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Calculated Shard: 150 (from laksa157)

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curl -X POST \
  'http://laksa150.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, ml_categories_json AS ml_categories_json, ml_types_json AS ml_types_json, ml_intent_types_json AS ml_intent_types_json, meta_language AS meta_language, attrs_author AS attrs_author, ifNull(toUnixTimestamp(attrs_publish_time), 0) AS attrs_publish_time, ifNull(toUnixTimestamp(attrs_original_publish_time), 0) AS attrs_original_publish_time, ifNull(attrs_is_republished, 0) AS attrs_is_republished, ifNull(attrs_nr_words, 0) AS attrs_nr_words, ifNull(attrs_boilerpipe_nr_words, 0) AS attrs_boilerpipe_nr_words, ifNull(body_ext_links_number, 0) AS body_ext_links_number, ifNull(body_int_links_number, 0) AS body_int_links_number, ifNull(meta_nofollow, 0) AS meta_nofollow, ifNull(meta_noarchive, 0) AS meta_noarchive, ifNull(props_was_rendered, 0) AS props_was_rendered, ifNull(src_redirect, \'\') AS src_redirect, ifNull(download_time_msec, 0) AS download_time_msec, ifNull(download_ttfb_msec, 0) AS download_ttfb_msec, ifNull(download_size, 0) AS download_size FROM crawler3.page_info_local FINAL PREWHERE (src_root_hash, src_unparsed) IN ((getAhrefsRootHashFromUnparsed(getAhrefsUnparsedNoserviceFromURL(\'https://www.statisticshowto.com/beta-distribution/\')), getAhrefsUnparsedNoserviceFromURL(\'https://www.statisticshowto.com/beta-distribution/\'))) FORMAT JSONEachRow'

Response:

{"found_url":"https:\/\/www.statisticshowto.com\/beta-distribution\/","crawl_time":1776844231,"first_indexed_time":1586054467,"http_code":200,"src_unparsed":"com,statisticshowto!www,\/beta-distribution\/ s443","src_root_hash":"12746512124316119150","history_drop_reason":null,"meta_title":"Beta Distribution: Definition, Calculation - Statistics How To","meta_descriptions":["Beta distribution definition. Represents a family of probabilities and is a way to represent outcomes for percentages or proportions."],"attrs_boilerpipe_text":"<\nProbability Distributions\n>\nBeta Distribution\nContents:\nWhat is a Beta Distribution?\nProperties\nSoftware Options: Mathematica and R\nExcel Beta Distribution: BETA.DIST & INV.BETA\nApplications of the Beta Density Function\nWhat is a Beta Distribution?\nThe\nbeta distribution\n(also called the \nbeta distribution of the first kind\n) is a family of\ncontinuous probability distributions\ndefined on [0, 1]. This distribution is similar to the\nbinomial distribution\n, except where the binomial models the\nnumber\nof successes (x), the beta models the\nprobability\n(p) of success.\nA beta distribution is a versatile way to represent outcomes for percentages or proportions. For example, how likely is it that a rogue candidate will win the next Presidential election? You might think the probability is 0.2. Your friend might think it’s 0.15. The beta distribution gives you a way to describe this. One reason that the beta distribution can be confusing to work with is that there are three “Betas” to contend with, and they all have different meanings:\nBeta(α, β): the\nname\nof the probability distribution.\nB(α, β ): the name of a function in the denominator of the\npdf\n. This acts as a “\nnormalizing constant\n” to ensure that the area under the curve of the pdf equals 1.\nβ: the name of the second\nshape parameter\nin the pdf.\nThe basic beta distribution is also called the\nbeta distribution of the first kind\n. Beta distribution of the second kind is another name for the\nbeta prime distribution\n.\nProperties\nThe general formula for the\nprobability density function (pdf)\nis:\nwhere:\nα and β are two positive\nshape parameters\nwhich control the shape of the distribution.\nThe graph of the beta density function can take on a variety of shapes.  For example, if α < 1 and Β < 1, the graph will be a U shaped distribution, and if α = 1 and Β = 2, the graph is a straight line.\nThe Beta Distribution pdf.\n  The general formula for the\ncumulative distribution function (CDF)\nis:\nwhich includes the\nregularized beta function\n:\nSoftware Options\nMost software packages have options for the beta distribution.\nMathematica\nImplement a beta distribution by typing BetaDistribution[alpha,beta].\nR\ndbeta(x, shape1, shape2, ncp = 0, log = FALSE) pbeta(q, shape1, shape2, ncp = 0, lower.tail = TRUE, log.p = FALSE) qbeta(p, shape1, shape2, ncp = 0, lower.tail = TRUE, log.p = FALSE) rbeta(n, shape1, shape2, ncp = 0)\nwhere:\nx, q = vector of\nquantiles\n. p = vector of probabilities.\nn = # of observations.\nshape1, shape2 =\nshape parameters\nα and β\nncp =\nnon-centrality parameter\n.\nlog, log.p = logical; if TRUE, probabilities p are given as log(p).\nlower.tail = logical; if TRUE (default), probabilities are P[X ≤ x], otherwise, P[X > x].\nExcel Beta Distribution: BETA.DIST & INV.BETA\nWatch the video for an example of BETA.DIST:\nCan’t see the video?\nClick here\nto watch it on YouTube. The Microsoft Excel beta distribution format depends on which version of\nExcel\nyou are using; Excel 2010 and later uses the BETA.DIST function, while earlier versions use BETADIST. You’ll need to know alpha and beta, both of which refer to the parameters describing the distribution. Five inputs are needed:\nThe value where you want to\nevaluate the function\n.\nAlpha and Beta, the parameters of the distribution which determine shape.\nThe lower and upper bound.\nAn extra input is needed for the Excel 2010 Beta Distribution: cumulative. Cumulative is a logical value that determines the function’s form and can either be TRUE (returns the cumulative distribution function) or FALSE (returns the probability density function).\nExample problem:\nCalculate a\ncumulative distribution function\nfor a beta distribution in Excel at 0.5 with an alpha of 9, a beta of 10, a\nlower bound\nof 0 and an\nupper bound\nof 1.\nExcel 2007 and Earlier:\nType the value where you want to evaluate the function in cell A1.\nFor this example, type “.5” in cell A1.\nType the value for alpha in cell A2\nand then type the value for beta in cell A1. For this example, type “9” in cell A2 and then type “10” in cell A3.\nType the lower bound in cell A4\nand then\ntype the upper bound in cell A5.\nFor this example, type “0” in cell A4 and then type “1” in cell A5\nType the beta distribution function into cell A6.\nThe format of the function is =BETADIST(value,alpha,beta,lower bound,upper bound). For this example, type “=BETADIST(A1,A2,A3,A4,A5)” into cell A6. Press “Enter” to see the result for the beta distribution, which is 0.592735.\nExcel 2010 and later\nExcel 2010 and later uses the BETA.DIST function. If you’re using earlier versions, use BETADIST instead. Inputs are as follows:\nA value where you want to evaluate the function.\nAlpha (α) and Beta (β), which determine the distribution’s shape.\nThe lower and upper bound.\n“Cumulative” — a logical value (TRUE\/FALSE) that determines the function’s form. TRUE returns the CDF and FALSE returns the PDF.\nExample problem:\n \nCalculate a \ncumulative distribution function\n for a beta distribution at 0.5 with α\n=\n9, β\n=\n10,\nlower bound = 0 and upper bound = 1.\nApplications of the Beta Density Function\nThe beta distribution is used for a variety of applications for modeling the behavior of\nrandom variables\nlimited to intervals of finite length. Uses include:\nBayesian inference, as the conjugate prior probability distribution for the Bernoulli, binomial, geometric  and negative binomial distributions.\nThe Rule of Succession (such as Pierre-Simon Laplace’s treatment of the sunrise problem),\nTask duration modeling.\nProject\/planning control systems like PERT and CPM.\nIn project management for the “three-point technique,” (also called the beta distribution technique), which recognizes uncertainty in estimated project time [1]. The three-point estimation technique is used to account for uncertainty in project time estimates. By applying this method, project managers can leverage powerful quantitative tools to identify tasks with the highest risk. It also facilitates effective management of overall project completion time.\nReferences\n3-Points Estimating\nhttps:\/\/www.projectmanagement.com\/contentPages\/wiki.cfm?ID=368763&thisPageURL=\/wikis\/368763\/3-Points-Estimating#_=_","attrs_markdown":"[Skip to content](https:\/\/www.statisticshowto.com\/beta-distribution\/#content \"Skip to content\")\n\n[Statistics How To](https:\/\/www.statisticshowto.com\/)\n\nMenu\n\n- [Home](https:\/\/www.statisticshowto.com\/)\n- [Tables](https:\/\/www.statisticshowto.com\/tables\/)\n  - [Binomial Distribution Table](https:\/\/www.statisticshowto.com\/tables\/binomial-distribution-table\/)\n  - [F Table](https:\/\/www.statisticshowto.com\/tables\/f-table\/)\n  - [Inverse T Distribution Table](https:\/\/www.statisticshowto.com\/tables\/inverse-t-distribution-table\/)\n  - [PPMC Critical Values](https:\/\/www.statisticshowto.com\/tables\/ppmc-critical-values\/)\n  - [T-Distribution Table (One Tail and Two-Tails)](https:\/\/www.statisticshowto.com\/tables\/t-distribution-table\/)\n  - 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[Hypothesis Testing](https:\/\/www.statisticshowto.com\/probability-and-statistics\/hypothesis-testing\/)\n  - [Non Normal Distribution](https:\/\/www.statisticshowto.com\/probability-and-statistics\/non-normal-distributions\/)\n  - [Normal Distributions](https:\/\/www.statisticshowto.com\/probability-and-statistics\/normal-distributions\/)\n  - [Probability](https:\/\/www.statisticshowto.com\/probability-and-statistics\/probability-main-index\/)\n  - [Regression Analysis](https:\/\/www.statisticshowto.com\/probability-and-statistics\/regression-analysis\/)\n  - [Statistics Basics](https:\/\/www.statisticshowto.com\/statistics-basics\/)\n  - [T-Distribution](https:\/\/www.statisticshowto.com\/probability-and-statistics\/t-distribution\/)\n  - [Multivariate Analysis & Independent Component](https:\/\/www.statisticshowto.com\/probability-and-statistics\/multivariate-analysis\/)\n  - [Sampling](https:\/\/www.statisticshowto.com\/probability-and-statistics\/sampling-in-statistics\/)\n- [Calculators](https:\/\/www.statisticshowto.com\/calculators\/)\n  - [Variance and Standard Deviation Calculator](https:\/\/www.statisticshowto.com\/calculators\/variance-and-standard-deviation-calculator\/)\n  - [Tdist Calculator](https:\/\/www.statisticshowto.com\/calculators\/tdist-calculator\/)\n  - [Permutation Calculator \/ Combination Calculator](https:\/\/www.statisticshowto.com\/calculators\/permutation-calculator-and-combination-calculator\/)\n  - [Interquartile Range Calculator](https:\/\/www.statisticshowto.com\/calculators\/interquartile-range-calculator\/)\n  - [Linear Regression Calculator](https:\/\/www.statisticshowto.com\/calculators\/linear-regression-calculator\/)\n  - [Expected Value Calculator](https:\/\/www.statisticshowto.com\/calculators\/expected-value-calculator\/)\n  - [Binomial Distribution Calculator](https:\/\/www.statisticshowto.com\/calculators\/binomial-distribution-calculator\/)\n- [Matrices](https:\/\/www.statisticshowto.com\/matrices-and-matrix-algebra\/)\n- [Experimental Design](https:\/\/www.statisticshowto.com\/experimental-design\/)\n- [Calculus Based Statistics](https:\/\/www.statisticshowto.com\/probability-and-statistics\/calculus-based-statistics\/)\n\n# Beta Distribution: Definition, Calculation\n\\< [Probability Distributions](https:\/\/www.statisticshowto.com\/probability-and-statistics\/statistics-definitions\/probability-distribution\/) \\> *Beta Distribution*\n\n**Contents:**\n\n1. 1. [What is a Beta Distribution?](https:\/\/www.statisticshowto.com\/beta-distribution\/#whatis)\n   2. [Properties](https:\/\/www.statisticshowto.com\/beta-distribution\/#properties)\n   3. [Software Options: Mathematica and R](https:\/\/www.statisticshowto.com\/beta-distribution\/#software)\n   4. [Excel Beta Distribution: BETA.DIST & INV.BETA](https:\/\/www.statisticshowto.com\/beta-distribution\/#excel)\n   5. [Applications of the Beta Density Function](https:\/\/www.statisticshowto.com\/beta-distribution\/#applications)\n## What is a Beta Distribution?\nThe **beta distribution** (also called the *beta distribution of the first kind*) is a family of [continuous probability distributions](https:\/\/www.statisticshowto.com\/continuous-probability-distribution\/) defined on \\[0, 1\\]. This distribution is similar to the [binomial distribution](https:\/\/www.statisticshowto.com\/probability-and-statistics\/binomial-theorem\/binomial-distribution-formula\/), except where the binomial models the *number* of successes (x), the beta models the *probability* (p) of success.\n\nA beta distribution is a versatile way to represent outcomes for percentages or proportions. For example, how likely is it that a rogue candidate will win the next Presidential election? You might think the probability is 0.2. Your friend might think it’s 0.15. The beta distribution gives you a way to describe this. One reason that the beta distribution can be confusing to work with is that there are three “Betas” to contend with, and they all have different meanings:\n\n1. 1. - Beta(α, β): the *name* of the probability distribution.\n      - B(α, β ): the name of a function in the denominator of the [pdf](https:\/\/www.statisticshowto.com\/probability-density-function\/). This acts as a “[normalizing constant](https:\/\/www.statisticshowto.com\/normalizing-constant-definition\/)” to ensure that the area under the curve of the pdf equals 1.\n      - β: the name of the second [shape parameter](https:\/\/www.statisticshowto.com\/shape-parameter\/) in the pdf.\n\nThe basic beta distribution is also called the *beta distribution of the first kind*. Beta distribution of the second kind is another name for the [beta prime distribution](https:\/\/www.statisticshowto.com\/beta-prime-distribution\/).\n\n## Properties\nThe general formula for the [probability density function (pdf)](https:\/\/www.statisticshowto.com\/probability-density-function-definition-examples\/) is: ![beta pdf](https:\/\/www.statisticshowto.com\/wp-content\/uploads\/2015\/08\/beta-pdf.png)   \nwhere:\n\n1. 1. - α and β are two positive [shape parameters](https:\/\/www.statisticshowto.com\/shape-parameter\/) which control the shape of the distribution.\n\nThe graph of the beta density function can take on a variety of shapes. For example, if α \\< 1 and Β \\< 1, the graph will be a U shaped distribution, and if α = 1 and Β = 2, the graph is a straight line.\n\n[![beta distribution](https:\/\/www.statisticshowto.com\/wp-content\/uploads\/2010\/09\/800px-Beta_distribution_pdf-300x225.png)](https:\/\/www.statisticshowto.com\/wp-content\/uploads\/2010\/09\/800px-Beta_distribution_pdf.png)\n\nThe Beta Distribution pdf.\n\nThe general formula for the [cumulative distribution function (CDF)](https:\/\/www.statisticshowto.com\/cumulative-distribution-function-cdf\/) is: ![beta distribution cdf](https:\/\/www.statisticshowto.com\/wp-content\/uploads\/2015\/08\/beta-distribution-cdf.png)![beta distribution cdf](https:\/\/www.statisticshowto.com\/wp-content\/uploads\/2015\/08\/beta-distribution-cdf.png) which includes the [regularized beta function](https:\/\/www.statisticshowto.com\/regularized-incomplete-beta-function\/): ![reg beta function](https:\/\/www.statisticshowto.com\/wp-content\/uploads\/2015\/08\/reg-beta-function.png)![reg beta function](https:\/\/www.statisticshowto.com\/wp-content\/uploads\/2015\/08\/reg-beta-function.png)\n\n## [Software Options]()\nMost software packages have options for the beta distribution.\n\n**[Mathematica](http:\/\/www.wolfram.com\/mathematica\/)**\n\nImplement a beta distribution by typing BetaDistribution\\[alpha,beta\\].\n\n**R**\n\ndbeta(x, shape1, shape2, ncp = 0, log = FALSE) pbeta(q, shape1, shape2, ncp = 0, lower.tail = TRUE, log.p = FALSE) qbeta(p, shape1, shape2, ncp = 0, lower.tail = TRUE, log.p = FALSE) rbeta(n, shape1, shape2, ncp = 0)\n\nwhere:\n\n1. 1. - x, q = vector of [quantiles](https:\/\/www.statisticshowto.com\/quantile-definition-find-easy-steps\/). p = vector of probabilities.\n      - n = \\# of observations.\n      - shape1, shape2 = [shape parameters](https:\/\/www.statisticshowto.com\/shape-parameter\/) α and β\n      - ncp = [non-centrality parameter](https:\/\/www.statisticshowto.com\/non-centrality-parameter-ncp\/).\n      - log, log.p = logical; if TRUE, probabilities p are given as log(p).\n      - lower.tail = logical; if TRUE (default), probabilities are P\\[X ≤ x\\], otherwise, P\\[X \\> x\\].\n## [Excel Beta Distribution: BETA.DIST & INV.BETA]()\nWatch the video for an example of BETA.DIST:\n\n[![YouTube video thumbnail](https:\/\/www.statisticshowto.com\/wp-content\/plugins\/wp-youtube-lyte\/lyteCache.php?origThumbUrl=https%3A%2F%2Fi.ytimg.com%2Fvi%2Fct1rhj4M54o%2F0.jpg) Watch this video on YouTube](https:\/\/youtu.be\/ct1rhj4M54o)\n\nCan’t see the video? [Click here](https:\/\/youtu.be\/ct1rhj4M54o?si=zzBQWjgDzHSHJJlg) to watch it on YouTube. The Microsoft Excel beta distribution format depends on which version of [Excel](http:\/\/office.microsoft.com\/en-us\/excel\/) you are using; Excel 2010 and later uses the BETA.DIST function, while earlier versions use BETADIST. You’ll need to know alpha and beta, both of which refer to the parameters describing the distribution. Five inputs are needed:\n\n1. 1. - The value where you want to [evaluate the function](https:\/\/www.statisticshowto.com\/evaluating-a-function\/).\n      - Alpha and Beta, the parameters of the distribution which determine shape.\n      - The lower and upper bound.\n\nAn extra input is needed for the Excel 2010 Beta Distribution: cumulative. Cumulative is a logical value that determines the function’s form and can either be TRUE (returns the cumulative distribution function) or FALSE (returns the probability density function). **Example problem:** *Calculate a [cumulative distribution function](https:\/\/www.statisticshowto.com\/cumulative-distribution-function-cdf\/) for a beta distribution in Excel at 0.5 with an alpha of 9, a beta of 10, a [lower bound](https:\/\/www.statisticshowto.com\/lower-bound-greatest-glb-infimum\/) of 0 and an [upper bound](https:\/\/www.statisticshowto.com\/upper-bound\/) of 1.*\n\n## Excel 2007 and Earlier:\n1. 1. 1. **Type the value where you want to evaluate the function in cell A1.** For this example, type “.5” in cell A1.\n      2. **Type the value for alpha in cell A2** and then type the value for beta in cell A1. For this example, type “9” in cell A2 and then type “10” in cell A3.\n      3. **Type the lower bound in cell A4** and then **type the upper bound in cell A5.** For this example, type “0” in cell A4 and then type “1” in cell A5\n      4. **Type the beta distribution function into cell A6.** The format of the function is =BETADIST(value,alpha,beta,lower bound,upper bound). For this example, type “=BETADIST(A1,A2,A3,A4,A5)” into cell A6. Press “Enter” to see the result for the beta distribution, which is 0.592735.\n\n[![Excel Beta Distribution](https:\/\/www.statisticshowto.com\/wp-content\/uploads\/2013\/01\/betadist1.jpg)![Excel Beta Distribution](https:\/\/www.statisticshowto.com\/wp-content\/uploads\/2013\/01\/betadist1.jpg)](https:\/\/www.statisticshowto.com\/how-to-calculate-a-microsoft-excel-beta-distribution\/)\n\n## Excel 2010 and later\nExcel 2010 and later uses the BETA.DIST function. If you’re using earlier versions, use BETADIST instead. Inputs are as follows:\n\n1. 1. - A value where you want to evaluate the function.\n      - Alpha (α) and Beta (β), which determine the distribution’s shape.\n      - The lower and upper bound.\n      - “Cumulative” — a logical value (TRUE\/FALSE) that determines the function’s form. TRUE returns the CDF and FALSE returns the PDF.\n\n**Example problem:** *Calculate a [cumulative distribution function](https:\/\/probabilityhowto.com\/cumulative-distribution-function-2\/) for a beta distribution at 0.5 with α* = *9, β* = *10,* lower bound = 0 and upper bound = 1.\n\n## [Applications of the Beta Density Function]()\nThe beta distribution is used for a variety of applications for modeling the behavior of [random variables](https:\/\/www.statisticshowto.com\/random-variable\/) limited to intervals of finite length. Uses include:\n\n1. 1. - Bayesian inference, as the conjugate prior probability distribution for the Bernoulli, binomial, geometric and negative binomial distributions.\n      - The Rule of Succession (such as Pierre-Simon Laplace’s treatment of the sunrise problem),\n      - Task duration modeling.\n      - Project\/planning control systems like PERT and CPM.\n      - In project management for the “three-point technique,” (also called the beta distribution technique), which recognizes uncertainty in estimated project time \\[1\\]. The three-point estimation technique is used to account for uncertainty in project time estimates. By applying this method, project managers can leverage powerful quantitative tools to identify tasks with the highest risk. It also facilitates effective management of overall project completion time.\n\nReferences\n\n1. 1. 3-Points Estimating <https:\/\/www.projectmanagement.com\/contentPages\/wiki.cfm?ID=368763&thisPageURL=\/wikis\/368763\/3-Points-Estimating#_=_>\n\n[Bipolar Distribution](https:\/\/www.statisticshowto.com\/bipolar-distribution\/)\n\n[Exponential Distribution \/ Negative Exponential: Definition, Examples](https:\/\/www.statisticshowto.com\/exponential-distribution\/)\n\n**Comments? Need to post a correction?** Please [***Contact Us***](https:\/\/www.statisticshowto.com\/contact\/).\n\nCheck out our **[\"Practically Cheating Statistics Handbook](https:\/\/www.statisticshowto.com\/?page_id=15743&preview=true\")**, which gives you hundreds of easy-to-follow answers in a convenient e-book.\n\n[![Statistics Handbook](https:\/\/www.statisticshowto.com\/wp-content\/uploads\/2024\/06\/BookCoverSHT-1.jpg)![Statistics Handbook](https:\/\/www.statisticshowto.com\/wp-content\/uploads\/2024\/06\/BookCoverSHT-1.jpg)](https:\/\/www.statisticshowto.com\/?page_id=15743&preview=true)\n***\n\n**Readers are calling it suspenseful, haunting, and unforgettable. Check out Stephanie’s debut post-apocalyptic novel — available now on Amazon.**\n\n[![an image of S.E. Glen's book Fallout in Georgia](https:\/\/www.statisticshowto.com\/wp-content\/uploads\/2026\/02\/novel-ad.jpg)![an image of S.E. Glen's book Fallout in Georgia](https:\/\/www.statisticshowto.com\/wp-content\/uploads\/2026\/02\/novel-ad.jpg)](https:\/\/amzn.to\/3LLzICB)\n\nLooking for elementary [**statistics help**](https:\/\/www.statisticshowto.com\/help-with-statistics-equations\/)? You’ve come to the right place. **Statistics How To** has more than 1,000 articles and **videos** for elementary statistics, probability, AP and advanced statistics topics. **Looking for a specific topic?** Type it into the search box at the top of the page.\n\nLatest articles\n\n- [Hedges’ g: Definition, Formula](https:\/\/www.statisticshowto.com\/hedges-g\/)\n- [Kaiser-Meyer-Olkin (KMO) Test for Sampling Adequacy](https:\/\/www.statisticshowto.com\/kaiser-meyer-olkin\/)\n- [Benjamini-Hochberg Procedure](https:\/\/www.statisticshowto.com\/benjamini-hochberg-procedure\/)\n- [Tetrachoric Correlation: Definition, Examples, Formula](https:\/\/www.statisticshowto.com\/tetrachoric-correlation\/)\n- [Random Variable: What is it in Statistics?](https:\/\/www.statisticshowto.com\/random-variable\/)\n- [Maximum Likelihood and Maximum Likelihood Estimation](https:\/\/www.statisticshowto.com\/maximum-likelihood-estimation\/)\n- [What is a Parameter in Statistics?](https:\/\/www.statisticshowto.com\/what-is-a-parameter-in-statistics\/)\n- [Beta Level: Definition & Examples](https:\/\/www.statisticshowto.com\/beta-level\/)\n- [Pairwise Independent, Mutually Independent: Definition, Example](https:\/\/www.statisticshowto.com\/pairwise-independent-mutually\/)\n- [Population Mean Definition, Example, Formula](https:\/\/www.statisticshowto.com\/population-mean\/)\n- [Dispersion \/ Measures of Dispersion: Definition](https:\/\/www.statisticshowto.com\/dispersion\/)\n- [Serial Correlation \/ Autocorrelation: Definition, Tests](https:\/\/www.statisticshowto.com\/serial-correlation-autocorrelation\/)\n\n© 2026 [**Statistics How To**](https:\/\/www.statisticshowto.com\/) \\| [About Us](https:\/\/www.statisticshowto.com\/contact\/) \\| [Privacy Policy](https:\/\/www.statisticshowto.com\/privacy-policy\/) \\| [Terms of Use](https:\/\/www.statisticshowto.com\/terms-of-use\/)","attrs_readable_markdown":"\\< [Probability Distributions](https:\/\/www.statisticshowto.com\/probability-and-statistics\/statistics-definitions\/probability-distribution\/) \\> *Beta Distribution*\n\n**Contents:**\n\n1. 1. [What is a Beta Distribution?](https:\/\/www.statisticshowto.com\/beta-distribution\/#whatis)\n   2. [Properties](https:\/\/www.statisticshowto.com\/beta-distribution\/#properties)\n   3. [Software Options: Mathematica and R](https:\/\/www.statisticshowto.com\/beta-distribution\/#software)\n   4. [Excel Beta Distribution: BETA.DIST & INV.BETA](https:\/\/www.statisticshowto.com\/beta-distribution\/#excel)\n   5. [Applications of the Beta Density Function](https:\/\/www.statisticshowto.com\/beta-distribution\/#applications)\n## What is a Beta Distribution?\nThe **beta distribution** (also called the *beta distribution of the first kind*) is a family of [continuous probability distributions](https:\/\/www.statisticshowto.com\/continuous-probability-distribution\/) defined on \\[0, 1\\]. This distribution is similar to the [binomial distribution](https:\/\/www.statisticshowto.com\/probability-and-statistics\/binomial-theorem\/binomial-distribution-formula\/), except where the binomial models the *number* of successes (x), the beta models the *probability* (p) of success.\n\nA beta distribution is a versatile way to represent outcomes for percentages or proportions. For example, how likely is it that a rogue candidate will win the next Presidential election? You might think the probability is 0.2. Your friend might think it’s 0.15. The beta distribution gives you a way to describe this. One reason that the beta distribution can be confusing to work with is that there are three “Betas” to contend with, and they all have different meanings:\n\n1. 1. - Beta(α, β): the *name* of the probability distribution.\n      - B(α, β ): the name of a function in the denominator of the [pdf](https:\/\/www.statisticshowto.com\/probability-density-function\/). This acts as a “[normalizing constant](https:\/\/www.statisticshowto.com\/normalizing-constant-definition\/)” to ensure that the area under the curve of the pdf equals 1.\n      - β: the name of the second [shape parameter](https:\/\/www.statisticshowto.com\/shape-parameter\/) in the pdf.\n\nThe basic beta distribution is also called the *beta distribution of the first kind*. Beta distribution of the second kind is another name for the [beta prime distribution](https:\/\/www.statisticshowto.com\/beta-prime-distribution\/).\n\n## Properties\nThe general formula for the [probability density function (pdf)](https:\/\/www.statisticshowto.com\/probability-density-function-definition-examples\/) is: ![beta pdf](https:\/\/www.statisticshowto.com\/wp-content\/uploads\/2015\/08\/beta-pdf.png)   \nwhere:\n\n1. 1. - α and β are two positive [shape parameters](https:\/\/www.statisticshowto.com\/shape-parameter\/) which control the shape of the distribution.\n\nThe graph of the beta density function can take on a variety of shapes. For example, if α \\< 1 and Β \\< 1, the graph will be a U shaped distribution, and if α = 1 and Β = 2, the graph is a straight line.\n\n[![beta distribution](https:\/\/www.statisticshowto.com\/wp-content\/uploads\/2010\/09\/800px-Beta_distribution_pdf-300x225.png)](https:\/\/www.statisticshowto.com\/wp-content\/uploads\/2010\/09\/800px-Beta_distribution_pdf.png)\n\nThe Beta Distribution pdf.\n\nThe general formula for the [cumulative distribution function (CDF)](https:\/\/www.statisticshowto.com\/cumulative-distribution-function-cdf\/) is: ![beta distribution cdf](https:\/\/www.statisticshowto.com\/wp-content\/uploads\/2015\/08\/beta-distribution-cdf.png) which includes the [regularized beta function](https:\/\/www.statisticshowto.com\/regularized-incomplete-beta-function\/): ![reg beta function](https:\/\/www.statisticshowto.com\/wp-content\/uploads\/2015\/08\/reg-beta-function.png)\n\n## [Software Options]()\nMost software packages have options for the beta distribution.\n\n**[Mathematica](http:\/\/www.wolfram.com\/mathematica\/)**\n\nImplement a beta distribution by typing BetaDistribution\\[alpha,beta\\].\n\n**R**\n\ndbeta(x, shape1, shape2, ncp = 0, log = FALSE) pbeta(q, shape1, shape2, ncp = 0, lower.tail = TRUE, log.p = FALSE) qbeta(p, shape1, shape2, ncp = 0, lower.tail = TRUE, log.p = FALSE) rbeta(n, shape1, shape2, ncp = 0)\n\nwhere:\n\n1. 1. - x, q = vector of [quantiles](https:\/\/www.statisticshowto.com\/quantile-definition-find-easy-steps\/). p = vector of probabilities.\n      - n = \\# of observations.\n      - shape1, shape2 = [shape parameters](https:\/\/www.statisticshowto.com\/shape-parameter\/) α and β\n      - ncp = [non-centrality parameter](https:\/\/www.statisticshowto.com\/non-centrality-parameter-ncp\/).\n      - log, log.p = logical; if TRUE, probabilities p are given as log(p).\n      - lower.tail = logical; if TRUE (default), probabilities are P\\[X ≤ x\\], otherwise, P\\[X \\> x\\].\n## [Excel Beta Distribution: BETA.DIST & INV.BETA]()\nWatch the video for an example of BETA.DIST:\n\nCan’t see the video? [Click here](https:\/\/youtu.be\/ct1rhj4M54o?si=zzBQWjgDzHSHJJlg) to watch it on YouTube. The Microsoft Excel beta distribution format depends on which version of [Excel](http:\/\/office.microsoft.com\/en-us\/excel\/) you are using; Excel 2010 and later uses the BETA.DIST function, while earlier versions use BETADIST. You’ll need to know alpha and beta, both of which refer to the parameters describing the distribution. Five inputs are needed:\n\n1. 1. - The value where you want to [evaluate the function](https:\/\/www.statisticshowto.com\/evaluating-a-function\/).\n      - Alpha and Beta, the parameters of the distribution which determine shape.\n      - The lower and upper bound.\n\nAn extra input is needed for the Excel 2010 Beta Distribution: cumulative. Cumulative is a logical value that determines the function’s form and can either be TRUE (returns the cumulative distribution function) or FALSE (returns the probability density function). **Example problem:** *Calculate a [cumulative distribution function](https:\/\/www.statisticshowto.com\/cumulative-distribution-function-cdf\/) for a beta distribution in Excel at 0.5 with an alpha of 9, a beta of 10, a [lower bound](https:\/\/www.statisticshowto.com\/lower-bound-greatest-glb-infimum\/) of 0 and an [upper bound](https:\/\/www.statisticshowto.com\/upper-bound\/) of 1.*\n\n## Excel 2007 and Earlier:\n1. 1. 1. **Type the value where you want to evaluate the function in cell A1.** For this example, type “.5” in cell A1.\n      2. **Type the value for alpha in cell A2** and then type the value for beta in cell A1. For this example, type “9” in cell A2 and then type “10” in cell A3.\n      3. **Type the lower bound in cell A4** and then **type the upper bound in cell A5.** For this example, type “0” in cell A4 and then type “1” in cell A5\n      4. **Type the beta distribution function into cell A6.** The format of the function is =BETADIST(value,alpha,beta,lower bound,upper bound). For this example, type “=BETADIST(A1,A2,A3,A4,A5)” into cell A6. Press “Enter” to see the result for the beta distribution, which is 0.592735.\n\n[![Excel Beta Distribution](https:\/\/www.statisticshowto.com\/wp-content\/uploads\/2013\/01\/betadist1.jpg)](https:\/\/www.statisticshowto.com\/how-to-calculate-a-microsoft-excel-beta-distribution\/)\n\n## Excel 2010 and later\nExcel 2010 and later uses the BETA.DIST function. If you’re using earlier versions, use BETADIST instead. Inputs are as follows:\n\n1. 1. - A value where you want to evaluate the function.\n      - Alpha (α) and Beta (β), which determine the distribution’s shape.\n      - The lower and upper bound.\n      - “Cumulative” — a logical value (TRUE\/FALSE) that determines the function’s form. TRUE returns the CDF and FALSE returns the PDF.\n\n**Example problem:** *Calculate a [cumulative distribution function](https:\/\/probabilityhowto.com\/cumulative-distribution-function-2\/) for a beta distribution at 0.5 with α* = *9, β* = *10,* lower bound = 0 and upper bound = 1.\n\n## [Applications of the Beta Density Function]()\nThe beta distribution is used for a variety of applications for modeling the behavior of [random variables](https:\/\/www.statisticshowto.com\/random-variable\/) limited to intervals of finite length. Uses include:\n\n1. 1. - Bayesian inference, as the conjugate prior probability distribution for the Bernoulli, binomial, geometric and negative binomial distributions.\n      - The Rule of Succession (such as Pierre-Simon Laplace’s treatment of the sunrise problem),\n      - Task duration modeling.\n      - Project\/planning control systems like PERT and CPM.\n      - In project management for the “three-point technique,” (also called the beta distribution technique), which recognizes uncertainty in estimated project time \\[1\\]. The three-point estimation technique is used to account for uncertainty in project time estimates. By applying this method, project managers can leverage powerful quantitative tools to identify tasks with the highest risk. It also facilitates effective management of overall project completion time.\n\nReferences\n\n1. 1. 3-Points Estimating <https:\/\/www.projectmanagement.com\/contentPages\/wiki.cfm?ID=368763&thisPageURL=\/wikis\/368763\/3-Points-Estimating#_=_>","meta_canonical":null,"ml_categories_json":"{\"\/Science\":919,\"\/Science\/Mathematics\":910,\"\/Science\/Mathematics\/Statistics\":909}","ml_types_json":"{\"\/Article\":996,\"\/Article\/Definitions\":697}","ml_intent_types_json":"{\"Informational\":999}","meta_language":"en-us","attrs_author":null,"attrs_publish_time":1734858183,"attrs_original_publish_time":1586054467,"attrs_is_republished":1,"attrs_nr_words":"1383","attrs_boilerpipe_nr_words":"999","body_ext_links_number":16,"body_int_links_number":74,"meta_nofollow":0,"meta_noarchive":0,"props_was_rendered":1,"src_redirect":"","download_time_msec":580,"download_ttfb_msec":579,"download_size":20927}

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< Probability Distributions > Beta Distribution Contents: What is a Beta Distribution? Properties Software Options: Mathematica and R Excel Beta Distribution: BETA.DIST & INV.BETA Applications of the Beta Density Function What is a Beta Distribution? The beta distribution (also called the beta distribution of the first kind ) is a family of continuous probability distributions defined on [0, 1]. This distribution is similar to the binomial distribution , except where the binomial models the number of successes (x), the beta models the probability (p) of success. A beta distribution is a versatile way to represent outcomes for percentages or proportions. For example, how likely is it that a rogue candidate will win the next Presidential election? You might think the probability is 0.2. Your friend might think it’s 0.15. The beta distribution gives you a way to describe this. One reason that the beta distribution can be confusing to work with is that there are three “Betas” to contend with, and they all have different meanings: Beta(α, β): the name of the probability distribution. B(α, β ): the name of a function in the denominator of the pdf . This acts as a “ normalizing constant ” to ensure that the area under the curve of the pdf equals 1. β: the name of the second shape parameter in the pdf. The basic beta distribution is also called the beta distribution of the first kind . Beta distribution of the second kind is another name for the beta prime distribution . Properties The general formula for the probability density function (pdf) is: where: α and β are two positive shape parameters which control the shape of the distribution. The graph of the beta density function can take on a variety of shapes. For example, if α < 1 and Β < 1, the graph will be a U shaped distribution, and if α = 1 and Β = 2, the graph is a straight line. The Beta Distribution pdf. The general formula for the cumulative distribution function (CDF) is: which includes the regularized beta function : Software Options Most software packages have options for the beta distribution. Mathematica Implement a beta distribution by typing BetaDistribution[alpha,beta]. R dbeta(x, shape1, shape2, ncp = 0, log = FALSE) pbeta(q, shape1, shape2, ncp = 0, lower.tail = TRUE, log.p = FALSE) qbeta(p, shape1, shape2, ncp = 0, lower.tail = TRUE, log.p = FALSE) rbeta(n, shape1, shape2, ncp = 0) where: x, q = vector of quantiles . p = vector of probabilities. n = # of observations. shape1, shape2 = shape parameters α and β ncp = non-centrality parameter . log, log.p = logical; if TRUE, probabilities p are given as log(p). lower.tail = logical; if TRUE (default), probabilities are P[X ≤ x], otherwise, P[X > x]. Excel Beta Distribution: BETA.DIST & INV.BETA Watch the video for an example of BETA.DIST: Can’t see the video? Click here to watch it on YouTube. The Microsoft Excel beta distribution format depends on which version of Excel you are using; Excel 2010 and later uses the BETA.DIST function, while earlier versions use BETADIST. You’ll need to know alpha and beta, both of which refer to the parameters describing the distribution. Five inputs are needed: The value where you want to evaluate the function . Alpha and Beta, the parameters of the distribution which determine shape. The lower and upper bound. An extra input is needed for the Excel 2010 Beta Distribution: cumulative. Cumulative is a logical value that determines the function’s form and can either be TRUE (returns the cumulative distribution function) or FALSE (returns the probability density function). Example problem: Calculate a cumulative distribution function for a beta distribution in Excel at 0.5 with an alpha of 9, a beta of 10, a lower bound of 0 and an upper bound of 1. Excel 2007 and Earlier: Type the value where you want to evaluate the function in cell A1. For this example, type “.5” in cell A1. Type the value for alpha in cell A2 and then type the value for beta in cell A1. For this example, type “9” in cell A2 and then type “10” in cell A3. Type the lower bound in cell A4 and then type the upper bound in cell A5. For this example, type “0” in cell A4 and then type “1” in cell A5 Type the beta distribution function into cell A6. The format of the function is =BETADIST(value,alpha,beta,lower bound,upper bound). For this example, type “=BETADIST(A1,A2,A3,A4,A5)” into cell A6. Press “Enter” to see the result for the beta distribution, which is 0.592735. Excel 2010 and later Excel 2010 and later uses the BETA.DIST function. If you’re using earlier versions, use BETADIST instead. Inputs are as follows: A value where you want to evaluate the function. Alpha (α) and Beta (β), which determine the distribution’s shape. The lower and upper bound. “Cumulative” — a logical value (TRUE/FALSE) that determines the function’s form. TRUE returns the CDF and FALSE returns the PDF. Example problem: Calculate a cumulative distribution function for a beta distribution at 0.5 with α = 9, β = 10, lower bound = 0 and upper bound = 1. Applications of the Beta Density Function The beta distribution is used for a variety of applications for modeling the behavior of random variables limited to intervals of finite length. Uses include: Bayesian inference, as the conjugate prior probability distribution for the Bernoulli, binomial, geometric and negative binomial distributions. The Rule of Succession (such as Pierre-Simon Laplace’s treatment of the sunrise problem), Task duration modeling. Project/planning control systems like PERT and CPM. In project management for the “three-point technique,” (also called the beta distribution technique), which recognizes uncertainty in estimated project time [1]. The three-point estimation technique is used to account for uncertainty in project time estimates. By applying this method, project managers can leverage powerful quantitative tools to identify tasks with the highest risk. It also facilitates effective management of overall project completion time. References 3-Points Estimating https://www.projectmanagement.com/contentPages/wiki.cfm?ID=368763&thisPageURL=/wikis/368763/3-Points-Estimating#_=_

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[Matrices](https://www.statisticshowto.com/matrices-and-matrix-algebra/) - [Experimental Design](https://www.statisticshowto.com/experimental-design/) - [Calculus Based Statistics](https://www.statisticshowto.com/probability-and-statistics/calculus-based-statistics/) # Beta Distribution: Definition, Calculation \< [Probability Distributions](https://www.statisticshowto.com/probability-and-statistics/statistics-definitions/probability-distribution/) \> *Beta Distribution* **Contents:** 1. 1. [What is a Beta Distribution?](https://www.statisticshowto.com/beta-distribution/#whatis) 2. [Properties](https://www.statisticshowto.com/beta-distribution/#properties) 3. [Software Options: Mathematica and R](https://www.statisticshowto.com/beta-distribution/#software) 4. [Excel Beta Distribution: BETA.DIST & INV.BETA](https://www.statisticshowto.com/beta-distribution/#excel) 5. [Applications of the Beta Density Function](https://www.statisticshowto.com/beta-distribution/#applications) ## What is a Beta Distribution? The **beta distribution** (also called the *beta distribution of the first kind*) is a family of [continuous probability distributions](https://www.statisticshowto.com/continuous-probability-distribution/) defined on \[0, 1\]. This distribution is similar to the [binomial distribution](https://www.statisticshowto.com/probability-and-statistics/binomial-theorem/binomial-distribution-formula/), except where the binomial models the *number* of successes (x), the beta models the *probability* (p) of success. A beta distribution is a versatile way to represent outcomes for percentages or proportions. For example, how likely is it that a rogue candidate will win the next Presidential election? You might think the probability is 0.2. Your friend might think it’s 0.15. The beta distribution gives you a way to describe this. One reason that the beta distribution can be confusing to work with is that there are three “Betas” to contend with, and they all have different meanings: 1. 1. - Beta(α, β): the *name* of the probability distribution. - B(α, β ): the name of a function in the denominator of the [pdf](https://www.statisticshowto.com/probability-density-function/). This acts as a “[normalizing constant](https://www.statisticshowto.com/normalizing-constant-definition/)” to ensure that the area under the curve of the pdf equals 1. - β: the name of the second [shape parameter](https://www.statisticshowto.com/shape-parameter/) in the pdf. The basic beta distribution is also called the *beta distribution of the first kind*. Beta distribution of the second kind is another name for the [beta prime distribution](https://www.statisticshowto.com/beta-prime-distribution/). ## Properties The general formula for the [probability density function (pdf)](https://www.statisticshowto.com/probability-density-function-definition-examples/) is: ![beta pdf](https://www.statisticshowto.com/wp-content/uploads/2015/08/beta-pdf.png) where: 1. 1. - α and β are two positive [shape parameters](https://www.statisticshowto.com/shape-parameter/) which control the shape of the distribution. The graph of the beta density function can take on a variety of shapes. For example, if α \< 1 and Β \< 1, the graph will be a U shaped distribution, and if α = 1 and Β = 2, the graph is a straight line. [![beta distribution](https://www.statisticshowto.com/wp-content/uploads/2010/09/800px-Beta_distribution_pdf-300x225.png)](https://www.statisticshowto.com/wp-content/uploads/2010/09/800px-Beta_distribution_pdf.png) The Beta Distribution pdf. The general formula for the [cumulative distribution function (CDF)](https://www.statisticshowto.com/cumulative-distribution-function-cdf/) is: ![beta distribution cdf](https://www.statisticshowto.com/wp-content/uploads/2015/08/beta-distribution-cdf.png)![beta distribution cdf](https://www.statisticshowto.com/wp-content/uploads/2015/08/beta-distribution-cdf.png) which includes the [regularized beta function](https://www.statisticshowto.com/regularized-incomplete-beta-function/): ![reg beta function](https://www.statisticshowto.com/wp-content/uploads/2015/08/reg-beta-function.png)![reg beta function](https://www.statisticshowto.com/wp-content/uploads/2015/08/reg-beta-function.png) ## [Software Options]() Most software packages have options for the beta distribution. **[Mathematica](http://www.wolfram.com/mathematica/)** Implement a beta distribution by typing BetaDistribution\[alpha,beta\]. **R** dbeta(x, shape1, shape2, ncp = 0, log = FALSE) pbeta(q, shape1, shape2, ncp = 0, lower.tail = TRUE, log.p = FALSE) qbeta(p, shape1, shape2, ncp = 0, lower.tail = TRUE, log.p = FALSE) rbeta(n, shape1, shape2, ncp = 0) where: 1. 1. - x, q = vector of [quantiles](https://www.statisticshowto.com/quantile-definition-find-easy-steps/). p = vector of probabilities. - n = \# of observations. - shape1, shape2 = [shape parameters](https://www.statisticshowto.com/shape-parameter/) α and β - ncp = [non-centrality parameter](https://www.statisticshowto.com/non-centrality-parameter-ncp/). - log, log.p = logical; if TRUE, probabilities p are given as log(p). - lower.tail = logical; if TRUE (default), probabilities are P\[X ≤ x\], otherwise, P\[X \> x\]. ## [Excel Beta Distribution: BETA.DIST & INV.BETA]() Watch the video for an example of BETA.DIST: [![YouTube video thumbnail](https://www.statisticshowto.com/wp-content/plugins/wp-youtube-lyte/lyteCache.php?origThumbUrl=https%3A%2F%2Fi.ytimg.com%2Fvi%2Fct1rhj4M54o%2F0.jpg) Watch this video on YouTube](https://youtu.be/ct1rhj4M54o) Can’t see the video? [Click here](https://youtu.be/ct1rhj4M54o?si=zzBQWjgDzHSHJJlg) to watch it on YouTube. The Microsoft Excel beta distribution format depends on which version of [Excel](http://office.microsoft.com/en-us/excel/) you are using; Excel 2010 and later uses the BETA.DIST function, while earlier versions use BETADIST. You’ll need to know alpha and beta, both of which refer to the parameters describing the distribution. Five inputs are needed: 1. 1. - The value where you want to [evaluate the function](https://www.statisticshowto.com/evaluating-a-function/). - Alpha and Beta, the parameters of the distribution which determine shape. - The lower and upper bound. An extra input is needed for the Excel 2010 Beta Distribution: cumulative. Cumulative is a logical value that determines the function’s form and can either be TRUE (returns the cumulative distribution function) or FALSE (returns the probability density function). **Example problem:** *Calculate a [cumulative distribution function](https://www.statisticshowto.com/cumulative-distribution-function-cdf/) for a beta distribution in Excel at 0.5 with an alpha of 9, a beta of 10, a [lower bound](https://www.statisticshowto.com/lower-bound-greatest-glb-infimum/) of 0 and an [upper bound](https://www.statisticshowto.com/upper-bound/) of 1.* ## Excel 2007 and Earlier: 1. 1. 1. **Type the value where you want to evaluate the function in cell A1.** For this example, type “.5” in cell A1. 2. **Type the value for alpha in cell A2** and then type the value for beta in cell A1. For this example, type “9” in cell A2 and then type “10” in cell A3. 3. **Type the lower bound in cell A4** and then **type the upper bound in cell A5.** For this example, type “0” in cell A4 and then type “1” in cell A5 4. **Type the beta distribution function into cell A6.** The format of the function is =BETADIST(value,alpha,beta,lower bound,upper bound). For this example, type “=BETADIST(A1,A2,A3,A4,A5)” into cell A6. Press “Enter” to see the result for the beta distribution, which is 0.592735. [![Excel Beta Distribution](https://www.statisticshowto.com/wp-content/uploads/2013/01/betadist1.jpg)![Excel Beta Distribution](https://www.statisticshowto.com/wp-content/uploads/2013/01/betadist1.jpg)](https://www.statisticshowto.com/how-to-calculate-a-microsoft-excel-beta-distribution/) ## Excel 2010 and later Excel 2010 and later uses the BETA.DIST function. If you’re using earlier versions, use BETADIST instead. Inputs are as follows: 1. 1. - A value where you want to evaluate the function. - Alpha (α) and Beta (β), which determine the distribution’s shape. - The lower and upper bound. - “Cumulative” — a logical value (TRUE/FALSE) that determines the function’s form. TRUE returns the CDF and FALSE returns the PDF. **Example problem:** *Calculate a [cumulative distribution function](https://probabilityhowto.com/cumulative-distribution-function-2/) for a beta distribution at 0.5 with α* = *9, β* = *10,* lower bound = 0 and upper bound = 1. ## [Applications of the Beta Density Function]() The beta distribution is used for a variety of applications for modeling the behavior of [random variables](https://www.statisticshowto.com/random-variable/) limited to intervals of finite length. Uses include: 1. 1. - Bayesian inference, as the conjugate prior probability distribution for the Bernoulli, binomial, geometric and negative binomial distributions. - The Rule of Succession (such as Pierre-Simon Laplace’s treatment of the sunrise problem), - Task duration modeling. - Project/planning control systems like PERT and CPM. - In project management for the “three-point technique,” (also called the beta distribution technique), which recognizes uncertainty in estimated project time \[1\]. The three-point estimation technique is used to account for uncertainty in project time estimates. By applying this method, project managers can leverage powerful quantitative tools to identify tasks with the highest risk. It also facilitates effective management of overall project completion time. References 1. 1. 3-Points Estimating <https://www.projectmanagement.com/contentPages/wiki.cfm?ID=368763&thisPageURL=/wikis/368763/3-Points-Estimating#_=_> [Bipolar Distribution](https://www.statisticshowto.com/bipolar-distribution/) [Exponential Distribution / Negative Exponential: Definition, Examples](https://www.statisticshowto.com/exponential-distribution/) **Comments? Need to post a correction?** Please [***Contact Us***](https://www.statisticshowto.com/contact/). Check out our **["Practically Cheating Statistics Handbook](https://www.statisticshowto.com/?page_id=15743&preview=true")**, which gives you hundreds of easy-to-follow answers in a convenient e-book. [![Statistics Handbook](https://www.statisticshowto.com/wp-content/uploads/2024/06/BookCoverSHT-1.jpg)![Statistics Handbook](https://www.statisticshowto.com/wp-content/uploads/2024/06/BookCoverSHT-1.jpg)](https://www.statisticshowto.com/?page_id=15743&preview=true) *** **Readers are calling it suspenseful, haunting, and unforgettable. Check out Stephanie’s debut post-apocalyptic novel — available now on Amazon.** [![an image of S.E. Glen's book Fallout in Georgia](https://www.statisticshowto.com/wp-content/uploads/2026/02/novel-ad.jpg)![an image of S.E. Glen's book Fallout in Georgia](https://www.statisticshowto.com/wp-content/uploads/2026/02/novel-ad.jpg)](https://amzn.to/3LLzICB) Looking for elementary [**statistics help**](https://www.statisticshowto.com/help-with-statistics-equations/)? You’ve come to the right place. **Statistics How To** has more than 1,000 articles and **videos** for elementary statistics, probability, AP and advanced statistics topics. **Looking for a specific topic?** Type it into the search box at the top of the page. 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\< [Probability Distributions](https://www.statisticshowto.com/probability-and-statistics/statistics-definitions/probability-distribution/) \> *Beta Distribution* **Contents:** 1. 1. [What is a Beta Distribution?](https://www.statisticshowto.com/beta-distribution/#whatis) 2. [Properties](https://www.statisticshowto.com/beta-distribution/#properties) 3. [Software Options: Mathematica and R](https://www.statisticshowto.com/beta-distribution/#software) 4. [Excel Beta Distribution: BETA.DIST & INV.BETA](https://www.statisticshowto.com/beta-distribution/#excel) 5. [Applications of the Beta Density Function](https://www.statisticshowto.com/beta-distribution/#applications) ## What is a Beta Distribution? The **beta distribution** (also called the *beta distribution of the first kind*) is a family of [continuous probability distributions](https://www.statisticshowto.com/continuous-probability-distribution/) defined on \[0, 1\]. This distribution is similar to the [binomial distribution](https://www.statisticshowto.com/probability-and-statistics/binomial-theorem/binomial-distribution-formula/), except where the binomial models the *number* of successes (x), the beta models the *probability* (p) of success. A beta distribution is a versatile way to represent outcomes for percentages or proportions. For example, how likely is it that a rogue candidate will win the next Presidential election? You might think the probability is 0.2. Your friend might think it’s 0.15. The beta distribution gives you a way to describe this. One reason that the beta distribution can be confusing to work with is that there are three “Betas” to contend with, and they all have different meanings: 1. 1. - Beta(α, β): the *name* of the probability distribution. - B(α, β ): the name of a function in the denominator of the [pdf](https://www.statisticshowto.com/probability-density-function/). This acts as a “[normalizing constant](https://www.statisticshowto.com/normalizing-constant-definition/)” to ensure that the area under the curve of the pdf equals 1. - β: the name of the second [shape parameter](https://www.statisticshowto.com/shape-parameter/) in the pdf. The basic beta distribution is also called the *beta distribution of the first kind*. Beta distribution of the second kind is another name for the [beta prime distribution](https://www.statisticshowto.com/beta-prime-distribution/). ## Properties The general formula for the [probability density function (pdf)](https://www.statisticshowto.com/probability-density-function-definition-examples/) is: ![beta pdf](https://www.statisticshowto.com/wp-content/uploads/2015/08/beta-pdf.png) where: 1. 1. - α and β are two positive [shape parameters](https://www.statisticshowto.com/shape-parameter/) which control the shape of the distribution. The graph of the beta density function can take on a variety of shapes. For example, if α \< 1 and Β \< 1, the graph will be a U shaped distribution, and if α = 1 and Β = 2, the graph is a straight line. [![beta distribution](https://www.statisticshowto.com/wp-content/uploads/2010/09/800px-Beta_distribution_pdf-300x225.png)](https://www.statisticshowto.com/wp-content/uploads/2010/09/800px-Beta_distribution_pdf.png) The Beta Distribution pdf. The general formula for the [cumulative distribution function (CDF)](https://www.statisticshowto.com/cumulative-distribution-function-cdf/) is: ![beta distribution cdf](https://www.statisticshowto.com/wp-content/uploads/2015/08/beta-distribution-cdf.png) which includes the [regularized beta function](https://www.statisticshowto.com/regularized-incomplete-beta-function/): ![reg beta function](https://www.statisticshowto.com/wp-content/uploads/2015/08/reg-beta-function.png) ## [Software Options]() Most software packages have options for the beta distribution. **[Mathematica](http://www.wolfram.com/mathematica/)** Implement a beta distribution by typing BetaDistribution\[alpha,beta\]. **R** dbeta(x, shape1, shape2, ncp = 0, log = FALSE) pbeta(q, shape1, shape2, ncp = 0, lower.tail = TRUE, log.p = FALSE) qbeta(p, shape1, shape2, ncp = 0, lower.tail = TRUE, log.p = FALSE) rbeta(n, shape1, shape2, ncp = 0) where: 1. 1. - x, q = vector of [quantiles](https://www.statisticshowto.com/quantile-definition-find-easy-steps/). p = vector of probabilities. - n = \# of observations. - shape1, shape2 = [shape parameters](https://www.statisticshowto.com/shape-parameter/) α and β - ncp = [non-centrality parameter](https://www.statisticshowto.com/non-centrality-parameter-ncp/). - log, log.p = logical; if TRUE, probabilities p are given as log(p). - lower.tail = logical; if TRUE (default), probabilities are P\[X ≤ x\], otherwise, P\[X \> x\]. ## [Excel Beta Distribution: BETA.DIST & INV.BETA]() Watch the video for an example of BETA.DIST: Can’t see the video? [Click here](https://youtu.be/ct1rhj4M54o?si=zzBQWjgDzHSHJJlg) to watch it on YouTube. The Microsoft Excel beta distribution format depends on which version of [Excel](http://office.microsoft.com/en-us/excel/) you are using; Excel 2010 and later uses the BETA.DIST function, while earlier versions use BETADIST. You’ll need to know alpha and beta, both of which refer to the parameters describing the distribution. Five inputs are needed: 1. 1. - The value where you want to [evaluate the function](https://www.statisticshowto.com/evaluating-a-function/). - Alpha and Beta, the parameters of the distribution which determine shape. - The lower and upper bound. An extra input is needed for the Excel 2010 Beta Distribution: cumulative. Cumulative is a logical value that determines the function’s form and can either be TRUE (returns the cumulative distribution function) or FALSE (returns the probability density function). **Example problem:** *Calculate a [cumulative distribution function](https://www.statisticshowto.com/cumulative-distribution-function-cdf/) for a beta distribution in Excel at 0.5 with an alpha of 9, a beta of 10, a [lower bound](https://www.statisticshowto.com/lower-bound-greatest-glb-infimum/) of 0 and an [upper bound](https://www.statisticshowto.com/upper-bound/) of 1.* ## Excel 2007 and Earlier: 1. 1. 1. **Type the value where you want to evaluate the function in cell A1.** For this example, type “.5” in cell A1. 2. **Type the value for alpha in cell A2** and then type the value for beta in cell A1. For this example, type “9” in cell A2 and then type “10” in cell A3. 3. **Type the lower bound in cell A4** and then **type the upper bound in cell A5.** For this example, type “0” in cell A4 and then type “1” in cell A5 4. **Type the beta distribution function into cell A6.** The format of the function is =BETADIST(value,alpha,beta,lower bound,upper bound). For this example, type “=BETADIST(A1,A2,A3,A4,A5)” into cell A6. Press “Enter” to see the result for the beta distribution, which is 0.592735. [![Excel Beta Distribution](https://www.statisticshowto.com/wp-content/uploads/2013/01/betadist1.jpg)](https://www.statisticshowto.com/how-to-calculate-a-microsoft-excel-beta-distribution/) ## Excel 2010 and later Excel 2010 and later uses the BETA.DIST function. If you’re using earlier versions, use BETADIST instead. Inputs are as follows: 1. 1. - A value where you want to evaluate the function. - Alpha (α) and Beta (β), which determine the distribution’s shape. - The lower and upper bound. - “Cumulative” — a logical value (TRUE/FALSE) that determines the function’s form. TRUE returns the CDF and FALSE returns the PDF. **Example problem:** *Calculate a [cumulative distribution function](https://probabilityhowto.com/cumulative-distribution-function-2/) for a beta distribution at 0.5 with α* = *9, β* = *10,* lower bound = 0 and upper bound = 1. ## [Applications of the Beta Density Function]() The beta distribution is used for a variety of applications for modeling the behavior of [random variables](https://www.statisticshowto.com/random-variable/) limited to intervals of finite length. Uses include: 1. 1. - Bayesian inference, as the conjugate prior probability distribution for the Bernoulli, binomial, geometric and negative binomial distributions. - The Rule of Succession (such as Pierre-Simon Laplace’s treatment of the sunrise problem), - Task duration modeling. - Project/planning control systems like PERT and CPM. - In project management for the “three-point technique,” (also called the beta distribution technique), which recognizes uncertainty in estimated project time \[1\]. The three-point estimation technique is used to account for uncertainty in project time estimates. By applying this method, project managers can leverage powerful quantitative tools to identify tasks with the highest risk. It also facilitates effective management of overall project completion time. References 1. 1. 3-Points Estimating <https://www.projectmanagement.com/contentPages/wiki.cfm?ID=368763&thisPageURL=/wikis/368763/3-Points-Estimating#_=_>

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