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{"found_url":"https:\/\/stat.ethz.ch\/R-manual\/R-devel\/library\/stats\/html\/Beta.html","crawl_time":1775805783,"first_indexed_time":1492697179,"http_code":200,"src_unparsed":"ch,ethz!stat,\/R-manual\/R-devel\/library\/stats\/html\/Beta.html s443","src_root_hash":"14177139736574070983","history_drop_reason":null,"meta_title":"R: The Beta Distribution","meta_descriptions":[],"attrs_boilerpipe_text":"Beta {stats}\nR Documentation\nDescription\nDensity, distribution function, quantile function and random\ngeneration for the Beta distribution with parameters\nshape1\nand\nshape2\n(and optional non-centrality parameter\nncp\n).\nUsage\ndbeta(x, shape1, shape2, ncp = 0, log = FALSE)\npbeta(q, shape1, shape2, ncp = 0, lower.tail = TRUE, log.p = FALSE)\nqbeta(p, shape1, shape2, ncp = 0, lower.tail = TRUE, log.p = FALSE)\nrbeta(n, shape1, shape2, ncp = 0)\nArguments\nx\n,\nq\nvector of quantiles.\np\nvector of probabilities.\nn\nnumber of observations. If\nlength(n) > 1\n, the length\nis taken to be the number required.\nshape1\n,\nshape2\nnon-negative parameters of the Beta distribution.\nncp\nnon-centrality parameter.\nlog\n,\nlog.p\nlogical; if\nTRUE\n, probabilities\/densities\nare given as logarithms.\nlower.tail\nlogical; if\nTRUE\n(default), probabilities are\nP[X \\le x]\n, otherwise,\nP[X > x]\n.\nDetails\nThe Beta distribution with parameters\nshape1\n= a\nand\nshape2\n= b\nhas density\nf(x)=\\frac{\\Gamma(a+b)}{\\Gamma(a)\\Gamma(b)}{x}^{a-1} {(1-x)}^{b-1}%\nfor\na > 0\n,\nb > 0\nand\n0 \\le x \\le 1\nwhere the boundary values at\nx=0\nor\nx=1\nare defined as\nby continuity (as limits).\nThe mean is\na\/(a+b)\nand the variance is\nab\/((a+b)^2 (a+b+1))\n.\nIf\na,b > 1\n, (or one of them\n=1\n), the mode is\n(a-1)\/(a+b-2)\n.\nThese and all other distributional properties can be defined as\nlimits (leading to point masses at 0, 1\/2, or 1) when\na\nor\nb\nare zero or infinite, and the corresponding\n[dpqr]beta()\nfunctions are defined correspondingly.\npbeta\nis closely related to the incomplete beta function.  As\ndefined by ⁠\nAbramowitz and Stegun (1972, section 6.6.1)\nB_x(a,b) = \\int_0^x t^{a-1} (1-t)^{b-1} dt,\nand 6.6.2\nI_x(a,b) = B_x(a,b) \/ B(a,b)\nwhere\nB(a,b) = B_1(a,b)\nis the Beta function (\nbeta\n).\nI_x(a,b)\nis\npbeta(x, a, b)\n.\nThe noncentral Beta distribution (with\nncp\n= \\lambda\n)\nis defined (Johnson\net al.\n, 1995, pp. 502) as the distribution of\nX\/(X+Y)\nwhere\nX \\sim \\chi^2_{2a}(\\lambda)\nand\nY \\sim \\chi^2_{2b}\n.\nThere,\n\\chi^2_n(\\lambda)\nis the noncentral\nchi-squared distribution with\nn\ndegrees of freedom and\nnon-centrality parameter\n\\lambda\n, see\nChisquare\n.\nValue\ndbeta\ngives the density,\npbeta\nis the cumulative distribution function, and\nqbeta\nis the quantile function of the Beta distribution.\nrbeta\ngenerates random deviates.\nInvalid arguments will result in return value\nNaN\n, with a warning.\nThe length of the result is determined by\nn\nfor\nrbeta\n, and is the maximum of the lengths of the\nnumerical arguments for the other functions.\nThe numerical arguments other than\nn\nare recycled to the\nlength of the result.  Only the first elements of the logical\narguments are used.\nNote\nSupplying\nncp = 0\nuses the algorithm for the non-central\ndistribution, which is not the same algorithm as when\nncp\nis\nomitted.  This is to give consistent behaviour in extreme cases with\nvalues of\nncp\nvery near zero.\nSource\nThe central\ndbeta\nis based on a binomial probability, using code\ncontributed by Catherine Loader (see\ndbinom\n) if either\nshape parameter is larger than one, otherwise directly from the definition.\nThe non-central case is based on the derivation as a Poisson\nmixture of betas (Johnson\net al.\n, 1995, pp. 502–3).\nThe central\npbeta\nfor the default (\nlog_p = FALSE\n)\nuses a C translation based on (\nDidonato and Morris 1992\n).\n(See also ⁠\nBrown and Levy (1994)\n.)\nWe have slightly tweaked the original “TOMS 708” algorithm, and\nenhanced for\nlog.p = TRUE\n.  For that (log-scale) case,\nunderflow to\n-Inf\n(i.e.,\nP = 0\n) or\n0\n, (i.e.,\nP = 1\n) still happens because the original algorithm was designed\nwithout log-scale considerations.  Underflow to\n-Inf\nnow\ntypically signals a\nwarning\n.\nThe non-central\npbeta\nuses a C translation of\n⁠\nLenth (1987)\nincorporating ⁠\nFrick (1990)\nand\n⁠\nLam (1995)\n.\nThis computes the lower tail only, so the upper tail suffers from\ncancellation and a warning will be given when this is likely to be\nsignificant.\nThe central case of\nqbeta\nis based on a C translation of\n⁠\nCran, Martin, and Thomas (1977)\nand subsequent remarks (AS83 and\ncorrection).\nEnhancements, notably for starting values and switching to a log-scale\nNewton search, by R Core.\nThe central case of\nrbeta\nis based on a C translation of\n⁠\nCheng (1978)\n.\nReferences\n⁠\nAbramowitz M, Stegun IA (1972).\nHandbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables\n.\nDover, New York.\nChapter 6: Gamma and\n    Related Functions.\n⁠\nBecker RA, Chambers JM, Wilks AR (1988).\nThe New S Language\n.\nChapman and Hall\/CRC, London.\n⁠\nBrown BW, Levy LB (1994).\n“Certification of Algorithm 708: Significant-digit Computation of the Incomplete Beta.”\nACM Transactions on Mathematical Software\n,\n20\n(3), 393–397.\ndoi:10.1145\/192115.192155\n.\n⁠\nCheng RCH (1978).\n“Generating Beta Variates with Nonintegral Shape Parameters.”\nCommunications of the ACM\n,\n21\n(4), 317–322.\ndoi:10.1145\/359460.359482\n.\n⁠\nCran GW, Martin KJ, Thomas GE (1977).\n“Remark AS R19 and Algorithm AS 109: A Remark on Algorithms AS 63: The Incomplete Beta Integral AS 64: Inverse of the Incomplete Beta Function Ratio.”\nApplied Statistics\n,\n26\n(1), 111.\ndoi:10.2307\/2346887\n.\n⁠\nDidonato AR, Morris AH (1992).\n“Algorithm 708: Significant Digit Computation of the Incomplete Beta Function Ratios.”\nACM Transactions on Mathematical Software\n,\n18\n(3), 360–373.\ndoi:10.1145\/131766.131776\n.\n⁠\nFrick H (1990).\n“Algorithm AS R84: A Remark on Algorithm AS 226: Computing Non-Central Beta Probabilities.”\nApplied Statistics\n,\n39\n(2), 311.\ndoi:10.2307\/2347780\n.\n⁠\nJohnson NL, Kotz S, Balakrishnan N (1995).\nContinuous Univariate Distributions\n, volume 2.\nWiley, New York.\nISBN 978-0-471-58494-0.\nEspecially\n    chapter 25.\n⁠\nLam ML (1995).\n“Remark AS R95: A Remark on Algorithm AS 226: Computing Non-Central Beta Probabilities.”\nApplied Statistics\n,\n44\n(4), 551.\ndoi:10.2307\/2986147\n.\n⁠\nLenth RV (1987).\n“Algorithm AS 226: Computing Noncentral Beta Probabilities.”\nApplied Statistics\n,\n36\n(2), 241.\ndoi:10.2307\/2347558\n.\nSee Also\nDistributions\nfor other standard distributions.\nbeta\nfor the Beta function.\nExamples\nx <- seq(0, 1, length.out = 21)\ndbeta(x, 1, 1)\npbeta(x, 1, 1)\n\n## Visualization, including limit cases:\npl.beta <- function(a,b, asp = if(isLim) 1, ylim = if(isLim) c(0,1.1)) {\n  if(isLim <- a == 0 || b == 0 || a == Inf || b == Inf) {\n    eps <- 1e-10\n    x <- c(0, eps, (1:7)\/16, 1\/2+c(-eps,0,eps), (9:15)\/16, 1-eps, 1)\n  } else {\n    x <- seq(0, 1, length.out = 1025)\n  }\n  fx <- cbind(dbeta(x, a,b), pbeta(x, a,b), qbeta(x, a,b))\n  f <- fx; f[fx == Inf] <- 1e100\n  matplot(x, f, ylab=\"\", type=\"l\", ylim=ylim, asp=asp,\n          main = sprintf(\"[dpq]beta(x, a=%g, b=%g)\", a,b))\n  abline(0,1,     col=\"gray\", lty=3)\n  abline(h = 0:1, col=\"gray\", lty=3)\n  legend(\"top\", paste0(c(\"d\",\"p\",\"q\"), \"beta(x, a,b)\"),\n         col=1:3, lty=1:3, bty = \"n\")\n  invisible(cbind(x, fx))\n}\npl.beta(3,1)\n\npl.beta(2, 4)\npl.beta(3, 7)\npl.beta(3, 7, asp=1)\n\npl.beta(0, 0)   ## point masses at  {0, 1}\n\npl.beta(0, 2)   ## point mass at 0 ; the same as\npl.beta(1, Inf)\n\npl.beta(Inf, 2) ## point mass at 1 ; the same as\npl.beta(3, 0)\n\npl.beta(Inf, Inf)# point mass at 1\/2\n[Package\nstats\nversion 4.6.0\nIndex\n]","attrs_markdown":"|   |   |\n|---|---|\n| Beta {stats} | R Documentation |\n## The Beta Distribution\n### Description\nDensity, distribution function, quantile function and random generation for the Beta distribution with parameters `shape1` and `shape2` (and optional non-centrality parameter `ncp`).\n\n### Usage\n```\ndbeta(x, shape1, shape2, ncp = 0, log = FALSE)\npbeta(q, shape1, shape2, ncp = 0, lower.tail = TRUE, log.p = FALSE)\nqbeta(p, shape1, shape2, ncp = 0, lower.tail = TRUE, log.p = FALSE)\nrbeta(n, shape1, shape2, ncp = 0)\n```\n### Arguments\n|   |   |\n|---|---|\n| `x`, `q` | vector of quantiles. |\n| `p` | vector of probabilities. |\n| `n` | number of observations. If `length(n) > 1`, the length is taken to be the number required. |\n| `shape1`, `shape2` | non-negative parameters of the Beta distribution. |\n| `ncp` | non-centrality parameter. |\n| `log`, `log.p` | logical; if `TRUE`, probabilities\/densities are given as logarithms. |\n| `lower.tail` | logical; if `TRUE` (default), probabilities are `P[X \\le x]`, otherwise, `P[X > x]`. |\n### Details\nThe Beta distribution with parameters `shape1` `= a` and `shape2` `= b` has density\n\n`f(x)=\\frac{\\Gamma(a+b)}{\\Gamma(a)\\Gamma(b)}{x}^{a-1} {(1-x)}^{b-1}%`\n\nfor `a > 0`, `b > 0` and `0 \\le x \\le 1` where the boundary values at `x=0` or `x=1` are defined as by continuity (as limits).   \n The mean is `a\/(a+b)` and the variance is `ab\/((a+b)^2 (a+b+1))`. If `a,b > 1`, (or one of them `=1`), the mode is `(a-1)\/(a+b-2)`. These and all other distributional properties can be defined as limits (leading to point masses at 0, 1\/2, or 1) when `a` or `b` are zero or infinite, and the corresponding `[dpqr]beta()` functions are defined correspondingly.\n\n`pbeta` is closely related to the incomplete beta function. As defined by ⁠[Abramowitz and Stegun (1972, section 6.6.1)](https:\/\/stat.ethz.ch\/R-manual\/R-devel\/library\/stats\/html\/Beta.html#reference+Beta.Rd+R+3AAbramowitz+2BStegun+3A1972)\n\n`B_x(a,b) = \\int_0^x t^{a-1} (1-t)^{b-1} dt,`\n\nand 6.6.2 `I_x(a,b) = B_x(a,b) \/ B(a,b)` where `B(a,b) = B_1(a,b)` is the Beta function (`beta`).\n\n`I_x(a,b)` is `pbeta(x, a, b)`.\n\nThe noncentral Beta distribution (with `ncp` ` = \\lambda`) is defined (Johnson et al., 1995, pp. 502) as the distribution of `X\/(X+Y)` where `X \\sim \\chi^2_{2a}(\\lambda)` and `Y \\sim \\chi^2_{2b}`. There, `\\chi^2_n(\\lambda)` is the noncentral chi-squared distribution with `n` degrees of freedom and non-centrality parameter `\\lambda`, see [Chisquare](https:\/\/stat.ethz.ch\/R-manual\/R-devel\/library\/stats\/help\/Chisquare.html).\n\n### Value\n`dbeta` gives the density, `pbeta` is the cumulative distribution function, and `qbeta` is the quantile function of the Beta distribution. `rbeta` generates random deviates.\n\nInvalid arguments will result in return value `NaN`, with a warning.\n\nThe length of the result is determined by `n` for `rbeta`, and is the maximum of the lengths of the numerical arguments for the other functions.\n\nThe numerical arguments other than `n` are recycled to the length of the result. Only the first elements of the logical arguments are used.\n\n### Note\nSupplying `ncp = 0` uses the algorithm for the non-central distribution, which is not the same algorithm as when `ncp` is omitted. This is to give consistent behaviour in extreme cases with values of `ncp` very near zero.\n\n### Source\n- The central `dbeta` is based on a binomial probability, using code contributed by Catherine Loader (see `dbinom`) if either shape parameter is larger than one, otherwise directly from the definition. The non-central case is based on the derivation as a Poisson mixture of betas (Johnson et al., 1995, pp. 502–3).\n- The central `pbeta` for the default (`log_p = FALSE`) uses a C translation based on ([Didonato and Morris 1992](https:\/\/stat.ethz.ch\/R-manual\/R-devel\/library\/stats\/html\/Beta.html#reference+Beta.Rd+R+3ADidonato+2BMorris+3A1992)). (See also ⁠[Brown and Levy (1994)](https:\/\/stat.ethz.ch\/R-manual\/R-devel\/library\/stats\/html\/Beta.html#reference+Beta.Rd+R+3ABrown+2BLevy+3A1994).)\n  We have slightly tweaked the original “TOMS 708” algorithm, and enhanced for `log.p = TRUE`. For that (log-scale) case, underflow to `-Inf` (i.e., `P = 0`) or `0`, (i.e., `P = 1`) still happens because the original algorithm was designed without log-scale considerations. Underflow to `-Inf` now typically signals a `warning`.\n- The non-central `pbeta` uses a C translation of ⁠[Lenth (1987)](https:\/\/stat.ethz.ch\/R-manual\/R-devel\/library\/stats\/html\/Beta.html#reference+Beta.Rd+R+3ALenth+3A1987) incorporating ⁠[Frick (1990)](https:\/\/stat.ethz.ch\/R-manual\/R-devel\/library\/stats\/html\/Beta.html#reference+Beta.Rd+R+3AFrick+3A1990) and ⁠[Lam (1995)](https:\/\/stat.ethz.ch\/R-manual\/R-devel\/library\/stats\/html\/Beta.html#reference+Beta.Rd+R+3ALam+3A1995).\n  This computes the lower tail only, so the upper tail suffers from cancellation and a warning will be given when this is likely to be significant.\n- The central case of `qbeta` is based on a C translation of ⁠[Cran, Martin, and Thomas (1977)](https:\/\/stat.ethz.ch\/R-manual\/R-devel\/library\/stats\/html\/Beta.html#reference+Beta.Rd+R+3ACran+2BMartin+2BThomas+3A1977) and subsequent remarks (AS83 and correction).\n  Enhancements, notably for starting values and switching to a log-scale Newton search, by R Core.\n- The central case of `rbeta` is based on a C translation of ⁠[Cheng (1978)](https:\/\/stat.ethz.ch\/R-manual\/R-devel\/library\/stats\/html\/Beta.html#reference+Beta.Rd+R+3ACheng+3A1978).\n### References\n⁠Abramowitz M, Stegun IA (1972). *Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables*. Dover, New York.  \n Chapter 6: Gamma and Related Functions.\n\n⁠Becker RA, Chambers JM, Wilks AR (1988). *The New S Language*. Chapman and Hall\/CRC, London.\n\n⁠Brown BW, Levy LB (1994). “Certification of Algorithm 708: Significant-digit Computation of the Incomplete Beta.” *ACM Transactions on Mathematical Software*, **20**(3), 393–397. [doi:10.1145\/192115.192155](https:\/\/doi.org\/10.1145\/192115.192155).\n\n⁠Cheng RCH (1978). “Generating Beta Variates with Nonintegral Shape Parameters.” *Communications of the ACM*, **21**(4), 317–322. [doi:10.1145\/359460.359482](https:\/\/doi.org\/10.1145\/359460.359482).\n\n⁠Cran GW, Martin KJ, Thomas GE (1977). “Remark AS R19 and Algorithm AS 109: A Remark on Algorithms AS 63: The Incomplete Beta Integral AS 64: Inverse of the Incomplete Beta Function Ratio.” *Applied Statistics*, **26**(1), 111. [doi:10.2307\/2346887](https:\/\/doi.org\/10.2307\/2346887).\n\n⁠Didonato AR, Morris AH (1992). “Algorithm 708: Significant Digit Computation of the Incomplete Beta Function Ratios.” *ACM Transactions on Mathematical Software*, **18**(3), 360–373. [doi:10.1145\/131766.131776](https:\/\/doi.org\/10.1145\/131766.131776).\n\n⁠Frick H (1990). “Algorithm AS R84: A Remark on Algorithm AS 226: Computing Non-Central Beta Probabilities.” *Applied Statistics*, **39**(2), 311. [doi:10.2307\/2347780](https:\/\/doi.org\/10.2307\/2347780).\n\n⁠Johnson NL, Kotz S, Balakrishnan N (1995). *Continuous Univariate Distributions*, volume 2. Wiley, New York. ISBN 978-0-471-58494-0.  \n Especially chapter 25.\n\n⁠Lam ML (1995). “Remark AS R95: A Remark on Algorithm AS 226: Computing Non-Central Beta Probabilities.” *Applied Statistics*, **44**(4), 551. [doi:10.2307\/2986147](https:\/\/doi.org\/10.2307\/2986147).\n\n⁠Lenth RV (1987). “Algorithm AS 226: Computing Noncentral Beta Probabilities.” *Applied Statistics*, **36**(2), 241. [doi:10.2307\/2347558](https:\/\/doi.org\/10.2307\/2347558).\n\n### See Also\n[Distributions](https:\/\/stat.ethz.ch\/R-manual\/R-devel\/library\/stats\/help\/Distributions.html) for other standard distributions.\n\n`beta` for the Beta function.\n\n### Examples\n```\nx <- seq(0, 1, length.out = 21)\ndbeta(x, 1, 1)\npbeta(x, 1, 1)\n\n## Visualization, including limit cases:\npl.beta <- function(a,b, asp = if(isLim) 1, ylim = if(isLim) c(0,1.1)) {\n  if(isLim <- a == 0 || b == 0 || a == Inf || b == Inf) {\n    eps <- 1e-10\n    x <- c(0, eps, (1:7)\/16, 1\/2+c(-eps,0,eps), (9:15)\/16, 1-eps, 1)\n  } else {\n    x <- seq(0, 1, length.out = 1025)\n  }\n  fx <- cbind(dbeta(x, a,b), pbeta(x, a,b), qbeta(x, a,b))\n  f <- fx; f[fx == Inf] <- 1e100\n  matplot(x, f, ylab=\"\", type=\"l\", ylim=ylim, asp=asp,\n          main = sprintf(\"[dpq]beta(x, a=%g, b=%g)\", a,b))\n  abline(0,1,     col=\"gray\", lty=3)\n  abline(h = 0:1, col=\"gray\", lty=3)\n  legend(\"top\", paste0(c(\"d\",\"p\",\"q\"), \"beta(x, a,b)\"),\n         col=1:3, lty=1:3, bty = \"n\")\n  invisible(cbind(x, fx))\n}\npl.beta(3,1)\n\npl.beta(2, 4)\npl.beta(3, 7)\npl.beta(3, 7, asp=1)\n\npl.beta(0, 0)   ## point masses at  {0, 1}\n\npl.beta(0, 2)   ## point mass at 0 ; the same as\npl.beta(1, Inf)\n\npl.beta(Inf, 2) ## point mass at 1 ; the same as\npl.beta(3, 0)\n\npl.beta(Inf, Inf)# point mass at 1\/2\n```\n***\n\\[Package *stats* version 4.6.0 [Index](https:\/\/stat.ethz.ch\/R-manual\/R-devel\/library\/stats\/html\/00Index.html)\\]","attrs_readable_markdown":null,"meta_canonical":null}

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Boilerpipe Text	Beta {stats} R Documentation Description Density, distribution function, quantile function and random generation for the Beta distribution with parameters shape1 and shape2 (and optional non-centrality parameter ncp ). Usage 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) Arguments x , q vector of quantiles. p vector of probabilities. n number of observations. If length(n) > 1 , the length is taken to be the number required. shape1 , shape2 non-negative parameters of the Beta distribution. ncp non-centrality parameter. log , log.p logical; if TRUE , probabilities/densities are given as logarithms. lower.tail logical; if TRUE (default), probabilities are P[X \le x] , otherwise, P[X > x] . Details The Beta distribution with parameters shape1 = a and shape2 = b has density f(x)=\frac{\Gamma(a+b)}{\Gamma(a)\Gamma(b)}{x}^{a-1} {(1-x)}^{b-1}% for a > 0 , b > 0 and 0 \le x \le 1 where the boundary values at x=0 or x=1 are defined as by continuity (as limits). The mean is a/(a+b) and the variance is ab/((a+b)^2 (a+b+1)) . If a,b > 1 , (or one of them =1 ), the mode is (a-1)/(a+b-2) . These and all other distributional properties can be defined as limits (leading to point masses at 0, 1/2, or 1) when a or b are zero or infinite, and the corresponding [dpqr]beta() functions are defined correspondingly. pbeta is closely related to the incomplete beta function. As defined by ⁠ Abramowitz and Stegun (1972, section 6.6.1) B_x(a,b) = \int_0^x t^{a-1} (1-t)^{b-1} dt, and 6.6.2 I_x(a,b) = B_x(a,b) / B(a,b) where B(a,b) = B_1(a,b) is the Beta function ( beta ). I_x(a,b) is pbeta(x, a, b) . The noncentral Beta distribution (with ncp = \lambda ) is defined (Johnson et al. , 1995, pp. 502) as the distribution of X/(X+Y) where X \sim \chi^2_{2a}(\lambda) and Y \sim \chi^2_{2b} . There, \chi^2_n(\lambda) is the noncentral chi-squared distribution with n degrees of freedom and non-centrality parameter \lambda , see Chisquare . Value dbeta gives the density, pbeta is the cumulative distribution function, and qbeta is the quantile function of the Beta distribution. rbeta generates random deviates. Invalid arguments will result in return value NaN , with a warning. The length of the result is determined by n for rbeta , and is the maximum of the lengths of the numerical arguments for the other functions. The numerical arguments other than n are recycled to the length of the result. Only the first elements of the logical arguments are used. Note Supplying ncp = 0 uses the algorithm for the non-central distribution, which is not the same algorithm as when ncp is omitted. This is to give consistent behaviour in extreme cases with values of ncp very near zero. Source The central dbeta is based on a binomial probability, using code contributed by Catherine Loader (see dbinom ) if either shape parameter is larger than one, otherwise directly from the definition. The non-central case is based on the derivation as a Poisson mixture of betas (Johnson et al. , 1995, pp. 502–3). The central pbeta for the default ( log_p = FALSE ) uses a C translation based on ( Didonato and Morris 1992 ). (See also ⁠ Brown and Levy (1994) .) We have slightly tweaked the original “TOMS 708” algorithm, and enhanced for log.p = TRUE . For that (log-scale) case, underflow to -Inf (i.e., P = 0 ) or 0 , (i.e., P = 1 ) still happens because the original algorithm was designed without log-scale considerations. Underflow to -Inf now typically signals a warning . The non-central pbeta uses a C translation of ⁠ Lenth (1987) incorporating ⁠ Frick (1990) and ⁠ Lam (1995) . This computes the lower tail only, so the upper tail suffers from cancellation and a warning will be given when this is likely to be significant. The central case of qbeta is based on a C translation of ⁠ Cran, Martin, and Thomas (1977) and subsequent remarks (AS83 and correction). Enhancements, notably for starting values and switching to a log-scale Newton search, by R Core. The central case of rbeta is based on a C translation of ⁠ Cheng (1978) . References ⁠ Abramowitz M, Stegun IA (1972). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables . Dover, New York. Chapter 6: Gamma and Related Functions. ⁠ Becker RA, Chambers JM, Wilks AR (1988). The New S Language . Chapman and Hall/CRC, London. ⁠ Brown BW, Levy LB (1994). “Certification of Algorithm 708: Significant-digit Computation of the Incomplete Beta.” ACM Transactions on Mathematical Software , 20 (3), 393–397. doi:10.1145/192115.192155 . ⁠ Cheng RCH (1978). “Generating Beta Variates with Nonintegral Shape Parameters.” Communications of the ACM , 21 (4), 317–322. doi:10.1145/359460.359482 . ⁠ Cran GW, Martin KJ, Thomas GE (1977). “Remark AS R19 and Algorithm AS 109: A Remark on Algorithms AS 63: The Incomplete Beta Integral AS 64: Inverse of the Incomplete Beta Function Ratio.” Applied Statistics , 26 (1), 111. doi:10.2307/2346887 . ⁠ Didonato AR, Morris AH (1992). “Algorithm 708: Significant Digit Computation of the Incomplete Beta Function Ratios.” ACM Transactions on Mathematical Software , 18 (3), 360–373. doi:10.1145/131766.131776 . ⁠ Frick H (1990). “Algorithm AS R84: A Remark on Algorithm AS 226: Computing Non-Central Beta Probabilities.” Applied Statistics , 39 (2), 311. doi:10.2307/2347780 . ⁠ Johnson NL, Kotz S, Balakrishnan N (1995). Continuous Univariate Distributions , volume 2. Wiley, New York. ISBN 978-0-471-58494-0. Especially chapter 25. ⁠ Lam ML (1995). “Remark AS R95: A Remark on Algorithm AS 226: Computing Non-Central Beta Probabilities.” Applied Statistics , 44 (4), 551. doi:10.2307/2986147 . ⁠ Lenth RV (1987). “Algorithm AS 226: Computing Noncentral Beta Probabilities.” Applied Statistics , 36 (2), 241. doi:10.2307/2347558 . See Also Distributions for other standard distributions. beta for the Beta function. Examples x <- seq(0, 1, length.out = 21) dbeta(x, 1, 1) pbeta(x, 1, 1) ## Visualization, including limit cases: pl.beta <- function(a,b, asp = if(isLim) 1, ylim = if(isLim) c(0,1.1)) { if(isLim <- a == 0 \|\| b == 0 \|\| a == Inf \|\| b == Inf) { eps <- 1e-10 x <- c(0, eps, (1:7)/16, 1/2+c(-eps,0,eps), (9:15)/16, 1-eps, 1) } else { x <- seq(0, 1, length.out = 1025) } fx <- cbind(dbeta(x, a,b), pbeta(x, a,b), qbeta(x, a,b)) f <- fx; f[fx == Inf] <- 1e100 matplot(x, f, ylab="", type="l", ylim=ylim, asp=asp, main = sprintf("[dpq]beta(x, a=%g, b=%g)", a,b)) abline(0,1, col="gray", lty=3) abline(h = 0:1, col="gray", lty=3) legend("top", paste0(c("d","p","q"), "beta(x, a,b)"), col=1:3, lty=1:3, bty = "n") invisible(cbind(x, fx)) } pl.beta(3,1) pl.beta(2, 4) pl.beta(3, 7) pl.beta(3, 7, asp=1) pl.beta(0, 0) ## point masses at {0, 1} pl.beta(0, 2) ## point mass at 0 ; the same as pl.beta(1, Inf) pl.beta(Inf, 2) ## point mass at 1 ; the same as pl.beta(3, 0) pl.beta(Inf, Inf)# point mass at 1/2 [Package stats version 4.6.0 Index ]
Markdown	\| \| \| \|---\|---\| \| Beta {stats} \| R Documentation \| ## The Beta Distribution ### Description Density, distribution function, quantile function and random generation for the Beta distribution with parameters `shape1` and `shape2` (and optional non-centrality parameter `ncp`). ### Usage ``` 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) ``` ### Arguments \| \| \| \|---\|---\| \| `x`, `q` \| vector of quantiles. \| \| `p` \| vector of probabilities. \| \| `n` \| number of observations. If `length(n) > 1`, the length is taken to be the number required. \| \| `shape1`, `shape2` \| non-negative parameters of the Beta distribution. \| \| `ncp` \| non-centrality parameter. \| \| `log`, `log.p` \| logical; if `TRUE`, probabilities/densities are given as logarithms. \| \| `lower.tail` \| logical; if `TRUE` (default), probabilities are `P[X \le x]`, otherwise, `P[X > x]`. \| ### Details The Beta distribution with parameters `shape1` `= a` and `shape2` `= b` has density `f(x)=\frac{\Gamma(a+b)}{\Gamma(a)\Gamma(b)}{x}^{a-1} {(1-x)}^{b-1}%` for `a > 0`, `b > 0` and `0 \le x \le 1` where the boundary values at `x=0` or `x=1` are defined as by continuity (as limits). The mean is `a/(a+b)` and the variance is `ab/((a+b)^2 (a+b+1))`. If `a,b > 1`, (or one of them `=1`), the mode is `(a-1)/(a+b-2)`. These and all other distributional properties can be defined as limits (leading to point masses at 0, 1/2, or 1) when `a` or `b` are zero or infinite, and the corresponding `[dpqr]beta()` functions are defined correspondingly. `pbeta` is closely related to the incomplete beta function. As defined by ⁠[Abramowitz and Stegun (1972, section 6.6.1)](https://stat.ethz.ch/R-manual/R-devel/library/stats/html/Beta.html#reference+Beta.Rd+R+3AAbramowitz+2BStegun+3A1972) `B_x(a,b) = \int_0^x t^{a-1} (1-t)^{b-1} dt,` and 6.6.2 `I_x(a,b) = B_x(a,b) / B(a,b)` where `B(a,b) = B_1(a,b)` is the Beta function (`beta`). `I_x(a,b)` is `pbeta(x, a, b)`. The noncentral Beta distribution (with `ncp` ` = \lambda`) is defined (Johnson et al., 1995, pp. 502) as the distribution of `X/(X+Y)` where `X \sim \chi^2_{2a}(\lambda)` and `Y \sim \chi^2_{2b}`. There, `\chi^2_n(\lambda)` is the noncentral chi-squared distribution with `n` degrees of freedom and non-centrality parameter `\lambda`, see [Chisquare](https://stat.ethz.ch/R-manual/R-devel/library/stats/help/Chisquare.html). ### Value `dbeta` gives the density, `pbeta` is the cumulative distribution function, and `qbeta` is the quantile function of the Beta distribution. `rbeta` generates random deviates. Invalid arguments will result in return value `NaN`, with a warning. The length of the result is determined by `n` for `rbeta`, and is the maximum of the lengths of the numerical arguments for the other functions. The numerical arguments other than `n` are recycled to the length of the result. Only the first elements of the logical arguments are used. ### Note Supplying `ncp = 0` uses the algorithm for the non-central distribution, which is not the same algorithm as when `ncp` is omitted. This is to give consistent behaviour in extreme cases with values of `ncp` very near zero. ### Source - The central `dbeta` is based on a binomial probability, using code contributed by Catherine Loader (see `dbinom`) if either shape parameter is larger than one, otherwise directly from the definition. The non-central case is based on the derivation as a Poisson mixture of betas (Johnson et al., 1995, pp. 502–3). - The central `pbeta` for the default (`log_p = FALSE`) uses a C translation based on ([Didonato and Morris 1992](https://stat.ethz.ch/R-manual/R-devel/library/stats/html/Beta.html#reference+Beta.Rd+R+3ADidonato+2BMorris+3A1992)). (See also ⁠[Brown and Levy (1994)](https://stat.ethz.ch/R-manual/R-devel/library/stats/html/Beta.html#reference+Beta.Rd+R+3ABrown+2BLevy+3A1994).) We have slightly tweaked the original “TOMS 708” algorithm, and enhanced for `log.p = TRUE`. For that (log-scale) case, underflow to `-Inf` (i.e., `P = 0`) or `0`, (i.e., `P = 1`) still happens because the original algorithm was designed without log-scale considerations. Underflow to `-Inf` now typically signals a `warning`. - The non-central `pbeta` uses a C translation of ⁠[Lenth (1987)](https://stat.ethz.ch/R-manual/R-devel/library/stats/html/Beta.html#reference+Beta.Rd+R+3ALenth+3A1987) incorporating ⁠[Frick (1990)](https://stat.ethz.ch/R-manual/R-devel/library/stats/html/Beta.html#reference+Beta.Rd+R+3AFrick+3A1990) and ⁠[Lam (1995)](https://stat.ethz.ch/R-manual/R-devel/library/stats/html/Beta.html#reference+Beta.Rd+R+3ALam+3A1995). This computes the lower tail only, so the upper tail suffers from cancellation and a warning will be given when this is likely to be significant. - The central case of `qbeta` is based on a C translation of ⁠[Cran, Martin, and Thomas (1977)](https://stat.ethz.ch/R-manual/R-devel/library/stats/html/Beta.html#reference+Beta.Rd+R+3ACran+2BMartin+2BThomas+3A1977) and subsequent remarks (AS83 and correction). Enhancements, notably for starting values and switching to a log-scale Newton search, by R Core. - The central case of `rbeta` is based on a C translation of ⁠[Cheng (1978)](https://stat.ethz.ch/R-manual/R-devel/library/stats/html/Beta.html#reference+Beta.Rd+R+3ACheng+3A1978). ### References ⁠Abramowitz M, Stegun IA (1972). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Dover, New York. Chapter 6: Gamma and Related Functions. ⁠Becker RA, Chambers JM, Wilks AR (1988). The New S Language. Chapman and Hall/CRC, London. ⁠Brown BW, Levy LB (1994). “Certification of Algorithm 708: Significant-digit Computation of the Incomplete Beta.” ACM Transactions on Mathematical Software, 20(3), 393–397. [doi:10.1145/192115.192155](https://doi.org/10.1145/192115.192155). ⁠Cheng RCH (1978). “Generating Beta Variates with Nonintegral Shape Parameters.” Communications of the ACM, 21(4), 317–322. [doi:10.1145/359460.359482](https://doi.org/10.1145/359460.359482). ⁠Cran GW, Martin KJ, Thomas GE (1977). “Remark AS R19 and Algorithm AS 109: A Remark on Algorithms AS 63: The Incomplete Beta Integral AS 64: Inverse of the Incomplete Beta Function Ratio.” Applied Statistics, 26(1), 111. [doi:10.2307/2346887](https://doi.org/10.2307/2346887). ⁠Didonato AR, Morris AH (1992). “Algorithm 708: Significant Digit Computation of the Incomplete Beta Function Ratios.” ACM Transactions on Mathematical Software, 18(3), 360–373. [doi:10.1145/131766.131776](https://doi.org/10.1145/131766.131776). ⁠Frick H (1990). “Algorithm AS R84: A Remark on Algorithm AS 226: Computing Non-Central Beta Probabilities.” Applied Statistics, 39(2), 311. [doi:10.2307/2347780](https://doi.org/10.2307/2347780). ⁠Johnson NL, Kotz S, Balakrishnan N (1995). Continuous Univariate Distributions, volume 2. Wiley, New York. ISBN 978-0-471-58494-0. Especially chapter 25. ⁠Lam ML (1995). “Remark AS R95: A Remark on Algorithm AS 226: Computing Non-Central Beta Probabilities.” Applied Statistics, 44(4), 551. [doi:10.2307/2986147](https://doi.org/10.2307/2986147). ⁠Lenth RV (1987). “Algorithm AS 226: Computing Noncentral Beta Probabilities.” Applied Statistics, 36(2), 241. [doi:10.2307/2347558](https://doi.org/10.2307/2347558). ### See Also [Distributions](https://stat.ethz.ch/R-manual/R-devel/library/stats/help/Distributions.html) for other standard distributions. `beta` for the Beta function. ### Examples ``` x <- seq(0, 1, length.out = 21) dbeta(x, 1, 1) pbeta(x, 1, 1) ## Visualization, including limit cases: pl.beta <- function(a,b, asp = if(isLim) 1, ylim = if(isLim) c(0,1.1)) { if(isLim <- a == 0 \|\| b == 0 \|\| a == Inf \|\| b == Inf) { eps <- 1e-10 x <- c(0, eps, (1:7)/16, 1/2+c(-eps,0,eps), (9:15)/16, 1-eps, 1) } else { x <- seq(0, 1, length.out = 1025) } fx <- cbind(dbeta(x, a,b), pbeta(x, a,b), qbeta(x, a,b)) f <- fx; f[fx == Inf] <- 1e100 matplot(x, f, ylab="", type="l", ylim=ylim, asp=asp, main = sprintf("[dpq]beta(x, a=%g, b=%g)", a,b)) abline(0,1, col="gray", lty=3) abline(h = 0:1, col="gray", lty=3) legend("top", paste0(c("d","p","q"), "beta(x, a,b)"), col=1:3, lty=1:3, bty = "n") invisible(cbind(x, fx)) } pl.beta(3,1) pl.beta(2, 4) pl.beta(3, 7) pl.beta(3, 7, asp=1) pl.beta(0, 0) ## point masses at {0, 1} pl.beta(0, 2) ## point mass at 0 ; the same as pl.beta(1, Inf) pl.beta(Inf, 2) ## point mass at 1 ; the same as pl.beta(3, 0) pl.beta(Inf, Inf)# point mass at 1/2 ``` *** \[Package stats version 4.6.0 [Index](https://stat.ethz.ch/R-manual/R-devel/library/stats/html/00Index.html)\]
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