ref/FRatioDistribution.html) - [DirichletDistribution](https://reference.wolfram.com/language/ref/DirichletDistribution.html) - [Related Guides](https://reference.wolfram.com/language/ref/BetaDistribution.html) - [Bounded Domain Distributions](https://reference.wolfram.com/language/guide/BoundedDomainDistributions.html) - [Parametric Statistical Distributions](https://reference.wolfram.com/language/guide/ParametricStatisticalDistributions.html) - [Functions Used in Statistics](https://reference.wolfram.com/language/guide/FunctionsUsedInStatistics.html) - [Tech Notes](https://reference.wolfram.com/language/ref/BetaDistribution.html) - [Continuous Distributions](https://reference.wolfram.com/language/tutorial/NumericalOperationsOnData.html#11002) Examples Basic Examples Scope Applications Properties & Relations Possible Issues [BetaDistribution](https://reference.wolfram.com/language/ref/BetaDistribution.html)\[α,β\] Copy to clipboard. `BetaDistribution[α,β]` represents a continuous beta distribution with shape parameters α and β. ## Details ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/details_1.png) ## Background & Context - [BetaDistribution](https://reference.wolfram.com/language/ref/BetaDistribution.html)\[α,β\] represents a statistical distribution defined over the interval ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/6.png) and parametrized by two positive values α, β known as "shape parameters", which, roughly speaking, determine the "fatness" of the left and right tails in the probability density function (PDF). Depending on the values of α and β, the PDF of the beta distribution may be monotonic increasing, monotonic decreasing, or unimodal with potential singularities approaching the boundaries of its domain. - The beta distribution arises as a prior distribution for binomial proportions in Bayesian analysis. It is also commonly used to model random variables limited to a finite interval. For example, the distribution of the ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/7.png)![^(th)](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/8.png) smallest element in a continuous, independent, and uniformly distributed sample of size of ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/9.png) can be computed using [OrderDistribution](https://reference.wolfram.com/language/ref/OrderDistribution.html)\[{[UniformDistribution](https://reference.wolfram.com/language/ref/UniformDistribution.html)\[\],n},k\] and is precisely equal to [BetaDistribution](https://reference.wolfram.com/language/ref/BetaDistribution.html)\[k,n\-k\+1\]. In addition to its statistical significance, the beta distribution also plays a fundamental role in a number of scientific fields, including phenomena related to allele frequency distribution, soil property variability, geological mineral-to-rock ratios, and HIV transmission behavior. - [RandomVariate](https://reference.wolfram.com/language/ref/RandomVariate.html) can be used to give one or more machine- or arbitrary-precision (the latter via the [WorkingPrecision](https://reference.wolfram.com/language/ref/WorkingPrecision.html) option) pseudorandom variates from a beta distribution. [Distributed](https://reference.wolfram.com/language/ref/Distributed.html)\[x,BetaDistribution\[α,β\]\], written more concisely as xBetaDistribution\[α,β\], can be used to assert that a random variable x is distributed according to a beta distribution. Such an assertion can then be used in functions such as [Probability](https://reference.wolfram.com/language/ref/Probability.html), [NProbability](https://reference.wolfram.com/language/ref/NProbability.html), [Expectation](https://reference.wolfram.com/language/ref/Expectation.html), and [NExpectation](https://reference.wolfram.com/language/ref/NExpectation.html). - The probability density and cumulative distribution functions may be given using [PDF](https://reference.wolfram.com/language/ref/PDF.html)\[BetaDistribution\[α,β\],x\] and [CDF](https://reference.wolfram.com/language/ref/CDF.html)\[BetaDistribution\[α,β\],x\]. The mean, median, variance, raw moments, and central moments may be computed using [Mean](https://reference.wolfram.com/language/ref/Mean.html), [Median](https://reference.wolfram.com/language/ref/Median.html), [Variance](https://reference.wolfram.com/language/ref/Variance.html), [Moment](https://reference.wolfram.com/language/ref/Moment.html), and [CentralMoment](https://reference.wolfram.com/language/ref/CentralMoment.html), respectively. - [DistributionFitTest](https://reference.wolfram.com/language/ref/DistributionFitTest.html) can be used to test if a given dataset is consistent with a beta distribution, [EstimatedDistribution](https://reference.wolfram.com/language/ref/EstimatedDistribution.html) to estimate a beta parametric distribution from given data, and [FindDistributionParameters](https://reference.wolfram.com/language/ref/FindDistributionParameters.html) to fit data to a beta distribution. [ProbabilityPlot](https://reference.wolfram.com/language/ref/ProbabilityPlot.html) can be used to generate a plot of the CDF of given data against the CDF of a symbolic beta distribution and [QuantilePlot](https://reference.wolfram.com/language/ref/QuantilePlot.html) to generate a plot of the quantiles of given data against the quantiles of a symbolic beta distribution. - [TransformedDistribution](https://reference.wolfram.com/language/ref/TransformedDistribution.html) can be used to represent a transformed beta distribution, [CensoredDistribution](https://reference.wolfram.com/language/ref/CensoredDistribution.html) to represent the distribution of values censored between upper and lower values, and [TruncatedDistribution](https://reference.wolfram.com/language/ref/TruncatedDistribution.html) to represent the distribution of values truncated between upper and lower values. [CopulaDistribution](https://reference.wolfram.com/language/ref/CopulaDistribution.html) can be used to build higher-dimensional distributions that contain a beta distribution, and [ProductDistribution](https://reference.wolfram.com/language/ref/ProductDistribution.html) can be used to compute a joint distribution with independent component distributions involving beta distributions. - The beta distribution is related to a number of other distributions. For example, [BetaDistribution](https://reference.wolfram.com/language/ref/BetaDistribution.html) is the so-called "conjugate prior" for the parameters of a number of other distributions, including [BernoulliDistribution](https://reference.wolfram.com/language/ref/BernoulliDistribution.html), [BinomialDistribution](https://reference.wolfram.com/language/ref/BinomialDistribution.html), [NegativeBinomialDistribution](https://reference.wolfram.com/language/ref/NegativeBinomialDistribution.html), and [GeometricDistribution](https://reference.wolfram.com/language/ref/GeometricDistribution.html). Moreover, [BetaDistribution](https://reference.wolfram.com/language/ref/BetaDistribution.html) generalizes both [UniformDistribution](https://reference.wolfram.com/language/ref/UniformDistribution.html) and [PowerDistribution](https://reference.wolfram.com/language/ref/PowerDistribution.html) in the sense that (modulo inclusion of the endpoints ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/10.png) and ![](https://reference.wolfram.com/language/ref/Files/BetaDistribution.en/11.png)), [PDF](https://reference.wolfram.com/language/ref/PDF.html)\[BetaDistribution\[1,1\],x\] is equal to both [PDF](https://reference.wolfram.com/language/ref/PDF.html)\[[UniformDistribution](https://reference.wolfram.com/language/ref/UniformDistribution.html)\[\],x\] and [PDF](https://reference.wolfram.com/language/ref/PDF.html)\[[PowerDistribution](https://reference.wolfram.com/language/ref/PowerDistribution.html)\[1,1\],x\]. [BetaDistribution](https://reference.wolfram.com/language/ref/BetaDistribution.html) can also be obtained as transformations of [KumaraswamyDistribution](https://reference.wolfram.com/language/ref/KumaraswamyDistribution.html) and [NoncentralBetaDistribution](https://reference.wolfram.com/language/ref/NoncentralBetaDistribution.html) and is closely related to [PERTDistribution](https://reference.wolfram.com/language/ref/PERTDistribution.html), [PearsonDistribution](https://reference.wolfram.com/language/ref/PearsonDistribution.html), [ChiSquareDistribution](https://reference.wolfram.com/language/ref/ChiSquareDistribution.html), [GammaDistribution](https://reference.wolfram.com/language/ref/GammaDistribution.html), [FRatioDistribution](https://reference.wolfram.com/language/ref/FRatioDistribution.html), and [BetaPrimeDistribution](https://reference.wolfram.com/language/ref/BetaPrimeDistribution.html). ## Examples open all close all ## Basic Examples (4)Summary of the most common use cases ## Scope (8)Survey of the scope of standard use cases ## Applications (3)Sample problems that can be solved with this function ## Properties & Relations (21)Properties of the function, and connections to other functions ## Possible Issues (2)Common pitfalls and unexpected behavior Wolfram Research (2007), BetaDistribution, Wolfram Language function, https://reference.wolfram.com/language/ref/BetaDistribution.html (updated 2016). Copy to clipboard. `Wolfram Research (2007), BetaDistribution, Wolfram Language function, https://reference.wolfram.com/language/ref/BetaDistribution.html (updated 2016).` #### Text Wolfram Research (2007), BetaDistribution, Wolfram Language function, https://reference.wolfram.com/language/ref/BetaDistribution.html (updated 2016). Copy to clipboard. `Wolfram Research (2007), BetaDistribution, Wolfram Language function, https://reference.wolfram.com/language/ref/BetaDistribution.html (updated 2016).` #### CMS Wolfram Language. 2007. "BetaDistribution." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2016. https://reference.wolfram.com/language/ref/BetaDistribution.html. Copy to clipboard. `Wolfram Language. 2007. "BetaDistribution." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2016. https://reference.wolfram.com/language/ref/BetaDistribution.html.` #### APA Wolfram Language. (2007). BetaDistribution. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/BetaDistribution.html Copy to clipboard. `Wolfram Language. (2007). BetaDistribution. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/BetaDistribution.html` #### BibTeX @misc{reference.wolfram\_2025\_betadistribution, author="Wolfram Research", title="{BetaDistribution}", year="2016", howpublished="\\url{https://reference.wolfram.com/language/ref/BetaDistribution.html}", note=\[Accessed: 11-April-2026\]} Copy to clipboard. `@misc{reference.wolfram_2025_betadistribution, author="Wolfram Research", title="{BetaDistribution}", year="2016", howpublished="\url{https://reference.wolfram.com/language/ref/BetaDistribution.html}", note=[Accessed: 11-April-2026]}` #### BibLaTeX @online{reference.wolfram\_2025\_betadistribution, organization={Wolfram Research}, title={BetaDistribution}, year={2016}, url={https://reference.wolfram.com/language/ref/BetaDistribution.html}, note=\[Accessed: 11-April-2026\]} Copy to clipboard. `@online{reference.wolfram_2025_betadistribution, organization={Wolfram Research}, title={BetaDistribution}, year={2016}, url={https://reference.wolfram.com/language/ref/BetaDistribution.html}, note=[Accessed: 11-April-2026]}`
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