{"history":[],"article":{"id":"ba2afadc-d8e9-47af-b94d-fd1b5bc2a77f","title":"Beta distribution","slug":"beta-distribution","content":":::summary\n:::\n\n# \nIn [probability theory](https://bliptext.com/articles/probability-theory) and [statistics](https://bliptext.com/articles/statistics), the **beta distribution** is a family of continuous [probability distributions](https://bliptext.com/articles/probability-distribution) defined on the interval [0, 1] or (0, 1) in terms of two positive [parameters](https://bliptext.com/articles/statistical-parameter), denoted by alpha (α) and beta (β), that appear as exponents of the variable and its complement to 1, respectively, and control the [shape](https://bliptext.com/articles/shape-parameter) of the distribution.\nThe beta distribution has been applied to model the behavior of [random variables](https://bliptext.com/articles/random-variables) limited to intervals of finite length in a wide variety of disciplines.\nThe beta distribution is a suitable model for the random behavior of percentages and proportions.\nIn [Bayesian inference](https://bliptext.com/articles/bayesian-inference), the beta distribution is the [conjugate prior probability distribution](https://bliptext.com/articles/conjugate-prior-distribution) for the [Bernoulli](https://bliptext.com/articles/bernoulli-distribution), [binomial](https://bliptext.com/articles/binomial-distribution), negative [binomial](https://bliptext.com/articles/binomial-distribution), and [geometric](https://bliptext.com/articles/geometric-distribution) distributions.\nThe formulation of the beta distribution discussed here is also known as the **beta distribution of the first kind**, whereas beta distribution of the second kind is an alternative name for the [beta prime distribution](https://bliptext.com/articles/beta-prime-distribution).\nThe generalization to multiple variables is called a [Dirichlet distribution](https://bliptext.com/articles/dirichlet-distribution).\n\n# Definitions\n\n## Probability density function\nThe [probability density function](https://bliptext.com/articles/probability-density-function) (PDF) of the beta distribution, for or 0 < x < 1, and shape parameters \\alpha, \\beta > 0, is a [power function](https://bliptext.com/articles/power-function) of the variable x and of its [reflection](https://bliptext.com/articles/reflection-formula) (1-x) as follows:\nwhere \\Gamma(z) is the [gamma function](https://bliptext.com/articles/gamma-function).\nThe [beta function](https://bliptext.com/articles/beta-function), \\Beta, is a [normalization constant](https://bliptext.com/articles/normalization-constant) to ensure that the total probability is 1.\nIn the above equations x is a [realization](https://bliptext.com/articles/realization-probability)—an observed value that actually occurred—of a [random variable](https://bliptext.com/articles/random-variable) X.\nSeveral authors, including [N. L. Johnson](https://bliptext.com/articles/norman-lloyd-johnson) and [S. Kotz](https://bliptext.com/articles/samuel-kotz), use the symbols p and q (instead of \\alpha and \\beta) for the shape parameters of the beta distribution, reminiscent of the symbols traditionally used for the parameters of the [Bernoulli distribution](https://bliptext.com/articles/bernoulli-distribution), because the beta distribution approaches the [Bernoulli distribution](https://bliptext.com/articles/bernoulli-distribution) in the limit when both shape parameters \\alpha and \\beta approach the value of zero.\nIn the following, a random variable X beta-distributed with parameters \\alpha and \\beta will be denoted by:\nOther notations for beta-distributed random variables used in the statistical literature are and.\n\n## Cumulative distribution function\nThe [cumulative distribution function](https://bliptext.com/articles/cumulative-distribution-function) is\nwhere is the [incomplete beta function](https://bliptext.com/articles/beta-function) and is the regularized [incomplete beta function](https://bliptext.com/articles/beta-function).\n# [beta distribution | upload.wikimedia.org/wikipedia/commons/d/d3/CDF///for///symmetric///Beta///distribution///vs.///x///and///alpha%3Dbeta///-///J.///Rodal.jpg]\n# [beta distribution | upload.wikimedia.org/wikipedia/commons/9/9f/CDF///for///skewed///Beta///distribution///vs.///x///and///beta%3D///5///alpha///-///J.///Rodal.jpg]\n\n## Alternative parameterizations\n\n### Two parameters\n\n#### Mean and sample size\nThe beta distribution may also be reparameterized in terms of its mean μ (0 < μ < 1) and the sum of the two shape parameters ν = α + β > 0( p. 83).\nDenoting by αPosterior and βPosterior the shape parameters of the posterior beta distribution resulting from applying Bayes theorem to a binomial likelihood function and a prior probability, the interpretation of the addition of both shape parameters to be sample size = ν = α·Posterior + β·Posterior is only correct for the Haldane prior probability Beta(0,0).\nSpecifically, for the Bayes (uniform) prior Beta(1,1) the correct interpretation would be sample size = α·Posterior + β Posterior − 2, or ν = (sample size) + 2.\nFor sample size much larger than 2, the difference between these two priors becomes negligible.\n(See section [Bayesian inference](https://bliptext.com/articles/) for further details.) ν = α + β is referred to as the \"sample size\" of a beta distribution, but one should remember that it is, strictly speaking, the \"sample size\" of a binomial likelihood function only when using a Haldane Beta(0,0) prior in Bayes theorem.\nThis parametrization may be useful in Bayesian parameter estimation.\nFor example, one may administer a test to a number of individuals.\nIf it is assumed that each person's score (0 ≤ θ ≤ 1) is drawn from a population-level beta distribution, then an important statistic is the mean of this population-level distribution.\nThe mean and sample size parameters are related to the shape parameters α and β via\nUnder this [parametrization](https://bliptext.com/articles/statistical-parameter), one may place an [uninformative prior](https://bliptext.com/articles/uninformative-prior) probability over the mean, and a vague prior probability (such as an [exponential](https://bliptext.com/articles/exponential-distribution) or [gamma distribution](https://bliptext.com/articles/gamma-distribution)) over the positive reals for the sample size, if they are independent, and prior data and/or beliefs justify it.\n\n#### Mode and concentration\n[Concave](https://bliptext.com/articles/concave-function) beta distributions, which have, can be parametrized in terms of mode and \"concentration\".\nThe mode,, and concentration, , can be used to define the usual shape parameters as follows:\nFor the mode, 0<\\omega<1, to be well-defined, we need, or equivalently \\kappa>2.\nIf instead we define the concentration as, the condition simplifies to c>0 and the beta density at and can be written as:\nwhere c directly scales the [sufficient statistics](https://bliptext.com/articles/sufficient-statistics), \\log(x) and \\log(1-x).\nNote also that in the limit, c\\to0, the distribution becomes flat.\n\n#### Mean and variance\nSolving the system of (coupled) equations given in the above sections as the equations for the mean and the variance of the beta distribution in terms of the original parameters α and β, one can express the α and β parameters in terms of the mean (μ) and the variance (var):\nThis [parametrization](https://bliptext.com/articles/statistical-parameter) of the beta distribution may lead to a more intuitive understanding than the one based on the original parameters α and β.\nFor example, by expressing the mode, skewness, excess kurtosis and differential entropy in terms of the mean and the variance:\n# [325px | upload.wikimedia.org/wikipedia/commons/6/68/Mode///Beta///Distribution///for///both///alpha///and///beta///greater///than///1///-///J.///Rodal.jpg]\n# [325px | upload.wikimedia.org/wikipedia/commons/c/c2/Mode///Beta///Distribution///for///both///alpha///and///beta///greater///than///1///-///another///view///-///J.///Rodal.jpg]\n# [325px | upload.wikimedia.org/wikipedia/commons/a/a4/Skewness///Beta///Distribution///for///mean///full///range///and///variance///between///0.05///and///0.25///-///Dr.///J.///Rodal.jpg]\n# [325px | upload.wikimedia.org/wikipedia/commons/9/95/Skewness///Beta///Distribution///for///mean///and///variance///both///full///range///-///J.///Rodal.jpg]\n# [325px | upload.wikimedia.org/wikipedia/commons/0/01/Excess///Kurtosis///Beta///Distribution///with///mean///for///full///range///and///variance///from///0.05///to///0.25///-///J.///Rodal.jpg]\n# [325px | upload.wikimedia.org/wikipedia/commons/6/66/Excess///Kurtosis///Beta///Distribution///with///mean///and///variance///for///full///range///-///J.///Rodal.jpg]\n# [325px | upload.wikimedia.org/wikipedia/commons/0/02/Differential///Entropy///Beta///Distribution///with///mean///from///0.2///to///0.8///and///variance///from///0.01///to///0.09///-///J.///Rodal.jpg]\n# [325px | upload.wikimedia.org/wikipedia/commons/8/8f/Differential///Entropy///Beta///Distribution///with///mean///from///0.3///to///0.7///and///variance///from///0///to///0.2///-///J.///Rodal.jpg]\n\n### Four parameters\nA beta distribution with the two shape parameters α and β is supported on the range [0,1] or (0,1).\nIt is possible to alter the location and scale of the distribution by introducing two further parameters representing the minimum, a, and maximum c (c > a), values of the distribution, by a linear transformation substituting the non-dimensional variable x in terms of the new variable y (with support [a,c] or (a,c)) and the parameters a and c:\nThe [probability density function](https://bliptext.com/articles/probability-density-function) of the four parameter beta distribution is equal to the two parameter distribution, scaled by the range (c − a), (so that the total area under the density curve equals a probability of one), and with the \"y\" variable shifted and scaled as follows:\nThat a random variable Y is beta-distributed with four parameters α, β, a, and c will be denoted by:\nSome measures of central location are scaled (by (c − a)) and shifted (by a), as follows:\nNote: the geometric mean and harmonic mean cannot be transformed by a linear transformation in the way that the mean, median and mode can.\nThe shape parameters of Y can be written in term of its mean and variance as\nThe statistical dispersion measures are scaled (they do not need to be shifted because they are already centered on the mean) by the range (c − a), linearly for the mean deviation and nonlinearly for the variance:\nSince the [skewness](https://bliptext.com/articles/skewness) and [excess kurtosis](https://bliptext.com/articles/excess-kurtosis) are non-dimensional quantities (as [moments](https://bliptext.com/articles/moment-mathematics) centered on the mean and normalized by the [standard deviation](https://bliptext.com/articles/standard-deviation)), they are independent of the parameters a and c, and therefore equal to the expressions given above in terms of X (with support [0,1] or (0,1)):\n\n# Properties\n\n## Measures of central tendency\n\n### Mode\nThe [mode](https://bliptext.com/articles/mode-statistics) of a beta distributed [random variable](https://bliptext.com/articles/random-variable) X with α, β > 1 is the most likely value of the distribution (corresponding to the peak in the PDF), and is given by the following expression:\nWhen both parameters are less than one (α, β < 1), this is the anti-mode: the lowest point of the probability density curve.\nLetting α = β, the expression for the mode simplifies to 1/2, showing that for α = β > 1 the mode (resp. anti-mode when α, β < 1), is at the center of the distribution: it is symmetric in those cases.\nSee [Shapes](https://bliptext.com/articles/beta-distribution) section in this article for a full list of mode cases, for arbitrary values of α and β.\nFor several of these cases, the maximum value of the density function occurs at one or both ends.\nIn some cases the (maximum) value of the density function occurring at the end is finite.\nFor example, in the case of α = 2, β = 1 (or α = 1, β = 2), the density function becomes a [right-triangle distribution](https://bliptext.com/articles/triangular-distribution) which is finite at both ends.\nIn several other cases there is a [singularity](https://bliptext.com/articles/mathematical-singularity) at one end, where the value of the density function approaches infinity.\nFor example, in the case α = β = 1/2, the beta distribution simplifies to become the [arcsine distribution](https://bliptext.com/articles/arcsine-distribution).\nThere is debate among mathematicians about some of these cases and whether the ends (x = 0, and x = 1) can be called modes or not.\n# [Mode for beta distribution for 1 ≤ α ≤ 5 and 1 ≤ β ≤ 5 | upload.wikimedia.org/wikipedia/commons/5/56/Mode///Beta///Distribution///for///alpha///and///beta///from///1///to///5///-///J.///Rodal.jpg]\n\n### Median\nThe median of the beta distribution is the unique real number for which the [regularized incomplete beta function](https://bliptext.com/articles/regularized-incomplete-beta-function).\nThere is no general [closed-form expression](https://bliptext.com/articles/closed-form-expression) for the [median](https://bliptext.com/articles/median) of the beta distribution for arbitrary values of α and β.\n[Closed-form expressions](https://bliptext.com/articles/closed-form-expression) for particular values of the parameters α and β follow:\nThe following are the limits with one parameter finite (non-zero) and the other approaching these limits:\nA reasonable approximation of the value of the median of the beta distribution, for both α and β greater or equal to one, is given by the formula\nWhen α, β ≥ 1, the [relative error](https://bliptext.com/articles/relative-error) (the [absolute error](https://bliptext.com/articles/approximation-error) divided by the median) in this approximation is less than 4% and for both α ≥ 2 and β ≥ 2 it is less than 1%.\nThe [absolute error](https://bliptext.com/articles/approximation-error) divided by the difference between the mean and the mode is similarly small:\n# [Median for beta distribution for 0 ≤ α ≤ 5 and 0 ≤ β ≤ 5 | upload.wikimedia.org/wikipedia/commons/4/42/Median///Beta///Distribution///for///alpha///and///beta///from///0///to///5///-///J.///Rodal.jpg]\n# [(Mean–median) for beta distribution versus alpha and beta from 0 to 2 | upload.wikimedia.org/wikipedia/commons/e/e9/%28Mean///-///Median%29///for///Beta///distribution///versus///alpha///and///beta///from///0///to///2///-///J.///Rodal.jpg]\n# [Abs[(Median-Appr.)/Median] for beta distribution for 1 ≤ α ≤ 5 and 1 ≤ β ≤ 5 | upload.wikimedia.org/wikipedia/commons/a/af/Relative///Error///for///Approximation///to///Median///of///Beta///Distribution///for///alpha///and///beta///from///1///to///5///-///J.///Rodal.jpg]\n# [Abs[(Median-Appr.)/(Mean-Mode)] for beta distribution for 1≤α≤5 and 1≤β≤5 | upload.wikimedia.org/wikipedia/commons/e/e8/Error///in///Median///Apprx.///relative///to///Mean-Mode///distance///for///Beta///Distribution///with///alpha///and///beta///from///1///to///5///-///J.///Rodal.jpg]\n\n### Mean\nThe [expected value](https://bliptext.com/articles/expected-value) (mean) (μ) of a beta distribution [random variable](https://bliptext.com/articles/random-variable) X with two parameters α and β is a function of only the ratio β/α of these parameters:\nLetting α = β in the above expression one obtains μ = 1/2, showing that for α = β the mean is at the center of the distribution: it is symmetric.\nAlso, the following limits can be obtained from the above expression:\nTherefore, for β/α → 0, or for α/β → ∞, the mean is located at the right end, x = 1.\nFor these limit ratios, the beta distribution becomes a one-point [degenerate distribution](https://bliptext.com/articles/degenerate-distribution) with a [Dirac delta function](https://bliptext.com/articles/dirac-delta-function) spike at the right end, x = 1, with probability 1, and zero probability everywhere else.\nThere is 100% probability (absolute certainty) concentrated at the right end, x = 1.\nSimilarly, for β/α → ∞, or for α/β → 0, the mean is located at the left end, x = 0.\nThe beta distribution becomes a 1-point [Degenerate distribution](https://bliptext.com/articles/degenerate-distribution) with a [Dirac delta function](https://bliptext.com/articles/dirac-delta-function) spike at the left end, x = 0, with probability 1, and zero probability everywhere else.\nThere is 100% probability (absolute certainty) concentrated at the left end, x = 0.\nFollowing are the limits with one parameter finite (non-zero) and the other approaching these limits:\nWhile for typical unimodal distributions (with centrally located modes, inflexion points at both sides of the mode, and longer tails) (with Beta(α, β) such that α, β > 2) it is known that the sample mean (as an estimate of location) is not as [robust](https://bliptext.com/articles/robust-statistics) as the sample median, the opposite is the case for uniform or \"U-shaped\" bimodal distributions (with Beta(α, β) such that α, β ≤ 1), with the modes located at the ends of the distribution.\nAs Mosteller and Tukey remark ( p. 207) \"the average of the two extreme observations uses all the sample information. This illustrates how, for short-tailed distributions, the extreme observations should get more weight.\"\nBy contrast, it follows that the median of \"U-shaped\" bimodal distributions with modes at the edge of the distribution (with Beta(α, β) such that α, β ≤ 1) is not robust, as the sample median drops the extreme sample observations from consideration.\nA practical application of this occurs for example for [random walks](https://bliptext.com/articles/random-walk), since the probability for the time of the last visit to the origin in a random walk is distributed as the [arcsine distribution](https://bliptext.com/articles/arcsine-distribution) Beta(1/2, 1/2): the mean of a number of [realizations](https://bliptext.com/articles/realization-probability) of a random walk is a much more robust estimator than the median (which is an inappropriate sample measure estimate in this case).\n# [Mean for beta distribution for 0 ≤ α ≤ 5 and 0 ≤ β ≤ 5 | upload.wikimedia.org/wikipedia/commons/1/12/Mean///Beta///Distribution///for///alpha///and///beta///from///0///to///5///-///J.///Rodal.jpg]\n\n### Geometric mean\nThe logarithm of the [geometric mean](https://bliptext.com/articles/geometric-mean) GX of a distribution with [random variable](https://bliptext.com/articles/random-variable) X is the arithmetic mean of ln(X), or, equivalently, its expected value:\nFor a beta distribution, the expected value integral gives:\nwhere ψ is the [digamma function](https://bliptext.com/articles/digamma-function).\nTherefore, the geometric mean of a beta distribution with shape parameters α and β is the exponential of the digamma functions of α and β as follows:\nWhile for a beta distribution with equal shape parameters α = β, it follows that skewness = 0 and mode = mean = median = 1/2, the geometric mean is less than 1/2: 0 < GX < 1/2.\nThe reason for this is that the logarithmic transformation strongly weights the values of X close to zero, as ln(X) strongly tends towards negative infinity as X approaches zero, while ln(X) flattens towards zero as X → 1.\nAlong a line α = β, the following limits apply:\nFollowing are the limits with one parameter finite (non-zero) and the other approaching these limits:\nThe accompanying plot shows the difference between the mean and the geometric mean for shape parameters α and β from zero to 2.\nBesides the fact that the difference between them approaches zero as α and β approach infinity and that the difference becomes large for values of α and β approaching zero, one can observe an evident asymmetry of the geometric mean with respect to the shape parameters α and β.\nThe difference between the geometric mean and the mean is larger for small values of α in relation to β than when exchanging the magnitudes of β and α.\n[N. L.Johnson](https://bliptext.com/articles/norman-lloyd-johnson) and [S. Kotz](https://bliptext.com/articles/samuel-kotz) suggest the logarithmic approximation to the digamma function ψ(α) ≈ ln(α − 1/2) which results in the following approximation to the geometric mean:\nNumerical values for the [relative error](https://bliptext.com/articles/relative-error) in this approximation follow: [(α = β = 1): 9.39%]; [(α = β = 2): 1.29%]; [(α = 2, β = 3): 1.51%]; [(α = 3, β = 2): 0.44%]; [(α = β = 3): 0.51%]; [(α = β = 4): 0.26%]; [(α = 3, β = 4): 0.55%]; [(α = 4, β = 3): 0.24%].\nSimilarly, one can calculate the value of shape parameters required for the geometric mean to equal 1/2.\nGiven the value of the parameter β, what would be the value of the other parameter, α, required for the geometric mean to equal 1/2?.\nThe answer is that (for β > 1), the value of α required tends towards β + 1/2 as β → ∞.\nFor example, all these couples have the same geometric mean of 1/2: [β = 1, α = 1.4427], [β = 2, α = 2.46958], [β = 3, α = 3.47943], [β = 4, α = 4.48449], [β = 5, α = 5.48756], [β = 10, α = 10.4938], [β = 100, α = 100.499].\nThe fundamental property of the geometric mean, which can be proven to be false for any other mean, is\nThis makes the geometric mean the only correct mean when averaging normalized results, that is results that are presented as ratios to reference values.\nThis is relevant because the beta distribution is a suitable model for the random behavior of percentages and it is particularly suitable to the statistical modelling of proportions.\nThe geometric mean plays a central role in maximum likelihood estimation, see section \"Parameter estimation, maximum likelihood.\"\nActually, when performing maximum likelihood estimation, besides the [geometric mean](https://bliptext.com/articles/geometric-mean) GX based on the random variable X, also another [geometric mean](https://bliptext.com/articles/geometric-mean) appears naturally: the [geometric mean](https://bliptext.com/articles/geometric-mean) based on the linear transformation ––(1 − X), the mirror-image of X, denoted by G(1−X):\nAlong a line α = β, the following limits apply:\nFollowing are the limits with one parameter finite (non-zero) and the other approaching these limits:\nIt has the following approximate value:\nAlthough both GX and G(1−X) are asymmetric, in the case that both shape parameters are equal α = β, the geometric means are equal: GX = G(1−X).\nThis equality follows from the following symmetry displayed between both geometric means:\n# [(Mean − GeometricMean) for beta distribution versus α and β from 0 to 2, showing the asymmetry between α and β for the geometric mean | upload.wikimedia.org/wikipedia/commons/9/9f/%28Mean///-///GeometricMean%29///for///Beta///Distribution///versus///alpha///and///beta///from///0///to///2///-///J.///Rodal.jpg]\n# [beta distribution | upload.wikimedia.org/wikipedia/commons/1/1e/Geometric///Means///for///Beta///distribution///Purple%3DG%28X%29%2C///Yellow%3DG%281-X%29%2C///smaller///values///alpha///and///beta///in///front///-///J.///Rodal.jpg]\n# [beta distribution | upload.wikimedia.org/wikipedia/commons/1/1e/Geometric///Means///for///Beta///distribution///Purple%3DG%28X%29%2C///Yellow%3DG%281-X%29%2C///larger///values///alpha///and///beta///in///front///-///J.///Rodal.jpg]\n\n### Harmonic mean\nThe inverse of the [harmonic mean](https://bliptext.com/articles/harmonic-mean) (HX) of a distribution with [random variable](https://bliptext.com/articles/random-variable) X is the arithmetic mean of 1/X, or, equivalently, its expected value.\nTherefore, the [harmonic mean](https://bliptext.com/articles/harmonic-mean) (HX) of a beta distribution with shape parameters α and β is:\nThe [harmonic mean](https://bliptext.com/articles/harmonic-mean) (HX) of a beta distribution with α < 1 is undefined, because its defining expression is not bounded in [0, 1] for shape parameter α less than unity.\nLetting α = β in the above expression one obtains\nshowing that for α = β the harmonic mean ranges from 0, for α = β = 1, to 1/2, for α = β → ∞.\nFollowing are the limits with one parameter finite (non-zero) and the other approaching these limits:\nThe harmonic mean plays a role in maximum likelihood estimation for the four parameter case, in addition to the geometric mean.\nActually, when performing maximum likelihood estimation for the four parameter case, besides the harmonic mean HX based on the random variable X, also another harmonic mean appears naturally: the harmonic mean based on the linear transformation (1 − X), the mirror-image of X, denoted by H1 − X:\nThe [harmonic mean](https://bliptext.com/articles/harmonic-mean) (H(1 − X)) of a beta distribution with β < 1 is undefined, because its defining expression is not bounded in [0, 1] for shape parameter β less than unity.\nLetting α = β in the above expression one obtains\nshowing that for α = β the harmonic mean ranges from 0, for α = β = 1, to 1/2, for α = β → ∞.\nFollowing are the limits with one parameter finite (non-zero) and the other approaching these limits:\nAlthough both HX and H1−X are asymmetric, in the case that both shape parameters are equal α = β, the harmonic means are equal: HX = H1−X.\nThis equality follows from the following symmetry displayed between both harmonic means:\n# [Harmonic mean for beta distribution for 0 < α < 5 and 0 < β < 5 | upload.wikimedia.org/wikipedia/commons/8/8d/Harmonic///mean///for///Beta///distribution///for///alpha///and///beta///ranging///from///0///to///5///-///J.///Rodal.jpg]\n# [Harmonic mean for beta distribution versus α and β from 0 to 2 | upload.wikimedia.org/wikipedia/commons/c/c0/%28Mean///-///HarmonicMean%29///for///Beta///distribution///versus///alpha///and///beta///from///0///to///2///-///J.///Rodal.jpg]\n# [beta distribution | upload.wikimedia.org/wikipedia/commons/3/32/Harmonic///Means///for///Beta///distribution///Purple%3DH%28X%29%2C///Yellow%3DH%281-X%29%2C///smaller///values///alpha///and///beta///in///front///-///J.///Rodal.jpg]\n# [Harmonic means for beta distribution: purple = H(X), yellow = H(1 − X), larger values α and β in front | upload.wikimedia.org/wikipedia/commons/0/00/Harmonic///Means///for///Beta///distribution///Purple%3DH%28X%29%2C///Yellow%3DH%281-X%29%2C///larger///values///alpha///and///beta///in///front///-///J.///Rodal.jpg]\n\n## Measures of statistical dispersion\n\n### Variance\nThe [variance](https://bliptext.com/articles/variance) (the second moment centered on the mean) of a beta distribution [random variable](https://bliptext.com/articles/random-variable) X with parameters α and β is:\nLetting α = β in the above expression one obtains\nshowing that for α = β the variance decreases monotonically as α = β increases.\nSetting α = β = 0 in this expression, one finds the maximum variance var(X) = 1/4 which only occurs approaching the limit, at α = β = 0.\nThe beta distribution may also be [parametrized](https://bliptext.com/articles/statistical-parameter) in terms of its mean μ (0 < μ < 1) and sample size ν = α + β (ν > 0) (see subsection [Mean and sample size](https://bliptext.com/articles/)):\nUsing this [parametrization](https://bliptext.com/articles/statistical-parameter), one can express the variance in terms of the mean μ and the sample size ν as follows:\nSince ν = α + β > 0, it follows that var(X) < μ(1 − μ).\nFor a symmetric distribution, the mean is at the middle of the distribution, μ = 1/2, and therefore:\nAlso, the following limits (with only the noted variable approaching the limit) can be obtained from the above expressions:\n# [325px | upload.wikimedia.org/wikipedia/commons/4/49/Variance///for///Beta///Distribution///for///alpha///and///beta///ranging///from///0///to///5///-///J.///Rodal.jpg]\n\n### Geometric variance and covariance\nThe logarithm of the geometric variance, ln(varGX), of a distribution with [random variable](https://bliptext.com/articles/random-variable) X is the second moment of the logarithm of X centered on the geometric mean of X, ln(GX):\nand therefore, the geometric variance is:\nIn the [Fisher information](https://bliptext.com/articles/fisher-information) matrix, and the curvature of the log [likelihood function](https://bliptext.com/articles/likelihood-function), the logarithm of the geometric variance of the [reflected](https://bliptext.com/articles/reflection-formula) variable 1 − X and the logarithm of the geometric covariance between X and 1 − X appear:\nFor a beta distribution, higher order logarithmic moments can be derived by using the representation of a beta distribution as a proportion of two gamma distributions and differentiating through the integral.\nThey can be expressed in terms of higher order poly-gamma functions.\nSee the section.\nThe [variance](https://bliptext.com/articles/variance) of the logarithmic variables and co[variance](https://bliptext.com/articles/variance) of ln X and ln(1−X) are:\nwhere the **[trigamma function](https://bliptext.com/articles/trigamma-function)**, denoted ψ1(α), is the second of the [polygamma functions](https://bliptext.com/articles/polygamma-function), and is defined as the derivative of the [digamma function](https://bliptext.com/articles/digamma-function):\nTherefore,\nThe accompanying plots show the log geometric variances and log geometric covariance versus the shape parameters α and β.\nThe plots show that the log geometric variances and log geometric covariance are close to zero for shape parameters α and β greater than 2, and that the log geometric variances rapidly rise in value for shape parameter values α and β less than unity.\nThe log geometric variances are positive for all values of the shape parameters.\nThe log geometric covariance is negative for all values of the shape parameters, and it reaches large negative values for α and β less than unity.\nFollowing are the limits with one parameter finite (non-zero) and the other approaching these limits:\nLimits with two parameters varying:\nAlthough both ln(varGX) and ln(varG(1 − X)) are asymmetric, when the shape parameters are equal, α = β, one has: ln(varGX) = ln(varG(1−X)).\nThis equality follows from the following symmetry displayed between both log geometric variances:\nThe log geometric covariance is symmetric:\n# [log geometric variances vs. α and β | upload.wikimedia.org/wikipedia/commons/3/36/Beta///distribution///log///geometric///variances///front///view///-///J.///Rodal.png]\n# [log geometric variances vs. α and β | upload.wikimedia.org/wikipedia/commons/0/06/Beta///distribution///log///geometric///variances///back///view///-///J.///Rodal.png]\n\n### Mean absolute deviation around the mean\nThe [mean absolute deviation](https://bliptext.com/articles/mean-absolute-deviation) around the mean for the beta distribution with shape parameters α and β is:\nThe mean absolute deviation around the mean is a more [robust](https://bliptext.com/articles/robust-statistics) [estimator](https://bliptext.com/articles/estimator) of [statistical dispersion](https://bliptext.com/articles/statistical-dispersion) than the standard deviation for beta distributions with tails and inflection points at each side of the mode, Beta(α, β) distributions with α,β > 2, as it depends on the linear (absolute) deviations rather than the square deviations from the mean.\nTherefore, the effect of very large deviations from the mean are not as overly weighted.\nUsing [Stirling's approximation](https://bliptext.com/articles/stirling-s-approximation) to the [Gamma function](https://bliptext.com/articles/gamma-function), [N.L.Johnson](https://bliptext.com/articles/norman-lloyd-johnson) and [S.Kotz](https://bliptext.com/articles/samuel-kotz) derived the following approximation for values of the shape parameters greater than unity (the relative error for this approximation is only −3.5% for α = β = 1, and it decreases to zero as α → ∞, β → ∞):\nAt the limit α → ∞, β → ∞, the ratio of the mean absolute deviation to the standard deviation (for the beta distribution) becomes equal to the ratio of the same measures for the normal distribution:.\nFor α = β = 1 this ratio equals, so that from α = β = 1 to α, β → ∞ the ratio decreases by 8.5%.\nFor α = β = 0 the standard deviation is exactly equal to the mean absolute deviation around the mean.\nTherefore, this ratio decreases by 15% from α = β = 0 to α = β = 1, and by 25% from α = β = 0 to α, β → ∞.\nHowever, for skewed beta distributions such that α → 0 or β → 0, the ratio of the standard deviation to the mean absolute deviation approaches infinity (although each of them, individually, approaches zero) because the mean absolute deviation approaches zero faster than the standard deviation.\nUsing the [parametrization](https://bliptext.com/articles/statistical-parameter) in terms of mean μ and sample size ν = α + β > 0:\none can express the mean [absolute deviation](https://bliptext.com/articles/absolute-deviation) around the mean in terms of the mean μ and the sample size ν as follows:\nFor a symmetric distribution, the mean is at the middle of the distribution, μ = 1/2, and therefore:\nAlso, the following limits (with only the noted variable approaching the limit) can be obtained from the above expressions:\n# [Ratio of ,ean abs.dev. to std.dev. for beta distribution with α and β ranging from 0 to 5 | upload.wikimedia.org/wikipedia/commons/6/60/Ratio///of///Mean///Abs.///Dev.///to///Std.Dev.///Beta///distribution///with///alpha///and///beta///from///0///to///5///-///J.///Rodal.jpg]\n# [Ratio of mean abs.dev. to std.dev. for beta distribution with mean 0 ≤ μ ≤ 1 and sample size 0 < ν ≤ 10 | upload.wikimedia.org/wikipedia/commons/1/10/Ratio///of///Mean///Abs.///Dev.///to///Std.Dev.///Beta///distribution///vs.///nu///from///0///to///10///and///vs.///mean///-///J.///Rodal.jpg]\n\n### Mean absolute difference\nThe [mean absolute difference](https://bliptext.com/articles/mean-absolute-difference) for the beta distribution is:\nThe [Gini coefficient](https://bliptext.com/articles/gini-coefficient) for the beta distribution is half of the relative mean absolute difference:\n\n## Skewness\nThe [skewness](https://bliptext.com/articles/skewness) (the third moment centered on the mean, normalized by the 3/2 power of the variance) of the beta distribution is\nLetting α = β in the above expression one obtains γ1 = 0, showing once again that for α = β the distribution is symmetric and hence the skewness is zero.\nPositive skew (right-tailed) for α < β, negative skew (left-tailed) for α > β.\nUsing the [parametrization](https://bliptext.com/articles/statistical-parameter) in terms of mean μ and sample size ν = α + β:\none can express the skewness in terms of the mean μ and the sample size ν as follows:\nThe skewness can also be expressed just in terms of the variance var and the mean μ as follows:\nThe accompanying plot of skewness as a function of variance and mean shows that maximum variance (1/4) is coupled with zero skewness and the symmetry condition (μ = 1/2), and that maximum skewness (positive or negative infinity) occurs when the mean is located at one end or the other, so that the \"mass\" of the probability distribution is concentrated at the ends (minimum variance).\nThe following expression for the square of the skewness, in terms of the sample size ν = α + β and the variance var, is useful for the method of moments estimation of four parameters:\nThis expression correctly gives a skewness of zero for α = β, since in that case (see ):.\nFor the symmetric case (α = β), skewness = 0 over the whole range, and the following limits apply:\nFor the asymmetric cases (α ≠ β) the following limits (with only the noted variable approaching the limit) can be obtained from the above expressions:\n# [Skewness for Beta Distribution as a function of variance and mean | upload.wikimedia.org/wikipedia/commons/1/1b/Skewness///for///Beta///Distribution///as///a///function///of///the///variance///and///the///mean///-///J.///Rodal.jpg]\n# [325px | upload.wikimedia.org/wikipedia/commons/4/43/Skewness///Beta///Distribution///for///alpha///and///beta///from///1///to///5///-///J.///Rodal.jpg]\n# [325px | upload.wikimedia.org/wikipedia/commons/4/48/Skewness///Beta///Distribution///for///alpha///and///beta///from///.1///to///5///-///J.///Rodal.jpg]\n\n## Kurtosis\nThe beta distribution has been applied in acoustic analysis to assess damage to gears, as the kurtosis of the beta distribution has been reported to be a good indicator of the condition of a gear.\nKurtosis has also been used to distinguish the seismic signal generated by a person's footsteps from other signals.\nAs persons or other targets moving on the ground generate continuous signals in the form of seismic waves, one can separate different targets based on the seismic waves they generate.\nKurtosis is sensitive to impulsive signals, so it's much more sensitive to the signal generated by human footsteps than other signals generated by vehicles, winds, noise, etc.\nUnfortunately, the notation for kurtosis has not been standardized.\nKenney and Keeping use the symbol γ2 for the [excess kurtosis](https://bliptext.com/articles/excess-kurtosis), but [Abramowitz and Stegun](https://bliptext.com/articles/abramowitz-and-stegun) use different terminology.\nTo prevent confusion between kurtosis (the fourth moment centered on the mean, normalized by the square of the variance) and excess kurtosis, when using symbols, they will be spelled out as follows:\nLetting α = β in the above expression one obtains\nTherefore, for symmetric beta distributions, the excess kurtosis is negative, increasing from a minimum value of −2 at the limit as {α = β} → 0, and approaching a maximum value of zero as {α = β} → ∞.\nThe value of −2 is the minimum value of excess kurtosis that any distribution (not just beta distributions, but any distribution of any possible kind) can ever achieve.\nThis minimum value is reached when all the probability density is entirely concentrated at each end x = 0 and x = 1, with nothing in between: a 2-point [Bernoulli distribution](https://bliptext.com/articles/bernoulli-distribution) with equal probability 1/2 at each end (a coin toss: see section below \"Kurtosis bounded by the square of the skewness\" for further discussion).\nThe description of [kurtosis](https://bliptext.com/articles/kurtosis) as a measure of the \"potential outliers\" (or \"potential rare, extreme values\") of the probability distribution, is correct for all distributions including the beta distribution.\nWhen rare, extreme values can occur in the beta distribution, the higher its kurtosis; otherwise, the kurtosis is lower.\nFor α ≠ β, skewed beta distributions, the excess kurtosis can reach unlimited positive values (particularly for α → 0 for finite β, or for β → 0 for finite α) because the side away from the mode will produce occasional extreme values.\nMinimum kurtosis takes place when the mass density is concentrated equally at each end (and therefore the mean is at the center), and there is no probability mass density in between the ends.\nUsing the [parametrization](https://bliptext.com/articles/statistical-parameter) in terms of mean μ and sample size ν = α + β:\none can express the excess kurtosis in terms of the mean μ and the sample size ν as follows:\nThe excess kurtosis can also be expressed in terms of just the following two parameters: the variance var, and the sample size ν as follows:\nand, in terms of the variance var and the mean μ as follows:\nThe plot of excess kurtosis as a function of the variance and the mean shows that the minimum value of the excess kurtosis (−2, which is the minimum possible value for excess kurtosis for any distribution) is intimately coupled with the maximum value of variance (1/4) and the symmetry condition: the mean occurring at the midpoint (μ = 1/2).\nThis occurs for the symmetric case of α = β = 0, with zero skewness.\nAt the limit, this is the 2 point [Bernoulli distribution](https://bliptext.com/articles/bernoulli-distribution) with equal probability 1/2 at each [Dirac delta function](https://bliptext.com/articles/dirac-delta-function) end x = 0 and x = 1 and zero probability everywhere else.\n(A coin toss: one face of the coin being x = 0 and the other face being x = 1.) Variance is maximum because the distribution is bimodal with nothing in between the two modes (spikes) at each end.\nExcess kurtosis is minimum: the probability density \"mass\" is zero at the mean and it is concentrated at the two peaks at each end.\nExcess kurtosis reaches the minimum possible value (for any distribution) when the probability density function has two spikes at each end: it is bi-\"peaky\" with nothing in between them.\nOn the other hand, the plot shows that for extreme skewed cases, where the mean is located near one or the other end (μ = 0 or μ = 1), the variance is close to zero, and the excess kurtosis rapidly approaches infinity when the mean of the distribution approaches either end.\nAlternatively, the excess kurtosis can also be expressed in terms of just the following two parameters: the square of the skewness, and the sample size ν as follows:\nFrom this last expression, one can obtain the same limits published over a century ago by [Karl Pearson](https://bliptext.com/articles/karl-pearson) for the beta distribution (see section below titled \"Kurtosis bounded by the square of the skewness\").\nSetting α + β = ν = 0 in the above expression, one obtains Pearson's lower boundary (values for the skewness and excess kurtosis below the boundary (excess kurtosis + 2 − skewness2 = 0) cannot occur for any distribution, and hence [Karl Pearson](https://bliptext.com/articles/karl-pearson) appropriately called the region below this boundary the \"impossible region\").\nThe limit of α + β = ν → ∞ determines Pearson's upper boundary.\ntherefore:\nValues of ν = α + β such that ν ranges from zero to infinity, 0 < ν < ∞, span the whole region of the beta distribution in the plane of excess kurtosis versus squared skewness.\nFor the symmetric case (α = β), the following limits apply:\nFor the unsymmetric cases (α ≠ β) the following limits (with only the noted variable approaching the limit) can be obtained from the above expressions:\n# [Excess Kurtosis for Beta Distribution as a function of variance and mean | upload.wikimedia.org/wikipedia/commons/6/69/Excess///Kurtosis///for///Beta///Distribution///as///a///function///of///variance///and///mean///-///J.///Rodal.jpg]\n# [325px | upload.wikimedia.org/wikipedia/commons/c/c9/Excess///Kurtosis///for///Beta///Distribution///with///alpha///and///beta///ranging///from///1///to///5///-///J.///Rodal.jpg]\n# [325px | upload.wikimedia.org/wikipedia/commons/2/26/Excess///Kurtosis///for///Beta///Distribution///with///alpha///and///beta///ranging///from///0.1///to///5///-///J.///Rodal.jpg]\n\n## Characteristic function\nThe [characteristic function](https://bliptext.com/articles/characteristic-function-probability-theory) is the [Fourier transform](https://bliptext.com/articles/fourier-transform) of the probability density function.\nThe characteristic function of the beta distribution is [Kummer's confluent hypergeometric function](https://bliptext.com/articles/confluent-hypergeometric-function) (of the first kind):\nwhere\nis the [rising factorial](https://bliptext.com/articles/rising-factorial), also called the \"Pochhammer symbol\".\nThe value of the characteristic function for t = 0, is one:\nAlso, the real and imaginary parts of the characteristic function enjoy the following symmetries with respect to the origin of variable t:\nThe symmetric case α = β simplifies the characteristic function of the beta distribution to a [Bessel function](https://bliptext.com/articles/bessel-function), since in the special case α + β = 2α the [confluent hypergeometric function](https://bliptext.com/articles/confluent-hypergeometric-function) (of the first kind) reduces to a [Bessel function](https://bliptext.com/articles/bessel-function) (the modified [Bessel function](https://bliptext.com/articles/bessel-function) of the first kind ) using [Kummer's](https://bliptext.com/articles/ernst-kummer) second transformation as follows:\nIn the accompanying plots, the [real part](https://bliptext.com/articles/complex-number) (Re) of the [characteristic function](https://bliptext.com/articles/characteristic-function-probability-theory) of the beta distribution is displayed for symmetric (α = β) and skewed (α ≠ β) cases.\n# [Re(characteristic function) symmetric case α = β ranging from 25 to 0 | upload.wikimedia.org/wikipedia/commons/2/24/Re%28CharacteristicFunction%29///Beta///Distr///alpha%3Dbeta///from///0///to///25///Back///-///J.///Rodal.jpg]\n# [Re(characteristic function) symmetric case α = β ranging from 0 to 25 | upload.wikimedia.org/wikipedia/commons/f/f3/Re%28CharacteristicFunc%29///Beta///Distr///alpha%3Dbeta///from///0///to///25///Front-///J.///Rodal.jpg]\n# [Re(characteristic function) β = α + 1/2; α ranging from 25 to 0 | upload.wikimedia.org/wikipedia/commons/2/24/Re%28CharacteristFunc%29///Beta///Distr///alpha///from///0///to///25///and///beta%3Dalpha%2B0.5///Back///-///J.///Rodal.jpg]\n# [Re(characteristic function) α = β + 1/2; β ranging from 25 to 0 | upload.wikimedia.org/wikipedia/commons/1/1e/Re%28CharacterFunc%29///Beta///Distrib.///beta///from///0///to///25%2C///alpha%3Dbeta%2B0.5///Back///-///J.///Rodal.jpg]\n# [Re(characteristic function) α = β + 1/2; β ranging from 0 to 25 | upload.wikimedia.org/wikipedia/commons/0/0f/Re%28CharacterFunc%29///Beta///Distr.///beta///from///0///to///25%2C///alpha%3Dbeta%2B0.5///Front///-///J.///Rodal.jpg]\n\n## Other moments\n\n### Moment generating function\nIt also follows that the [moment generating function](https://bliptext.com/articles/moment-generating-function) is\nIn particular MX(α; β; 0) = 1.\n\n### Higher moments\nUsing the [moment generating function](https://bliptext.com/articles/moment-generating-function), the k-th [raw moment](https://bliptext.com/articles/raw-moment) is given by the factor\nmultiplying the (exponential series) term in the series of the [moment generating function](https://bliptext.com/articles/moment-generating-function)\nwhere (x)(k) is a [Pochhammer symbol](https://bliptext.com/articles/pochhammer-symbol) representing rising factorial.\nIt can also be written in a recursive form as\nSince the moment generating function has a positive radius of convergence, the beta distribution is [determined by its moments](https://bliptext.com/articles/moment-problem).\n\n### Moments of transformed random variables\n\n#### Moments of linearly transformed, product and inverted random variables\nOne can also show the following expectations for a transformed random variable, where the random variable X is Beta-distributed with parameters α and β: X ~ Beta(α, β).\nThe expected value of the variable 1 − X is the mirror-symmetry of the expected value based on X:\nDue to the mirror-symmetry of the probability density function of the beta distribution, the variances based on variables X and 1 − X are identical, and the covariance on X(1 − X is the negative of the variance:\nThese are the expected values for inverted variables, (these are related to the harmonic means, see ):\nThe following transformation by dividing the variable X by its mirror-image X/(1 − X) results in the expected value of the \"inverted beta distribution\" or [beta prime distribution](https://bliptext.com/articles/beta-prime-distribution) (also known as beta distribution of the second kind or [Pearson's Type VI](https://bliptext.com/articles/pearson-distribution)):\nVariances of these transformed variables can be obtained by integration, as the expected values of the second moments centered on the corresponding variables:\nThe following variance of the variable X divided by its mirror-image (X/(1−X) results in the variance of the \"inverted beta distribution\" or [beta prime distribution](https://bliptext.com/articles/beta-prime-distribution) (also known as beta distribution of the second kind or [Pearson's Type VI](https://bliptext.com/articles/pearson-distribution)):\nThe covariances are:\nThese expectations and variances appear in the four-parameter Fisher information matrix (.)\n\n#### Moments of logarithmically transformed random variables\nExpected values for [logarithmic transformations](https://bliptext.com/articles/logarithm-transformation) (useful for [maximum likelihood](https://bliptext.com/articles/maximum-likelihood) estimates, see ) are discussed in this section.\nThe following logarithmic linear transformations are related to the geometric means GX and G(1−X) (see ):\nWhere the **[digamma function](https://bliptext.com/articles/digamma-function)** ψ(α) is defined as the [logarithmic derivative](https://bliptext.com/articles/logarithmic-derivative) of the [gamma function](https://bliptext.com/articles/gamma-function):\n[Logit](https://bliptext.com/articles/logit) transformations are interesting, as they usually transform various shapes (including J-shapes) into (usually skewed) bell-shaped densities over the logit variable, and they may remove the end singularities over the original variable:\nJohnson considered the distribution of the [logit](https://bliptext.com/articles/logit) – transformed variable ln(X/1 − X), including its moment generating function and approximations for large values of the shape parameters.\nThis transformation extends the finite support [0, 1] based on the original variable X to infinite support in both directions of the real line (−∞, +∞).\nThe logit of a beta variate has the [logistic-beta distribution](https://bliptext.com/articles/logistic-beta-distribution).\nHigher order logarithmic moments can be derived by using the representation of a beta distribution as a proportion of two gamma distributions and differentiating through the integral.\nThey can be expressed in terms of higher order poly-gamma functions as follows:\ntherefore the [variance](https://bliptext.com/articles/variance) of the logarithmic variables and co[variance](https://bliptext.com/articles/variance) of ln(X) and ln(1−X) are:\nwhere the **[trigamma function](https://bliptext.com/articles/trigamma-function)**, denoted ψ1(α), is the second of the [polygamma functions](https://bliptext.com/articles/polygamma-function), and is defined as the derivative of the [digamma](https://bliptext.com/articles/digamma) function:\nThe variances and covariance of the logarithmically transformed variables X and (1 − X) are different, in general, because the logarithmic transformation destroys the mirror-symmetry of the original variables X and (1 − X), as the logarithm approaches negative infinity for the variable approaching zero.\nThese logarithmic variances and covariance are the elements of the [Fisher information](https://bliptext.com/articles/fisher-information) matrix for the beta distribution.\nThey are also a measure of the curvature of the log likelihood function (see section on Maximum likelihood estimation).\nThe variances of the log inverse variables are identical to the variances of the log variables:\nIt also follows that the variances of the [logit](https://bliptext.com/articles/logit)-transformed variables are\n# [350px | upload.wikimedia.org/wikipedia/commons/c/c8/Logit.svg]\n\n## Quantities of information (entropy)\nGiven a beta distributed random variable, X ~ Beta(α, β), the [differential entropy](https://bliptext.com/articles/information-entropy) of X is (measured in [nats](https://bliptext.com/articles/nat-unit)), the expected value of the negative of the logarithm of the [probability density function](https://bliptext.com/articles/probability-density-function):\nwhere f(x; α, β) is the [probability density function](https://bliptext.com/articles/probability-density-function) of the beta distribution:\nThe [digamma function](https://bliptext.com/articles/digamma-function) ψ appears in the formula for the differential entropy as a consequence of Euler's integral formula for the [harmonic numbers](https://bliptext.com/articles/harmonic-number) which follows from the integral:\nThe [differential entropy](https://bliptext.com/articles/information-entropy) of the beta distribution is negative for all values of α and β greater than zero, except at α = β = 1 (for which values the beta distribution is the same as the [uniform distribution](https://bliptext.com/articles/uniform-distribution-continuous)), where the [differential entropy](https://bliptext.com/articles/information-entropy) reaches its [maximum](https://bliptext.com/articles/maxima-and-minima) value of zero.\nIt is to be expected that the maximum entropy should take place when the beta distribution becomes equal to the uniform distribution, since uncertainty is maximal when all possible events are equiprobable.\nFor α or β approaching zero, the [differential entropy](https://bliptext.com/articles/information-entropy) approaches its [minimum](https://bliptext.com/articles/maxima-and-minima) value of negative infinity.\nFor (either or both) α or β approaching zero, there is a maximum amount of order: all the probability density is concentrated at the ends, and there is zero probability density at points located between the ends.\nSimilarly for (either or both) α or β approaching infinity, the differential entropy approaches its minimum value of negative infinity, and a maximum amount of order.\nIf either α or β approaches infinity (and the other is finite) all the probability density is concentrated at an end, and the probability density is zero everywhere else.\nIf both shape parameters are equal (the symmetric case), α = β, and they approach infinity simultaneously, the probability density becomes a spike ([Dirac delta function](https://bliptext.com/articles/dirac-delta-function)) concentrated at the middle x = 1/2, and hence there is 100% probability at the middle x = 1/2 and zero probability everywhere else.\nThe (continuous case) [differential entropy](https://bliptext.com/articles/information-entropy) was introduced by Shannon in his original paper (where he named it the \"entropy of a continuous distribution\"), as the concluding part of the same paper where he defined the [discrete entropy](https://bliptext.com/articles/information-entropy).\nIt is known since then that the differential entropy may differ from the infinitesimal limit of the discrete entropy by an infinite offset, therefore the differential entropy can be negative (as it is for the beta distribution).\nWhat really matters is the relative value of entropy.\nGiven two beta distributed random variables, X1 ~ Beta(α, β) and X2 ~ Beta(**', '**), the [cross-entropy](https://bliptext.com/articles/cross-entropy) is (measured in nats)\nThe [cross entropy](https://bliptext.com/articles/cross-entropy) has been used as an error metric to measure the distance between two hypotheses.\nIts absolute value is minimum when the two distributions are identical.\nIt is the information measure most closely related to the log maximum likelihood (see section on \"Parameter estimation. Maximum likelihood estimation\")).\nThe relative entropy, or [Kullback–Leibler divergence](https://bliptext.com/articles/kullback-leibler-divergence) DKL(X1 || X2), is a measure of the inefficiency of assuming that the distribution is X2 ~ Beta(**', '**) when the distribution is really X1 ~ Beta(α, β).\nIt is defined as follows (measured in nats).\nThe relative entropy, or [Kullback–Leibler divergence](https://bliptext.com/articles/kullback-leibler-divergence), is always non-negative.\nA few numerical examples follow:\nThe [Kullback–Leibler divergence](https://bliptext.com/articles/kullback-leibler-divergence) is not symmetric DKL(X1 || X2) ≠ DKL(X2 || X1) for the case in which the individual beta distributions Beta(1, 1) and Beta(3, 3) are symmetric, but have different entropies h(X1) ≠ h(X2).\nThe value of the Kullback divergence depends on the direction traveled: whether going from a higher (differential) entropy to a lower (differential) entropy or the other way around.\nIn the numerical example above, the Kullback divergence measures the inefficiency of assuming that the distribution is (bell-shaped) Beta(3, 3), rather than (uniform) Beta(1, 1).\nThe \"h\" entropy of Beta(1, 1) is higher than the \"h\" entropy of Beta(3, 3) because the uniform distribution Beta(1, 1) has a maximum amount of disorder.\nThe Kullback divergence is more than two times higher (0.598803 instead of 0.267864) when measured in the direction of decreasing entropy: the direction that assumes that the (uniform) Beta(1, 1) distribution is (bell-shaped) Beta(3, 3) rather than the other way around.\nIn this restricted sense, the Kullback divergence is consistent with the [second law of thermodynamics](https://bliptext.com/articles/second-law-of-thermodynamics).\nThe [Kullback–Leibler divergence](https://bliptext.com/articles/kullback-leibler-divergence) is symmetric DKL(X1 || X2) = DKL(X2 || X1) for the skewed cases Beta(3, 0.5) and Beta(0.5, 3) that have equal differential entropy h(X1) = h(X2).\nThe symmetry condition:\nfollows from the above definitions and the mirror-symmetry f(x; α, β) = f(1 − x; α, β) enjoyed by the beta distribution.\n# [325px | upload.wikimedia.org/wikipedia/commons/e/e2/Differential///Entropy///Beta///Distribution///for///alpha///and///beta///from///1///to///5///-///J.///Rodal.jpg]\n# [325px | upload.wikimedia.org/wikipedia/commons/2/2a/Differential///Entropy///Beta///Distribution///for///alpha///and///beta///from///0.1///to///5///-///J.///Rodal.jpg]\n\n## Relationships between statistical measures\n\n### Mean, mode and median relationship\nIf 1 < α < β then mode ≤ median ≤ mean.\nExpressing the mode (only for α, β > 1), and the mean in terms of α and β:\nIf 1 < β < α then the order of the inequalities are reversed.\nFor α, β > 1 the absolute distance between the mean and the median is less than 5% of the distance between the maximum and minimum values of x.\nOn the other hand, the absolute distance between the mean and the mode can reach 50% of the distance between the [maximum](https://bliptext.com/articles/maxima-and-minima) and minimum values of x, for the ([pathological](https://bliptext.com/articles/pathological-mathematics)) case of α = 1 and β = 1, for which values the beta distribution approaches the uniform distribution and the [differential entropy](https://bliptext.com/articles/information-entropy) approaches its [maximum](https://bliptext.com/articles/maxima-and-minima) value, and hence [maximum](https://bliptext.com/articles/maxima-and-minima) \"disorder\".\nFor example, for α = 1.0001 and β = 1.00000001:\nwhere PDF stands for the value of the [probability density function](https://bliptext.com/articles/probability-density-function).\n# [325px | upload.wikimedia.org/wikipedia/commons/f/ff/Mean///Median///Difference///-///Beta///Distribution///for///alpha///and///beta///from///1///to///5///-///J.///Rodal.jpg]\n# [325px | upload.wikimedia.org/wikipedia/commons/8/88/Mean///Mode///Difference///-///Beta///Distribution///for///alpha///and///beta///from///1///to///5///-///J.///Rodal.jpg]\n\n### Mean, geometric mean and harmonic mean relationship\nIt is known from the [inequality of arithmetic and geometric means](https://bliptext.com/articles/inequality-of-arithmetic-and-geometric-means) that the geometric mean is lower than the mean.\nSimilarly, the harmonic mean is lower than the geometric mean.\nThe accompanying plot shows that for α = β, both the mean and the median are exactly equal to 1/2, regardless of the value of α = β, and the mode is also equal to 1/2 for α = β > 1, however the geometric and harmonic means are lower than 1/2 and they only approach this value asymptotically as α = β → ∞.\n# [:Mean, median, geometric mean and harmonic mean for beta distribution with 0 < α = β < 5 | upload.wikimedia.org/wikipedia/commons/a/a0/Mean%2C///Median%2C///Geometric///Mean///and///Harmonic///Mean///for///Beta///distribution///with///alpha///%3D///beta///from///0///to///5///-///J.///Rodal.png]\n\n### Kurtosis bounded by the square of the skewness\nAs remarked by [Feller](https://bliptext.com/articles/william-feller), in the [Pearson system](https://bliptext.com/articles/pearson-distribution) the beta probability density appears as [type I](https://bliptext.com/articles/pearson-distribution) (any difference between the beta distribution and Pearson's [type I](https://bliptext.com/articles/pearson-distribution) distribution is only superficial and it makes no difference for the following discussion regarding the relationship between kurtosis and skewness).\n[Karl Pearson](https://bliptext.com/articles/karl-pearson) showed, in Plate 1 of his paper published in 1916, a graph with the [kurtosis](https://bliptext.com/articles/kurtosis) as the vertical axis ([ordinate](https://bliptext.com/articles/ordinate)) and the square of the [skewness](https://bliptext.com/articles/skewness) as the horizontal axis ([abscissa](https://bliptext.com/articles/abscissa)), in which a number of distributions were displayed.\nThe region occupied by the beta distribution is bounded by the following two [lines](https://bliptext.com/articles/line-geometry) in the (skewness2,kurtosis) [plane](https://bliptext.com/articles/cartesian-coordinate-system), or the (skewness2,excess kurtosis) [plane](https://bliptext.com/articles/cartesian-coordinate-system):\nor, equivalently,\nAt a time when there were no powerful digital computers, [Karl Pearson](https://bliptext.com/articles/karl-pearson) accurately computed further boundaries, for example, separating the \"U-shaped\" from the \"J-shaped\" distributions.\nThe lower boundary line (excess kurtosis + 2 − skewness2 = 0) is produced by skewed \"U-shaped\" beta distributions with both values of shape parameters α and β close to zero.\nThe upper boundary line (excess kurtosis − (3/2) skewness2 = 0) is produced by extremely skewed distributions with very large values of one of the parameters and very small values of the other parameter.\n[Karl Pearson](https://bliptext.com/articles/karl-pearson) showed that this upper boundary line (excess kurtosis − (3/2) skewness2 = 0) is also the intersection with Pearson's distribution III, which has unlimited support in one direction (towards positive infinity), and can be bell-shaped or J-shaped.\nHis son, [Egon Pearson](https://bliptext.com/articles/egon-pearson), showed that the region (in the kurtosis/squared-skewness plane) occupied by the beta distribution (equivalently, Pearson's distribution I) as it approaches this boundary (excess kurtosis − (3/2) skewness2 = 0) is shared with the [noncentral chi-squared distribution](https://bliptext.com/articles/noncentral-chi-squared-distribution).\nKarl Pearson (Pearson 1895, pp. 357, 360, 373–376) also showed that the [gamma distribution](https://bliptext.com/articles/gamma-distribution) is a Pearson type III distribution.\nHence this boundary line for Pearson's type III distribution is known as the gamma line.\n(This can be shown from the fact that the excess kurtosis of the gamma distribution is 6/k and the square of the skewness is 4/k, hence (excess kurtosis − (3/2) skewness2 = 0) is identically satisfied by the gamma distribution regardless of the value of the parameter \"k\").\nPearson later noted that the [chi-squared distribution](https://bliptext.com/articles/chi-squared-distribution) is a special case of Pearson's type III and also shares this boundary line (as it is apparent from the fact that for the [chi-squared distribution](https://bliptext.com/articles/chi-squared-distribution) the excess kurtosis is 12/k and the square of the skewness is 8/k, hence (excess kurtosis − (3/2) skewness2 = 0) is identically satisfied regardless of the value of the parameter \"k\").\nThis is to be expected, since the chi-squared distribution X ~ χ2(k) is a special case of the gamma distribution, with parametrization X ~ Γ(k/2, 1/2) where k is a positive integer that specifies the \"number of degrees of freedom\" of the chi-squared distribution.\nAn example of a beta distribution near the upper boundary (excess kurtosis − (3/2) skewness2 = 0) is given by α = 0.1, β = 1000, for which the ratio (excess kurtosis)/(skewness2) = 1.49835 approaches the upper limit of 1.5 from below.\nAn example of a beta distribution near the lower boundary (excess kurtosis + 2 − skewness2 = 0) is given by α= 0.0001, β = 0.1, for which values the expression (excess kurtosis + 2)/(skewness2) = 1.01621 approaches the lower limit of 1 from above.\nIn the infinitesimal limit for both α and β approaching zero symmetrically, the excess kurtosis reaches its minimum value at −2.\nThis minimum value occurs at the point at which the lower boundary line intersects the vertical axis ([ordinate](https://bliptext.com/articles/ordinate)).\n(However, in Pearson's original chart, the ordinate is kurtosis, instead of excess kurtosis, and it increases downwards rather than upwards).\nValues for the skewness and excess kurtosis below the lower boundary (excess kurtosis + 2 − skewness2 = 0) cannot occur for any distribution, and hence [Karl Pearson](https://bliptext.com/articles/karl-pearson) appropriately called the region below this boundary the \"impossible region\".\nThe boundary for this \"impossible region\" is determined by (symmetric or skewed) bimodal U-shaped distributions for which the parameters α and β approach zero and hence all the probability density is concentrated at the ends: x = 0, 1 with practically nothing in between them.\nSince for α ≈ β ≈ 0 the probability density is concentrated at the two ends x = 0 and x = 1, this \"impossible boundary\" is determined by a [Bernoulli distribution](https://bliptext.com/articles/bernoulli-distribution), where the two only possible outcomes occur with respective probabilities p and q = 1−p.\nFor cases approaching this limit boundary with symmetry α = β, skewness ≈ 0, excess kurtosis ≈ −2 (this is the lowest excess kurtosis possible for any distribution), and the probabilities are p ≈ q ≈ 1/2.\nFor cases approaching this limit boundary with skewness, excess kurtosis ≈ −2 + skewness2, and the probability density is concentrated more at one end than the other end (with practically nothing in between), with probabilities at the left end x = 0 and at the right end x = 1.\n# [Beta distribution α and β parameters vs. excess kurtosis and squared skewness | upload.wikimedia.org/wikipedia/commons/0/06/%28alpha///and///beta%29///Parameter///estimates///vs.///excess///Kurtosis///and///%28squared%29///Skewness///Beta///distribution///-///J.///Rodal.png]\n\n## Symmetry\nAll statements are conditional on α, β > 0:\n\n## Geometry of the probability density function\n\n### Inflection points\nFor certain values of the shape parameters α and β, the [probability density function](https://bliptext.com/articles/probability-density-function) has [inflection points](https://bliptext.com/articles/inflection-points), at which the [curvature](https://bliptext.com/articles/curvature) changes sign.\nThe position of these inflection points can be useful as a measure of the [dispersion](https://bliptext.com/articles/statistical-dispersion) or spread of the distribution.\nDefining the following quantity:\nPoints of inflection occur, depending on the value of the shape parameters α and β, as follows:\nThere are no inflection points in the remaining (symmetric and skewed) regions: U-shaped: (α, β < 1) upside-down-U-shaped: (1 < α < 2, 1 < β < 2), reverse-J-shaped (α < 1, β > 2) or J-shaped: (α > 2, β < 1)\nThe accompanying plots show the inflection point locations (shown vertically, ranging from 0 to 1) versus α and β (the horizontal axes ranging from 0 to 5).\nThere are large cuts at surfaces intersecting the lines α = 1, β = 1, α = 2, and β = 2 because at these values the beta distribution change from 2 modes, to 1 mode to no mode.\n# [Inflection point location versus α and β showing regions with one inflection point | upload.wikimedia.org/wikipedia/commons/e/e0/Inflexion///points///Beta///Distribution///alpha///and///beta///ranging///from///0///to///5///large///ptl///view///-///J.///Rodal.jpg]\n# [Inflection point location versus α and β showing region with two inflection points | upload.wikimedia.org/wikipedia/commons/3/37/Inflexion///points///Beta///Distribution///alpha///and///beta///ranging///from///0///to///5///large///ptr///view///-///J.///Rodal.jpg]\n\n### Shapes\nThe beta density function can take a wide variety of different shapes depending on the values of the two parameters α and β.\nThe ability of the beta distribution to take this great diversity of shapes (using only two parameters) is partly responsible for finding wide application for modeling actual measurements:\n# [beta distribution | upload.wikimedia.org/wikipedia/commons/b/b9/PDF///for///symmetric///beta///distribution///vs.///x///and///alpha%3Dbeta///from///0///to///30///-///J.///Rodal.jpg]\n# [beta distribution | upload.wikimedia.org/wikipedia/commons/4/4e/PDF///for///symmetric///beta///distribution///vs.///x///and///alpha%3Dbeta///from///0///to///2///-///J.///Rodal.jpg]\n# [beta distribution | upload.wikimedia.org/wikipedia/commons/d/de/PDF///for///skewed///beta///distribution///vs.///x///and///beta%3D///2.5///alpha///from///0///to///9///-///J.///Rodal.jpg]\n# [beta distribution | upload.wikimedia.org/wikipedia/commons/d/dc/PDF///for///skewed///beta///distribution///vs.///x///and///beta%3D///5.5///alpha///from///0///to///9///-///J.///Rodal.jpg]\n# [beta distribution | upload.wikimedia.org/wikipedia/commons/8/85/PDF///for///skewed///beta///distribution///vs.///x///and///beta%3D///8///alpha///from///0///to///10///-///J.///Rodal.jpg]\n\n#### Symmetric (α = β)\n\n#### Skewed (α ≠ β)\nThe density function is [skewed](https://bliptext.com/articles/skewness).\nAn interchange of parameter values yields the [mirror image](https://bliptext.com/articles/mirror-image) (the reverse) of the initial curve, some more specific cases:\n\n# Related distributions\n\n## Transformations\n\n## Special and limiting cases\n# [Example of eight realizations of a random walk in one dimension starting at 0: the probability for the time of the last visit to the origin is distributed as Beta(1/2, 1/2) | upload.wikimedia.org/wikipedia/commons/d/da/Random///Walk///example.svg]\n# [Beta(1/2, 1/2): The arcsine distribution probability density was proposed by Harold Jeffreys to represent uncertainty for a Bernoulli or a binomial distribution in Bayesian inference, and is now commonly referred to as Jeffreys prior: p−1/2(1 − p)−1/2. This distribution also appears in several random walk fundamental theorems | upload.wikimedia.org/wikipedia/commons/d/db/Arcsin///density.svg]\n\n## Derived from other distributions\n\n## Combination with other distributions\n\n## Compounding with other distributions\n\n## Generalisations\n\n# Statistical inference\n\n## Parameter estimation\n\n### Method of moments\n\n#### Two unknown parameters\nTwo unknown parameters ( of a beta distribution supported in the [0,1] interval) can be estimated, using the method of moments, with the first two moments (sample mean and sample variance) as follows.\nLet:\nbe the [sample mean](https://bliptext.com/articles/sample-mean) estimate and\nbe the [sample variance](https://bliptext.com/articles/sample-variance) estimate.\nThe [method-of-moments](https://bliptext.com/articles/method-of-moments-statistics) estimates of the parameters are\nWhen the distribution is required over a known interval other than [0, 1] with random variable X, say [a, c] with random variable Y, then replace \\bar{x} with and \\bar{v} with in the above couple of equations for the shape parameters (see the \"Four unknown parameters\" section below), where:\n\n#### Four unknown parameters\nAll four parameters ( of a beta distribution supported in the [a, c] interval, see section [\"Alternative parametrizations, Four parameters\"](https://bliptext.com/articles/beta-distribution)) can be estimated, using the method of moments developed by [Karl Pearson](https://bliptext.com/articles/karl-pearson), by equating sample and population values of the first four central moments (mean, variance, skewness and excess kurtosis).\nThe excess kurtosis was expressed in terms of the square of the skewness, and the sample size ν = α + β, (see previous section [\"Kurtosis\"](https://bliptext.com/articles/beta-distribution)) as follows:\nOne can use this equation to solve for the sample size ν= α + β in terms of the square of the skewness and the excess kurtosis as follows:\nThis is the ratio (multiplied by a factor of 3) between the previously derived limit boundaries for the beta distribution in a space (as originally done by Karl Pearson ) defined with coordinates of the square of the skewness in one axis and the excess kurtosis in the other axis (see ):\nThe case of zero skewness, can be immediately solved because for zero skewness, α = β and hence ν = 2α = 2β, therefore α = β = ν/2\n(Excess kurtosis is negative for the beta distribution with zero skewness, ranging from -2 to 0, so that \\hat{\\nu} -and therefore the sample shape parameters- is positive, ranging from zero when the shape parameters approach zero and the excess kurtosis approaches -2, to infinity when the shape parameters approach infinity and the excess kurtosis approaches zero).\nFor non-zero sample skewness one needs to solve a system of two coupled equations.\nSince the skewness and the excess kurtosis are independent of the parameters, the parameters can be uniquely determined from the sample skewness and the sample excess kurtosis, by solving the coupled equations with two known variables (sample skewness and sample excess kurtosis) and two unknowns (the shape parameters):\nresulting in the following solution:\nWhere one should take the solutions as follows: for (negative) sample skewness < 0, and for (positive) sample skewness > 0.\nThe accompanying plot shows these two solutions as surfaces in a space with horizontal axes of (sample excess kurtosis) and (sample squared skewness) and the shape parameters as the vertical axis.\nThe surfaces are constrained by the condition that the sample excess kurtosis must be bounded by the sample squared skewness as stipulated in the above equation.\nThe two surfaces meet at the right edge defined by zero skewness.\nAlong this right edge, both parameters are equal and the distribution is symmetric U-shaped for α = β < 1, uniform for α = β = 1, upside-down-U-shaped for 1 < α = β < 2 and bell-shaped for α = β > 2.\nThe surfaces also meet at the front (lower) edge defined by \"the impossible boundary\" line (excess kurtosis + 2 - skewness2 = 0).\nAlong this front (lower) boundary both shape parameters approach zero, and the probability density is concentrated more at one end than the other end (with practically nothing in between), with probabilities at the left end x = 0 and at the right end x = 1.\nThe two surfaces become further apart towards the rear edge.\nAt this rear edge the surface parameters are quite different from each other.\nAs remarked, for example, by Bowman and Shenton, sampling in the neighborhood of the line (sample excess kurtosis - (3/2)(sample skewness)2 = 0) (the just-J-shaped portion of the rear edge where blue meets beige), \"is dangerously near to chaos\", because at that line the denominator of the expression above for the estimate ν = α + β becomes zero and hence ν approaches infinity as that line is approached.\nBowman and Shenton write that \"the higher moment parameters (kurtosis and skewness) are extremely fragile (near that line). However, the mean and standard deviation are fairly reliable.\"\nTherefore, the problem is for the case of four parameter estimation for very skewed distributions such that the excess kurtosis approaches (3/2) times the square of the skewness.\nThis boundary line is produced by extremely skewed distributions with very large values of one of the parameters and very small values of the other parameter.\nSee for a numerical example and further comments about this rear edge boundary line (sample excess kurtosis - (3/2)(sample skewness)2 = 0).\nAs remarked by Karl Pearson himself this issue may not be of much practical importance as this trouble arises only for very skewed J-shaped (or mirror-image J-shaped) distributions with very different values of shape parameters that are unlikely to occur much in practice). The usual skewed-bell-shape distributions that occur in practice do not have this parameter estimation problem.\nThe remaining two parameters can be determined using the sample mean and the sample variance using a variety of equations.\nOne alternative is to calculate the support interval range based on the sample variance and the sample kurtosis.\nFor this purpose one can solve, in terms of the range, the equation expressing the excess kurtosis in terms of the sample variance, and the sample size ν (see and ):\nto obtain:\nAnother alternative is to calculate the support interval range based on the sample variance and the sample skewness.\nFor this purpose one can solve, in terms of the range, the equation expressing the squared skewness in terms of the sample variance, and the sample size ν (see section titled \"Skewness\" and \"Alternative parametrizations, four parameters\"):\nto obtain:\nThe remaining parameter can be determined from the sample mean and the previously obtained parameters: :\nand finally,.\nIn the above formulas one may take, for example, as estimates of the sample moments:\nThe estimators G1 for [sample skewness](https://bliptext.com/articles/skewness) and G2 for [sample kurtosis](https://bliptext.com/articles/kurtosis) are used by [DAP](https://bliptext.com/articles/dap-software)/[SAS](https://bliptext.com/articles/sas-system), [PSPP](https://bliptext.com/articles/pspp)/[SPSS](https://bliptext.com/articles/spss), and [Excel](https://bliptext.com/articles/microsoft-excel).\nHowever, they are not used by [BMDP](https://bliptext.com/articles/bmdp) and (according to ) they were not used by [MINITAB](https://bliptext.com/articles/minitab) in 1998.\nActually, Joanes and Gill in their 1998 study concluded that the skewness and kurtosis estimators used in [BMDP](https://bliptext.com/articles/bmdp) and in [MINITAB](https://bliptext.com/articles/minitab) (at that time) had smaller variance and mean-squared error in normal samples, but the skewness and kurtosis estimators used in [DAP](https://bliptext.com/articles/dap-software)/[SAS](https://bliptext.com/articles/sas-system), [PSPP](https://bliptext.com/articles/pspp)/[SPSS](https://bliptext.com/articles/spss), namely G1 and G2, had smaller mean-squared error in samples from a very skewed distribution.\nIt is for this reason that we have spelled out \"sample skewness\", etc., in the above formulas, to make it explicit that the user should choose the best estimator according to the problem at hand, as the best estimator for skewness and kurtosis depends on the amount of skewness (as shown by Joanes and Gill ).\n# [Solutions for parameter estimates vs. (sample) excess Kurtosis and (sample) squared Skewness Beta distribution | upload.wikimedia.org/wikipedia/commons/0/06/%28alpha///and///beta%29///Parameter///estimates///vs.///excess///Kurtosis///and///%28squared%29///Skewness///Beta///distribution///-///J.///Rodal.png]\n\n### Maximum likelihood\n\n#### Two unknown parameters\nAs is also the case for [maximum likelihood](https://bliptext.com/articles/maximum-likelihood) estimates for the [gamma distribution](https://bliptext.com/articles/gamma-distribution), the [maximum likelihood](https://bliptext.com/articles/maximum-likelihood) estimates for the beta distribution do not have a general closed form solution for arbitrary values of the shape parameters.\nIf X1, ..., XN are independent random variables each having a beta distribution, the joint log likelihood function for N [iid](https://bliptext.com/articles/independent-and-identically-distributed-random-variables) observations is:\nFinding the maximum with respect to a shape parameter involves taking the partial derivative with respect to the shape parameter and setting the expression equal to zero yielding the [maximum likelihood](https://bliptext.com/articles/maximum-likelihood) estimator of the shape parameters:\nwhere:\nsince the **[digamma function](https://bliptext.com/articles/digamma-function)** denoted ψ(α) is defined as the [logarithmic derivative](https://bliptext.com/articles/logarithmic-derivative) of the [gamma function](https://bliptext.com/articles/gamma-function):\nTo ensure that the values with zero tangent slope are indeed a maximum (instead of a saddle-point or a minimum) one has to also satisfy the condition that the curvature is negative.\nThis amounts to satisfying that the second partial derivative with respect to the shape parameters is negative\nusing the previous equations, this is equivalent to:\nwhere the **[trigamma function](https://bliptext.com/articles/trigamma-function)**, denoted ψ1(α), is the second of the [polygamma functions](https://bliptext.com/articles/polygamma-function), and is defined as the derivative of the [digamma](https://bliptext.com/articles/digamma) function:\nThese conditions are equivalent to stating that the variances of the logarithmically transformed variables are positive, since:\nTherefore, the condition of negative curvature at a maximum is equivalent to the statements:\nAlternatively, the condition of negative curvature at a maximum is also equivalent to stating that the following [logarithmic derivatives](https://bliptext.com/articles/logarithmic-derivative) of the [geometric means](https://bliptext.com/articles/geometric-mean) GX and G(1−X) are positive, since:\nWhile these slopes are indeed positive, the other slopes are negative:\nThe slopes of the mean and the median with respect to α and β display similar sign behavior.\nFrom the condition that at a maximum, the partial derivative with respect to the shape parameter equals zero, we obtain the following system of coupled [maximum likelihood estimate](https://bliptext.com/articles/maximum-likelihood-estimate) equations (for the average log-likelihoods) that needs to be inverted to obtain the (unknown) shape parameter estimates in terms of the (known) average of logarithms of the samples X1, ..., XN:\nwhere we recognize as the logarithm of the sample [geometric mean](https://bliptext.com/articles/geometric-mean) and as the logarithm of the sample [geometric mean](https://bliptext.com/articles/geometric-mean) based on (1 − X), the mirror-image of X.\nFor, it follows that.\nThese coupled equations containing [digamma functions](https://bliptext.com/articles/digamma-function) of the shape parameter estimates must be solved by numerical methods as done, for example, by Beckman et al. Gnanadesikan et al. give numerical solutions for a few cases.\n[N.L.Johnson](https://bliptext.com/articles/norman-lloyd-johnson) and [S.Kotz](https://bliptext.com/articles/samuel-kotz) suggest that for \"not too small\" shape parameter estimates, the logarithmic approximation to the digamma function may be used to obtain initial values for an iterative solution, since the equations resulting from this approximation can be solved exactly:\nwhich leads to the following solution for the initial values (of the estimate shape parameters in terms of the sample geometric means) for an iterative solution:\nAlternatively, the estimates provided by the method of moments can instead be used as initial values for an iterative solution of the maximum likelihood coupled equations in terms of the digamma functions.\nWhen the distribution is required over a known interval other than [0, 1] with random variable X, say [a, c] with random variable Y, then replace ln(Xi) in the first equation with\nand replace ln(1−Xi) in the second equation with\n(see \"Alternative parametrizations, four parameters\" section below).\nIf one of the shape parameters is known, the problem is considerably simplified.\nThe following [logit](https://bliptext.com/articles/logit) transformation can be used to solve for the unknown shape parameter (for skewed cases such that, otherwise, if symmetric, both -equal- parameters are known when one is known):\nThis [logit](https://bliptext.com/articles/logit) transformation is the logarithm of the transformation that divides the variable X by its mirror-image (X/(1 - X) resulting in the \"inverted beta distribution\" or [beta prime distribution](https://bliptext.com/articles/beta-prime-distribution) (also known as beta distribution of the second kind or [Pearson's Type VI](https://bliptext.com/articles/pearson-distribution)) with support [0, +∞).\nAs previously discussed in the section \"Moments of logarithmically transformed random variables,\" the [logit](https://bliptext.com/articles/logit) transformation, studied by Johnson, extends the finite support [0, 1] based on the original variable X to infinite support in both directions of the real line (−∞, +∞).\nIf, for example, \\hat{\\beta} is known, the unknown parameter can be obtained in terms of the inverse digamma function of the right hand side of this equation:\nIn particular, if one of the shape parameters has a value of unity, for example for (the power function distribution with bounded support [0,1]), using the identity ψ(x + 1) = ψ(x) + 1/x in the equation, the maximum likelihood estimator for the unknown parameter is, exactly:\nThe beta has support [0, 1], therefore, and hence , and therefore\nIn conclusion, the maximum likelihood estimates of the shape parameters of a beta distribution are (in general) a complicated function of the sample [geometric mean](https://bliptext.com/articles/geometric-mean), and of the sample [geometric mean](https://bliptext.com/articles/geometric-mean) based on (1−X), the mirror-image of X.\nOne may ask, if the variance (in addition to the mean) is necessary to estimate two shape parameters with the method of moments, why is the (logarithmic or geometric) variance not necessary to estimate two shape parameters with the maximum likelihood method, for which only the geometric means suffice?\nThe answer is because the mean does not provide as much information as the geometric mean.\nFor a beta distribution with equal shape parameters α = β, the mean is exactly 1/2, regardless of the value of the shape parameters, and therefore regardless of the value of the statistical dispersion (the variance).\nOn the other hand, the geometric mean of a beta distribution with equal shape parameters α = β, depends on the value of the shape parameters, and therefore it contains more information.\nAlso, the geometric mean of a beta distribution does not satisfy the symmetry conditions satisfied by the mean, therefore, by employing both the geometric mean based on X and geometric mean based on (1 − X), the maximum likelihood method is able to provide best estimates for both parameters α = β, without need of employing the variance.\nOne can express the joint log likelihood per N [iid](https://bliptext.com/articles/independent-and-identically-distributed-random-variables) observations in terms of the [sufficient statistics](https://bliptext.com/articles/sufficient-statistic) (the sample geometric means) as follows:\nWe can plot the joint log likelihood per N observations for fixed values of the sample geometric means to see the behavior of the likelihood function as a function of the shape parameters α and β.\nIn such a plot, the shape parameter estimators correspond to the maxima of the likelihood function.\nSee the accompanying graph that shows that all the likelihood functions intersect at α = β = 1, which corresponds to the values of the shape parameters that give the maximum entropy (the maximum entropy occurs for shape parameters equal to unity: the uniform distribution).\nIt is evident from the plot that the likelihood function gives sharp peaks for values of the shape parameter estimators close to zero, but that for values of the shape parameters estimators greater than one, the likelihood function becomes quite flat, with less defined peaks.\nObviously, the maximum likelihood parameter estimation method for the beta distribution becomes less acceptable for larger values of the shape parameter estimators, as the uncertainty in the peak definition increases with the value of the shape parameter estimators.\nOne can arrive at the same conclusion by noticing that the expression for the curvature of the likelihood function is in terms of the geometric variances\nThese variances (and therefore the curvatures) are much larger for small values of the shape parameter α and β.\nHowever, for shape parameter values α, β > 1, the variances (and therefore the curvatures) flatten out.\nEquivalently, this result follows from the [Cramér–Rao bound](https://bliptext.com/articles/cram-r-rao-bound), since the [Fisher information](https://bliptext.com/articles/fisher-information) matrix components for the beta distribution are these logarithmic variances.\nThe [Cramér–Rao bound](https://bliptext.com/articles/cram-r-rao-bound) states that the [variance](https://bliptext.com/articles/variance) of any unbiased estimator of α is bounded by the [reciprocal](https://bliptext.com/articles/multiplicative-inverse) of the [Fisher information](https://bliptext.com/articles/fisher-information):\nso the variance of the estimators increases with increasing α and β, as the logarithmic variances decrease.\nAlso one can express the joint log likelihood per N [iid](https://bliptext.com/articles/independent-and-identically-distributed-random-variables) observations in terms of the [digamma function](https://bliptext.com/articles/digamma-function) expressions for the logarithms of the sample geometric means as follows:\nthis expression is identical to the negative of the cross-entropy (see section on \"Quantities of information (entropy)\").\nTherefore, finding the maximum of the joint log likelihood of the shape parameters, per N [iid](https://bliptext.com/articles/independent-and-identically-distributed-random-variables) observations, is identical to finding the minimum of the cross-entropy for the beta distribution, as a function of the shape parameters.\nwith the cross-entropy defined as follows:\n# [beta distribution | upload.wikimedia.org/wikipedia/commons/5/58/Max///%28Joint///Log///Likelihood///per///N%29///for///Beta///distribution///Maxima///at///alpha%3Dbeta%3D2///-///J.///Rodal.png]\n# [beta distribution | upload.wikimedia.org/wikipedia/commons/1/1d/Max///%28Joint///Log///Likelihood///per///N%29///for///Beta///distribution///Maxima///at///alpha%3Dbeta%3D///0.25%2C0.5%2C1%2C2%2C4%2C6%2C8///-///J.///Rodal.png]\n\n#### Four unknown parameters\nThe procedure is similar to the one followed in the two unknown parameter case.\nIf Y1, ..., YN are independent random variables each having a beta distribution with four parameters, the joint log likelihood function for N [iid](https://bliptext.com/articles/independent-and-identically-distributed-random-variables) observations is:\nFinding the maximum with respect to a shape parameter involves taking the partial derivative with respect to the shape parameter and setting the expression equal to zero yielding the [maximum likelihood](https://bliptext.com/articles/maximum-likelihood) estimator of the shape parameters:\nthese equations can be re-arranged as the following system of four coupled equations (the first two equations are geometric means and the second two equations are the harmonic means) in terms of the maximum likelihood estimates for the four parameters :\nwith sample geometric means:\nThe parameters are embedded inside the geometric mean expressions in a nonlinear way (to the power 1/N).\nThis precludes, in general, a closed form solution, even for an initial value approximation for iteration purposes.\nOne alternative is to use as initial values for iteration the values obtained from the method of moments solution for the four parameter case.\nFurthermore, the expressions for the harmonic means are well-defined only for, which precludes a maximum likelihood solution for shape parameters less than unity in the four-parameter case.\nFisher's information matrix for the four parameter case is [positive-definite](https://bliptext.com/articles/positive-definite-matrix) only for α, β > 2 (for further discussion, see section on Fisher information matrix, four parameter case), for bell-shaped (symmetric or unsymmetric) beta distributions, with inflection points located to either side of the mode.\nThe following Fisher information components (that represent the expectations of the curvature of the log likelihood function) have [singularities](https://bliptext.com/articles/mathematical-singularity) at the following values:\n(for further discussion see section on Fisher information matrix).\nThus, it is not possible to strictly carry on the maximum likelihood estimation for some well known distributions belonging to the four-parameter beta distribution family, like the [uniform distribution](https://bliptext.com/articles/continuous-uniform-distribution) (Beta(1, 1, a, c)), and the [arcsine distribution](https://bliptext.com/articles/arcsine-distribution) (Beta(1/2, 1/2, a, c)).\n[N.L.Johnson](https://bliptext.com/articles/norman-lloyd-johnson) and [S.Kotz](https://bliptext.com/articles/samuel-kotz) ignore the equations for the harmonic means and instead suggest \"If a and c are unknown, and maximum likelihood estimators of a, c, α and β are required, the above procedure (for the two unknown parameter case, with X transformed as X = (Y − a)/(c − a)) can be repeated using a succession of trial values of a and c, until the pair (a, c) for which maximum likelihood (given a and c) is as great as possible, is attained\" (where, for the purpose of clarity, their notation for the parameters has been translated into the present notation).\n\n### Fisher information matrix\nLet a random variable X have a probability density f(x;α).\nThe partial derivative with respect to the (unknown, and to be estimated) parameter α of the log [likelihood function](https://bliptext.com/articles/likelihood-function) is called the [score](https://bliptext.com/articles/score-statistics).\nThe second moment of the score is called the [Fisher information](https://bliptext.com/articles/fisher-information):\nThe [expectation](https://bliptext.com/articles/expected-value) of the [score](https://bliptext.com/articles/score-statistics) is zero, therefore the Fisher information is also the second moment centered on the mean of the [score](https://bliptext.com/articles/score-statistics): the [variance](https://bliptext.com/articles/variance) of the [score](https://bliptext.com/articles/score-statistics).\nIf the log [likelihood function](https://bliptext.com/articles/likelihood-function) is twice differentiable with respect to the parameter α, and under certain regularity conditions, then the Fisher information may also be written as follows (which is often a more convenient form for calculation purposes):\nThus, the Fisher information is the negative of the expectation of the second [derivative](https://bliptext.com/articles/derivative) with respect to the parameter α of the log [likelihood function](https://bliptext.com/articles/likelihood-function).\nTherefore, Fisher information is a measure of the [curvature](https://bliptext.com/articles/curvature) of the log likelihood function of α.\nA low [curvature](https://bliptext.com/articles/[curvature](https://bliptext.com/articles/curvature)) (and therefore high radius of [curvature](https://bliptext.com/articles/[curvature](https://bliptext.com/articles/curvature))), flatter log likelihood function curve has low Fisher information; while a log likelihood function curve with large [curvature](https://bliptext.com/articles/[curvature](https://bliptext.com/articles/curvature)) (and therefore low radius of [curvature](https://bliptext.com/articles/[curvature](https://bliptext.com/articles/curvature))) has high Fisher information.\nWhen the Fisher information matrix is computed at the evaluates of the parameters (\"the observed Fisher information matrix\") it is equivalent to the replacement of the true log likelihood surface by a Taylor's series approximation, taken as far as the quadratic terms.\nThe word information, in the context of Fisher information, refers to information about the parameters.\nInformation such as: estimation, sufficiency and properties of variances of estimators.\nThe [Cramér–Rao bound](https://bliptext.com/articles/cram-r-rao-bound) states that the inverse of the Fisher information is a lower bound on the variance of any [estimator](https://bliptext.com/articles/estimator) of a parameter α:\nThe precision to which one can estimate the estimator of a parameter α is limited by the Fisher Information of the log likelihood function.\nThe Fisher information is a measure of the minimum error involved in estimating a parameter of a distribution and it can be viewed as a measure of the resolving power of an experiment needed to discriminate between two alternative hypothesis of a parameter.\nWhen there are N parameters\nthen the Fisher information takes the form of an N×N [positive semidefinite](https://bliptext.com/articles/positive-semidefinite-matrix) [symmetric matrix](https://bliptext.com/articles/symmetric-matrix), the Fisher information matrix, with typical element:\nUnder certain regularity conditions, the Fisher Information Matrix may also be written in the following form, which is often more convenient for computation:\nWith X1, ..., XN [iid](https://bliptext.com/articles/iid) random variables, an N-dimensional \"box\" can be constructed with sides X1, ..., XN.\nCosta and Cover show that the (Shannon) differential entropy h(X) is related to the volume of the typical set (having the sample entropy close to the true entropy), while the Fisher information is related to the surface of this typical set.\n\n#### Two parameters\nFor X1, ..., XN independent random variables each having a beta distribution parametrized with shape parameters α and β, the joint log likelihood function for N [iid](https://bliptext.com/articles/independent-and-identically-distributed-random-variables) observations is:\ntherefore the joint log likelihood function per N [iid](https://bliptext.com/articles/independent-and-identically-distributed-random-variables) observations is\nFor the two parameter case, the Fisher information has 4 components: 2 diagonal and 2 off-diagonal.\nSince the Fisher information matrix is symmetric, one of these off diagonal components is independent.\nTherefore, the Fisher information matrix has 3 independent components (2 diagonal and 1 off diagonal).\nAryal and Nadarajah calculated Fisher's information matrix for the four-parameter case, from which the two parameter case can be obtained as follows:\nSince the Fisher information matrix is symmetric\nThe Fisher information components are equal to the log geometric variances and log geometric covariance.\nTherefore, they can be expressed as **[trigamma functions](https://bliptext.com/articles/trigamma-function)**, denoted ψ1(α), the second of the [polygamma functions](https://bliptext.com/articles/polygamma-function), defined as the derivative of the [digamma](https://bliptext.com/articles/digamma) function:\nThese derivatives are also derived in the and plots of the log likelihood function are also shown in that section.\ncontains plots and further discussion of the Fisher information matrix components: the log geometric variances and log geometric covariance as a function of the shape parameters α and β.\ncontains formulas for moments of logarithmically transformed random variables.\nImages for the Fisher information components and are shown in.\nThe determinant of Fisher's information matrix is of interest (for example for the calculation of [Jeffreys prior](https://bliptext.com/articles/jeffreys-prior) probability).\nFrom the expressions for the individual components of the Fisher information matrix, it follows that the determinant of Fisher's (symmetric) information matrix for the beta distribution is:\nFrom [Sylvester's criterion](https://bliptext.com/articles/sylvester-s-criterion) (checking whether the diagonal elements are all positive), it follows that the Fisher information matrix for the two parameter case is [positive-definite](https://bliptext.com/articles/positive-definite-matrix) (under the standard condition that the shape parameters are positive α > 0 and β > 0).\n\n#### Four parameters\nIf Y1, ..., YN are independent random variables each having a beta distribution with four parameters: the exponents α and β, and also a (the minimum of the distribution range), and c (the maximum of the distribution range) (section titled \"Alternative parametrizations\", \"Four parameters\"), with [probability density function](https://bliptext.com/articles/probability-density-function):\nthe joint log likelihood function per N [iid](https://bliptext.com/articles/independent-and-identically-distributed-random-variables) observations is:\nFor the four parameter case, the Fisher information has 4*4=16 components.\nIt has 12 off-diagonal components = (4×4 total − 4 diagonal).\nSince the Fisher information matrix is symmetric, half of these components (12/2=6) are independent.\nTherefore, the Fisher information matrix has 6 independent off-diagonal + 4 diagonal = 10 independent components.\nAryal and Nadarajah calculated Fisher's information matrix for the four parameter case as follows:\nIn the above expressions, the use of X instead of Y in the expressions var[ln(X)] = ln(varGX) is not an error.\nThe expressions in terms of the log geometric variances and log geometric covariance occur as functions of the two parameter X ~ Beta(α, β) parametrization because when taking the partial derivatives with respect to the exponents (α, β) in the four parameter case, one obtains the identical expressions as for the two parameter case: these terms of the four parameter Fisher information matrix are independent of the minimum a and maximum c of the distribution's range.\nThe only non-zero term upon double differentiation of the log likelihood function with respect to the exponents α and β is the second derivative of the log of the beta function: ln(B(α, β)).\nThis term is independent of the minimum a and maximum c of the distribution's range.\nDouble differentiation of this term results in trigamma functions.\nThe sections titled \"Maximum likelihood\", \"Two unknown parameters\" and \"Four unknown parameters\" also show this fact.\nThe Fisher information for N [i.i.d.](https://bliptext.com/articles/i-i-d) samples is N times the individual Fisher information (eq. 11.279, page 394 of Cover and Thomas ).\n(Aryal and Nadarajah take a single observation, N = 1, to calculate the following components of the Fisher information, which leads to the same result as considering the derivatives of the log likelihood per N observations. Moreover, below the erroneous expression for in Aryal and Nadarajah has been corrected.)\nThe lower two diagonal entries of the Fisher information matrix, with respect to the parameter a (the minimum of the distribution's range):, and with respect to the parameter c (the maximum of the distribution's range): are only defined for exponents α > 2 and β > 2 respectively.\nThe Fisher information matrix component for the minimum a approaches infinity for exponent α approaching 2 from above, and the Fisher information matrix component for the maximum c approaches infinity for exponent β approaching 2 from above.\nThe Fisher information matrix for the four parameter case does not depend on the individual values of the minimum a and the maximum c, but only on the total range (c − a).\nMoreover, the components of the Fisher information matrix that depend on the range (c − a), depend only through its inverse (or the square of the inverse), such that the Fisher information decreases for increasing range (c − a).\nThe accompanying images show the Fisher information components and.\nImages for the Fisher information components and are shown in.\nAll these Fisher information components look like a basin, with the \"walls\" of the basin being located at low values of the parameters.\nThe following four-parameter-beta-distribution Fisher information components can be expressed in terms of the two-parameter: X ~ Beta(α, β) expectations of the transformed ratio ((1 − X)/X) and of its mirror image (X/(1 − X)), scaled by the range (c − a), which may be helpful for interpretation:\nThese are also the expected values of the \"inverted beta distribution\" or [beta prime distribution](https://bliptext.com/articles/beta-prime-distribution) (also known as beta distribution of the second kind or [Pearson's Type VI](https://bliptext.com/articles/pearson-distribution)) and its mirror image, scaled by the range (c − a).\nAlso, the following Fisher information components can be expressed in terms of the harmonic (1/X) variances or of variances based on the ratio transformed variables ((1-X)/X) as follows:\nSee section \"Moments of linearly transformed, product and inverted random variables\" for these expectations.\nThe determinant of Fisher's information matrix is of interest (for example for the calculation of [Jeffreys prior](https://bliptext.com/articles/jeffreys-prior) probability).\nFrom the expressions for the individual components, it follows that the determinant of Fisher's (symmetric) information matrix for the beta distribution with four parameters is:\nUsing [Sylvester's criterion](https://bliptext.com/articles/sylvester-s-criterion) (checking whether the diagonal elements are all positive), and since diagonal components and have [singularities](https://bliptext.com/articles/mathematical-singularity) at α=2 and β=2 it follows that the Fisher information matrix for the four parameter case is [positive-definite](https://bliptext.com/articles/positive-definite-matrix) for α>2 and β>2.\nSince for α > 2 and β > 2 the beta distribution is (symmetric or unsymmetric) bell shaped, it follows that the Fisher information matrix is positive-definite only for bell-shaped (symmetric or unsymmetric) beta distributions, with inflection points located to either side of the mode.\nThus, important well known distributions belonging to the four-parameter beta distribution family, like the parabolic distribution (Beta(2,2,a,c)) and the [uniform distribution](https://bliptext.com/articles/continuous-uniform-distribution) (Beta(1,1,a,c)) have Fisher information components that blow up (approach infinity) in the four-parameter case (although their Fisher information components are all defined for the two parameter case).\nThe four-parameter [Wigner semicircle distribution](https://bliptext.com/articles/wigner-semicircle-distribution) (Beta(3/2,3/2,a,c)) and [arcsine distribution](https://bliptext.com/articles/arcsine-distribution) (Beta(1/2,1/2,a,c)) have negative Fisher information determinants for the four-parameter case.\n# [Fisher Information I(a,a) for α = β vs range (c − a) and exponent α = β | upload.wikimedia.org/wikipedia/commons/0/08/Fisher///Information///I%28a%2Ca%29///for///alpha%3Dbeta///vs///range///%28c-a%29///and///exponent///alpha%3Dbeta///-///J.///Rodal.png]\n# [Fisher Information I(α,a) for α = β, vs. range (c − a) and exponent α = β | upload.wikimedia.org/wikipedia/commons/6/64/Fisher///Information///I%28alpha%2Ca%29///for///alpha%3Dbeta%2C///vs.///range///%28c///-///a%29///and///exponent///alpha%3Dbeta///-///J.///Rodal.png]\n\n## Bayesian inference\nThe use of Beta distributions in [Bayesian inference](https://bliptext.com/articles/bayesian-inference) is due to the fact that they provide a family of [conjugate prior probability distributions](https://bliptext.com/articles/conjugate-prior-distribution) for [binomial](https://bliptext.com/articles/binomial-distribution) (including [Bernoulli](https://bliptext.com/articles/bernoulli-distribution)) and [geometric distributions](https://bliptext.com/articles/geometric-distribution).\nThe domain of the beta distribution can be viewed as a probability, and in fact the beta distribution is often used to describe the distribution of a probability value p:\nExamples of beta distributions used as prior probabilities to represent ignorance of prior parameter values in Bayesian inference are Beta(1,1), Beta(0,0) and Beta(1/2,1/2).\n# [Beta(1,1): The uniform distribution probability density was proposed by Thomas Bayes to represent ignorance of prior probabilities in Bayesian inference. | upload.wikimedia.org/wikipedia/commons/0/09/Beta%281%2C1%29///Uniform///distribution///-///J.///Rodal.png]\n\n### Rule of succession\nA classic application of the beta distribution is the [rule of succession](https://bliptext.com/articles/rule-of-succession), introduced in the 18th century by [Pierre-Simon Laplace](https://bliptext.com/articles/pierre-simon-laplace) in the course of treating the [sunrise problem](https://bliptext.com/articles/sunrise-problem).\nIt states that, given s successes in n [conditionally independent](https://bliptext.com/articles/conditional-independence) [Bernoulli trials](https://bliptext.com/articles/bernoulli-trial) with probability p, that the estimate of the expected value in the next trial is.\nThis estimate is the expected value of the posterior distribution over p, namely Beta(s+1, n−s+1), which is given by [Bayes' rule](https://bliptext.com/articles/bayes-rule) if one assumes a uniform prior probability over p (i.e., Beta(1, 1)) and then observes that p generated s successes in n trials.\nLaplace's rule of succession has been criticized by prominent scientists.\nR. T. Cox described Laplace's application of the rule of succession to the [sunrise problem](https://bliptext.com/articles/sunrise-problem) ( p. 89) as \"a travesty of the proper use of the principle\".\nKeynes remarks ( Ch.XXX, p. 382) \"indeed this is so foolish a theorem that to entertain it is discreditable\".\nKarl Pearson showed that the probability that the next (n + 1) trials will be successes, after n successes in n trials, is only 50%, which has been considered too low by scientists like Jeffreys and unacceptable as a representation of the scientific process of experimentation to test a proposed scientific law.\nAs pointed out by Jeffreys ( p. 128) (crediting [C. D. Broad](https://bliptext.com/articles/c-d-broad) ) Laplace's rule of succession establishes a high probability of success ((n+1)/(n+2)) in the next trial, but only a moderate probability (50%) that a further sample (n+1) comparable in size will be equally successful.\nAs pointed out by Perks, \"The rule of succession itself is hard to accept. It assigns a probability to the next trial which implies the assumption that the actual run observed is an average run and that we are always at the end of an average run. It would, one would think, be more reasonable to assume that we were in the middle of an average run. Clearly a higher value for both probabilities is necessary if they are to accord with reasonable belief.\"\nThese problems with Laplace's rule of succession motivated Haldane, Perks, Jeffreys and others to search for other forms of prior probability (see the next ).\nAccording to Jaynes, the main problem with the [rule of succession](https://bliptext.com/articles/rule-of-succession) is that it is not valid when s=0 or s=n (see [rule of succession](https://bliptext.com/articles/rule-of-succession), for an analysis of its validity).\n\n### Bayes–Laplace prior probability (Beta(1,1))\nThe beta distribution achieves maximum differential entropy for Beta(1,1): the [uniform](https://bliptext.com/articles/uniform-density) probability density, for which all values in the domain of the distribution have equal density.\nThis uniform distribution Beta(1,1) was suggested (\"with a great deal of doubt\") by [Thomas Bayes](https://bliptext.com/articles/thomas-bayes) as the prior probability distribution to express ignorance about the correct prior distribution.\nThis prior distribution was adopted (apparently, from his writings, with little sign of doubt ) by [Pierre-Simon Laplace](https://bliptext.com/articles/pierre-simon-laplace), and hence it was also known as the \"Bayes–Laplace rule\" or the \"Laplace rule\" of \"[inverse probability](https://bliptext.com/articles/inverse-probability)\" in publications of the first half of the 20th century.\nIn the later part of the 19th century and early part of the 20th century, scientists realized that the assumption of uniform \"equal\" probability density depended on the actual functions (for example whether a linear or a logarithmic scale was most appropriate) and parametrizations used.\nIn particular, the behavior near the ends of distributions with finite support (for example near x = 0, for a distribution with initial support at x = 0) required particular attention.\nKeynes ( Ch.XXX, p. 381) criticized the use of Bayes's uniform prior probability (Beta(1,1)) that all values between zero and one are equiprobable, as follows: \"Thus experience, if it shows anything, shows that there is a very marked clustering of statistical ratios in the neighborhoods of zero and unity, of those for positive theories and for correlations between positive qualities in the neighborhood of zero, and of those for negative theories and for correlations between negative qualities in the neighborhood of unity. \"\n\n### Haldane's prior probability (Beta(0,0))\nThe Beta(0,0) distribution was proposed by [J.B.S. Haldane](https://bliptext.com/articles/j-b-s-haldane), who suggested that the prior probability representing complete uncertainty should be proportional to p−1(1−p)−1.\nThe function p−1(1−p)−1 can be viewed as the limit of the numerator of the beta distribution as both shape parameters approach zero: α, β → 0.\nThe Beta function (in the denominator of the beta distribution) approaches infinity, for both parameters approaching zero, α, β → 0.\nTherefore, p−1(1−p)−1 divided by the Beta function approaches a 2-point [Bernoulli distribution](https://bliptext.com/articles/bernoulli-distribution) with equal probability 1/2 at each end, at 0 and 1, and nothing in between, as α, β → 0.\nA coin-toss: one face of the coin being at 0 and the other face being at 1.\nThe Haldane prior probability distribution Beta(0,0) is an \"[improper prior](https://bliptext.com/articles/improper-prior)\" because its integration (from 0 to 1) fails to strictly converge to 1 due to the singularities at each end.\nHowever, this is not an issue for computing posterior probabilities unless the sample size is very small.\nFurthermore, Zellner points out that on the [log-odds](https://bliptext.com/articles/log-odds) scale, (the [logit](https://bliptext.com/articles/logit) transformation ln(p/1 − p)), the Haldane prior is the uniformly flat prior.\nThe fact that a uniform prior probability on the [logit](https://bliptext.com/articles/logit) transformed variable ln(p/1 − p) (with domain (−∞, ∞)) is equivalent to the Haldane prior on the domain [0, 1] was pointed out by [Harold Jeffreys](https://bliptext.com/articles/harold-jeffreys) in the first edition (1939) of his book Theory of Probability ( p. 123).\nJeffreys writes \"Certainly if we take the Bayes–Laplace rule right up to the extremes we are led to results that do not correspond to anybody's way of thinking. The (Haldane) rule dx/(x(1 − x)) goes too far the other way. It would lead to the conclusion that if a sample is of one type with respect to some property there is a probability 1 that the whole population is of that type.\"\nThe fact that \"uniform\" depends on the parametrization, led Jeffreys to seek a form of prior that would be invariant under different parametrizations.\n# [Beta(0,0): The Haldane prior probability expressing total ignorance about prior information, where we are not even sure whether it is physically possible for an experiment to yield either a success or a failure. As α, β → 0, the beta distribution approaches a two-point Bernoulli distribution with all probability density concentrated at each end, at 0 and 1, and nothing in between. A coin-toss: one face of the coin being at 0 and the other face being at 1. | upload.wikimedia.org/wikipedia/commons/1/1b/Beta///distribution///for///alpha///and///beta///approaching///zero///-///J.///Rodal.png]\n\n### Jeffreys' prior probability (Beta(1/2,1/2) for a Bernoulli or for a binomial distribution)\n[Harold Jeffreys](https://bliptext.com/articles/harold-jeffreys) proposed to use an [uninformative prior](https://bliptext.com/articles/uninformative-prior) probability measure that should be [invariant under reparameterization](https://bliptext.com/articles/parametrization-invariance): proportional to the square root of the [determinant](https://bliptext.com/articles/determinant) of [Fisher's information](https://bliptext.com/articles/fisher-s-information) matrix.\nFor the [Bernoulli distribution](https://bliptext.com/articles/bernoulli-distribution), this can be shown as follows: for a coin that is \"heads\" with probability p ∈ [0, 1] and is \"tails\" with probability 1 − p, for a given (H,T) ∈ {(0,1), (1,0)} the probability is pH(1 − p)T.\nSince T = 1 − H, the [Bernoulli distribution](https://bliptext.com/articles/bernoulli-distribution) is pH(1 − p)1 − H. Considering p as the only parameter, it follows that the log likelihood for the [Bernoulli distribution](https://bliptext.com/articles/bernoulli-distribution) is\nThe Fisher information matrix has only one component (it is a scalar, because there is only one parameter: p), therefore:\nSimilarly, for the [Binomial distribution](https://bliptext.com/articles/binomial-distribution) with n [Bernoulli trials](https://bliptext.com/articles/bernoulli-trials), it can be shown that\nThus, for the [Bernoulli](https://bliptext.com/articles/bernoulli-distribution), and [Binomial distributions](https://bliptext.com/articles/binomial-distribution), [Jeffreys prior](https://bliptext.com/articles/jeffreys-prior) is proportional to, which happens to be proportional to a beta distribution with domain variable x = p, and shape parameters α = β = 1/2, the [arcsine distribution](https://bliptext.com/articles/arcsine-distribution):\nIt will be shown in the next section that the normalizing constant for Jeffreys prior is immaterial to the final result because the normalizing constant cancels out in Bayes theorem for the posterior probability.\nHence Beta(1/2,1/2) is used as the Jeffreys prior for both Bernoulli and binomial distributions.\nAs shown in the next section, when using this expression as a prior probability times the likelihood in [Bayes theorem](https://bliptext.com/articles/bayes-theorem), the posterior probability turns out to be a beta distribution.\nIt is important to realize, however, that Jeffreys prior is proportional to for the Bernoulli and binomial distribution, but not for the beta distribution.\nJeffreys prior for the beta distribution is given by the determinant of Fisher's information for the beta distribution, which, as shown in the is a function of the [trigamma function](https://bliptext.com/articles/trigamma-function) ψ1 of shape parameters α and β as follows:\nAs previously discussed, Jeffreys prior for the Bernoulli and binomial distributions is proportional to the [arcsine distribution](https://bliptext.com/articles/arcsine-distribution) Beta(1/2,1/2), a one-dimensional curve that looks like a basin as a function of the parameter p of the Bernoulli and binomial distributions.\nThe walls of the basin are formed by p approaching the singularities at the ends p → 0 and p → 1, where Beta(1/2,1/2) approaches infinity.\nJeffreys prior for the beta distribution is a 2-dimensional surface (embedded in a three-dimensional space) that looks like a basin with only two of its walls meeting at the corner α = β = 0 (and missing the other two walls) as a function of the shape parameters α and β of the beta distribution.\nThe two adjoining walls of this 2-dimensional surface are formed by the shape parameters α and β approaching the singularities (of the trigamma function) at α, β → 0.\nIt has no walls for α, β → ∞ because in this case the determinant of Fisher's information matrix for the beta distribution approaches zero.\nIt will be shown in the next section that Jeffreys prior probability results in posterior probabilities (when multiplied by the binomial likelihood function) that are intermediate between the posterior probability results of the Haldane and Bayes prior probabilities.\nJeffreys prior may be difficult to obtain analytically, and for some cases it just doesn't exist (even for simple distribution functions like the asymmetric [triangular distribution](https://bliptext.com/articles/triangular-distribution)).\nBerger, Bernardo and Sun, in a 2009 paper defined a reference prior probability distribution that (unlike Jeffreys prior) exists for the asymmetric [triangular distribution](https://bliptext.com/articles/triangular-distribution).\nThey cannot obtain a closed-form expression for their reference prior, but numerical calculations show it to be nearly perfectly fitted by the (proper) prior\nwhere θ is the vertex variable for the asymmetric [triangular distribution](https://bliptext.com/articles/triangular-distribution) with support [0, 1] (corresponding to the following parameter values in Wikipedia's article on the [triangular distribution](https://bliptext.com/articles/triangular-distribution): vertex c = θ, left end a = 0,and right end b = 1).\nBerger et al. also give a heuristic argument that Beta(1/2,1/2) could indeed be the exact Berger–Bernardo–Sun reference prior for the asymmetric triangular distribution.\nTherefore, Beta(1/2,1/2) not only is Jeffreys prior for the Bernoulli and binomial distributions, but also seems to be the Berger–Bernardo–Sun reference prior for the asymmetric triangular distribution (for which the Jeffreys prior does not exist), a distribution used in project management and [PERT](https://bliptext.com/articles/pert) analysis to describe the cost and duration of project tasks.\nClarke and Barron prove that, among continuous positive priors, Jeffreys prior (when it exists) asymptotically maximizes Shannon's [mutual information](https://bliptext.com/articles/mutual-information) between a sample of size n and the parameter, and therefore Jeffreys prior is the most uninformative prior (measuring information as Shannon information).\nThe proof rests on an examination of the [Kullback–Leibler divergence](https://bliptext.com/articles/kullback-leibler-divergence) between probability density functions for [iid](https://bliptext.com/articles/iid) random variables.\n# [Jeffreys prior probability for the beta distribution: the square root of the determinant of Fisher's information matrix: is a function of the trigamma function ψ1 of shape parameters α, β | upload.wikimedia.org/wikipedia/commons/3/30/Jeffreys///prior///probability///for///the///beta///distribution///-///J.///Rodal.png]\n# [beta distribution | upload.wikimedia.org/wikipedia/commons/3/33/Beta///distribution///for///3///different///prior///probability///functions///-///J.///Rodal.png]\n# [beta distribution | upload.wikimedia.org/wikipedia/commons/4/49/Beta///distribution///for///3///different///prior///probability///functions%2C///skewed///case///-///J.///Rodal.png]\n# [beta distribution | upload.wikimedia.org/wikipedia/commons/a/a5/Beta///distribution///for///3///different///prior///probability///functions%2C///skewed///case///sample///size///%3D///%284%2C12%2C40%29///-///J.///Rodal.png]\n\n### Effect of different prior probability choices on the posterior beta distribution\nIf samples are drawn from the population of a random variable X that result in s successes and f failures in n [Bernoulli trials](https://bliptext.com/articles/bernoulli-trial) n = s + f, then the [likelihood function](https://bliptext.com/articles/likelihood-function) for parameters s and f given x = p (the notation x = p in the expressions below will emphasize that the domain x stands for the value of the parameter p in the [binomial distribution](https://bliptext.com/articles/binomial-distribution)), is the following [binomial distribution](https://bliptext.com/articles/binomial-distribution):\nIf beliefs about [prior probability](https://bliptext.com/articles/prior-probability) information are reasonably well approximated by a beta distribution with parameters α Prior and β Prior, then:\nAccording to [Bayes' theorem](https://bliptext.com/articles/bayes-theorem) for a continuous event space, the [posterior probability](https://bliptext.com/articles/posterior-probability) density is given by the product of the [prior probability](https://bliptext.com/articles/prior-probability) and the likelihood function (given the evidence s and f = n − s), normalized so that the area under the curve equals one, as follows:\nThe [binomial coefficient](https://bliptext.com/articles/binomial-coefficient)\nappears both in the numerator and the denominator of the posterior probability, and it does not depend on the integration variable x, hence it cancels out, and it is irrelevant to the final result.\nSimilarly the normalizing factor for the prior probability, the beta function B(αPrior,βPrior) cancels out and it is immaterial to the final result.\nThe same posterior probability result can be obtained if one uses an un-normalized prior\nbecause the normalizing factors all cancel out.\nSeveral authors (including Jeffreys himself) thus use an un-normalized prior formula since the normalization constant cancels out.\nThe numerator of the posterior probability ends up being just the (un-normalized) product of the prior probability and the likelihood function, and the denominator is its integral from zero to one.\nThe beta function in the denominator, B(s + α Prior, n − s + β Prior), appears as a normalization constant to ensure that the total posterior probability integrates to unity.\nThe ratio s/n of the number of successes to the total number of trials is a [sufficient statistic](https://bliptext.com/articles/sufficient-statistic) in the binomial case, which is relevant for the following results.\nFor the **Bayes**' prior probability (Beta(1,1)), the posterior probability is:\nFor the **Jeffreys**' prior probability (Beta(1/2,1/2)), the posterior probability is:\nand for the **Haldane** prior probability (Beta(0,0)), the posterior probability is:\nFrom the above expressions it follows that for s/n = 1/2) all the above three prior probabilities result in the identical location for the posterior probability mean = mode = 1/2. For s/n < 1/2, the mean of the posterior probabilities, using the following priors, are such that: mean for Bayes prior > mean for Jeffreys prior > mean for Haldane prior. For s/n > 1/2 the order of these inequalities is reversed such that the Haldane prior probability results in the largest posterior mean. The Haldane prior probability Beta(0,0) results in a posterior probability density with mean (the expected value for the probability of success in the \"next\" trial) identical to the ratio s/n of the number of successes to the total number of trials. Therefore, the Haldane prior results in a posterior probability with expected value in the next trial equal to the maximum likelihood.\nThe Bayes prior probability Beta(1,1) results in a posterior probability density with mode identical to the ratio s/n (the maximum likelihood).\nIn the case that 100% of the trials have been successful s = n, the Bayes prior probability Beta(1,1) results in a posterior expected value equal to the rule of succession (n + 1)/(n + 2), while the Haldane prior Beta(0,0) results in a posterior expected value of 1 (absolute certainty of success in the next trial).\nJeffreys prior probability results in a posterior expected value equal to (n + 1/2)/(n + 1).\nPerks (p. 303) points out: \"This provides a new rule of succession and expresses a 'reasonable' position to take up, namely, that after an unbroken run of n successes we assume a probability for the next trial equivalent to the assumption that we are about half-way through an average run, i.e. that we expect a failure once in (2n + 2) trials. The Bayes–Laplace rule implies that we are about at the end of an average run or that we expect a failure once in (n + 2) trials. The comparison clearly favours the new result (what is now called Jeffreys prior) from the point of view of 'reasonableness'.\"\nConversely, in the case that 100% of the trials have resulted in failure (s = 0), the Bayes prior probability Beta(1,1) results in a posterior expected value for success in the next trial equal to 1/(n + 2), while the Haldane prior Beta(0,0) results in a posterior expected value of success in the next trial of 0 (absolute certainty of failure in the next trial).\nJeffreys prior probability results in a posterior expected value for success in the next trial equal to (1/2)/(n + 1), which Perks (p. 303) points out: \"is a much more reasonably remote result than the Bayes–Laplace result 1/(n + 2)\".\nJaynes questions (for the uniform prior Beta(1,1)) the use of these formulas for the cases s = 0 or s = n because the integrals do not converge (Beta(1,1) is an improper prior for s = 0 or s = n).\nIn practice, the conditions 0 1 and β > 1).\nThe above estimate for the [mean](https://bliptext.com/articles/mean) is known as the [PERT](https://bliptext.com/articles/pert) [three-point estimation](https://bliptext.com/articles/three-point-estimation) and it is exact for either of the following values of β (for arbitrary α within these ranges):\nor\n[skewness](https://bliptext.com/articles/skewness), and [excess kurtosis](https://bliptext.com/articles/excess-kurtosis)\nThe above estimate for the [standard deviation](https://bliptext.com/articles/standard-deviation) σ(X) = (c − a)/6 is exact for either of the following values of α and β:\nOtherwise, these can be poor approximations for beta distributions with other values of α and β, exhibiting average errors of 40% in the mean and 549% in the variance.\n# [beta distribution | upload.wikimedia.org/wikipedia/commons/0/0b/Beta///Distribution///beta%3Dalpha///from///1.05///to///4.95.svg]\n# [beta distribution | upload.wikimedia.org/wikipedia/commons/b/b9/Beta///Distribution///beta%3D6-alpha///from///1.05///to///4.95.svg]\n# [beta distribution | upload.wikimedia.org/wikipedia/commons/f/fc/Beta///Distribution///for///conjugate///alpha///beta.svg]\n\n# Random variate generation\nIf X and Y are independent, with and then\nSo one algorithm for generating beta variates is to generate, where X is a [gamma variate](https://bliptext.com/articles/gamma-distribution) with parameters (α, 1) and Y is an independent [gamma variate](https://bliptext.com/articles/gamma-distribution) with parameters (β, 1).\nIn fact, here and X+Y are independent, and.\nIf and Z is independent of X and Y, then and is independent of.\nThis shows that the product of independent and random variables is a random variable.\nAlso, the kth [order statistic](https://bliptext.com/articles/order-statistic) of n [uniformly distributed](https://bliptext.com/articles/uniform-distribution-continuous) variates is, so an alternative if α and β are small integers is to generate α + β − 1 uniform variates and choose the α-th smallest.\nAnother way to generate the Beta distribution is by [Pólya urn model](https://bliptext.com/articles/p-lya-urn-model).\nAccording to this method, one start with an \"urn\" with α \"black\" balls and β \"white\" balls and draw uniformly with replacement.\nEvery trial an additional ball is added according to the color of the last ball which was drawn.\nAsymptotically, the proportion of black and white balls will be distributed according to the Beta distribution, where each repetition of the experiment will produce a different value.\nIt is also possible to use the [inverse transform sampling](https://bliptext.com/articles/inverse-transform-sampling).\n\n# Normal approximation to the Beta distribution\nA beta distribution with α ~ β and α and β >> 1 is approximately normal with mean 1/2 and variance 1/(4(2α + 1)).\nIf α ≥ β the normal approximation can be improved by taking the cube-root of the logarithm of the reciprocal of\n\n# History\n[Thomas Bayes](https://bliptext.com/articles/thomas-bayes), in a posthumous paper published in 1763 by [Richard Price](https://bliptext.com/articles/richard-price), obtained a beta distribution as the density of the probability of success in Bernoulli trials (see ), but the paper does not analyze any of the moments of the beta distribution or discuss any of its properties.\nThe first systematic modern discussion of the beta distribution is probably due to [Karl Pearson](https://bliptext.com/articles/karl-pearson).\nIn Pearson's papers the beta distribution is couched as a solution of a differential equation: [Pearson's Type I distribution](https://bliptext.com/articles/pearson-distribution) which it is essentially identical to except for arbitrary shifting and re-scaling (the beta and Pearson Type I distributions can always be equalized by proper choice of parameters).\nIn fact, in several English books and journal articles in the few decades prior to World War II, it was common to refer to the beta distribution as Pearson's Type I distribution.\n[William P. Elderton](https://bliptext.com/articles/william-palin-elderton) in his 1906 monograph \"Frequency curves and correlation\" further analyzes the beta distribution as Pearson's Type I distribution, including a full discussion of the method of moments for the four parameter case, and diagrams of (what Elderton describes as) U-shaped, J-shaped, twisted J-shaped, \"cocked-hat\" shapes, horizontal and angled straight-line cases.\nElderton wrote \"I am chiefly indebted to Professor Pearson, but the indebtedness is of a kind for which it is impossible to offer formal thanks.\"\n[Elderton](https://bliptext.com/articles/william-palin-elderton) in his 1906 monograph provides an impressive amount of information on the beta distribution, including equations for the origin of the distribution chosen to be the mode, as well as for other Pearson distributions: types I through VII.\nElderton also included a number of appendixes, including one appendix (\"II\") on the beta and gamma functions.\nIn later editions, Elderton added equations for the origin of the distribution chosen to be the mean, and analysis of Pearson distributions VIII through XII.\nAs remarked by Bowman and Shenton \"Fisher and Pearson had a difference of opinion in the approach to (parameter) estimation, in particular relating to (Pearson's method of) moments and (Fisher's method of) maximum likelihood in the case of the Beta distribution.\"\nAlso according to Bowman and Shenton, \"the case of a Type I (beta distribution) model being the center of the controversy was pure serendipity. A more difficult model of 4 parameters would have been hard to find.\"\nThe long running public conflict of Fisher with Karl Pearson can be followed in a number of articles in prestigious journals.\nFor example, concerning the estimation of the four parameters for the beta distribution, and Fisher's criticism of Pearson's method of moments as being arbitrary, see Pearson's article \"Method of moments and method of maximum likelihood\" (published three years after his retirement from University College, London, where his position had been divided between Fisher and Pearson's son Egon) in which Pearson writes \"I read (Koshai's paper in the Journal of the Royal Statistical Society, 1933) which as far as I am aware is the only case at present published of the application of Professor Fisher's method. To my astonishment that method depends on first working out the constants of the frequency curve by the (Pearson) Method of Moments and then superposing on it, by what Fisher terms \"the Method of Maximum Likelihood\" a further approximation to obtain, what he holds, he will thus get, 'more efficient values' of the curve constants\".\nDavid and Edwards's treatise on the history of statistics cites the first modern treatment of the beta distribution, in 1911, using the beta designation that has become standard, due to [Corrado Gini](https://bliptext.com/articles/corrado-gini), an Italian [statistician](https://bliptext.com/articles/statistician), [demographer](https://bliptext.com/articles/demography), and [sociologist](https://bliptext.com/articles/sociology), who developed the [Gini coefficient](https://bliptext.com/articles/gini-coefficient).\n[N.L.Johnson](https://bliptext.com/articles/norman-lloyd-johnson) and [S.Kotz](https://bliptext.com/articles/samuel-kotz), in their comprehensive and very informative monograph on leading historical personalities in statistical sciences credit [Corrado Gini](https://bliptext.com/articles/corrado-gini) as \"an early Bayesian...who dealt with the problem of eliciting the parameters of an initial Beta distribution, by singling out techniques which anticipated the advent of the so-called empirical Bayes approach.\"\n# [Karl Pearson analyzed the beta distribution as the solution Type I of Pearson distributions | upload.wikimedia.org/wikipedia/commons/2/21/Karl///Pearson///2.jpg]","current_revision":null,"created_by":"kACnuQmH5k9Nr5VNVEpVeg7XpvUmMctD","created_at":"2025-01-25T18:41:35.768Z","updated_at":"2025-01-25T18:41:35.768Z","search_vector":"'beta':1 'distribut':2","search_content":" \n \n\n# \nIn [probability theory](https://bliptext.com/articles/probability-theory) and [statistics](https://bliptext.com/articles/statistics), the **beta distribution** is a family of continuous [probability distributions](https://bliptext.com/articles/probability-distribution) defined on the interval [0, 1] or (0, 1) in terms of two positive [parameters](https://bliptext.com/articles/statistical-parameter), denoted by alpha (α) and beta (β), that appear as exponents of the variable and its complement to 1, respectively, and control the [shape](https://bliptext.com/articles/shape-parameter) of the distribution.\nThe beta distribution has been applied to model the behavior of [random variables](https://bliptext.com/articles/random-variables) limited to intervals of finite length in a wide variety of disciplines.\nThe beta distribution is a suitable model for the random behavior of percentages and proportions.\nIn [Bayesian inference](https://bliptext.com/articles/bayesian-inference), the beta distribution is the [conjugate prior probability distribution](https://bliptext.com/articles/conjugate-prior-distribution) for the [Bernoulli](https://bliptext.com/articles/bernoulli-distribution), [binomial](https://bliptext.com/articles/binomial-distribution), negative [binomial](https://bliptext.com/articles/binomial-distribution), and [geometric](https://bliptext.com/articles/geometric-distribution) distributions.\nThe formulation of the beta distribution discussed here is also known as the **beta distribution of the first kind**, whereas beta distribution of the second kind is an alternative name for the [beta prime distribution](https://bliptext.com/articles/beta-prime-distribution).\nThe generalization to multiple variables is called a [Dirichlet distribution](https://bliptext.com/articles/dirichlet-distribution).\n\n# Definitions\n\n## Probability density function\nThe [probability density function](https://bliptext.com/articles/probability-density-function) (PDF) of the beta distribution, for or 0 0, is a [power function](https://bliptext.com/articles/power-function) of the variable x and of its [reflection](https://bliptext.com/articles/reflection-formula) (1-x) as follows:\nwhere \\Gamma(z) is the [gamma function](https://bliptext.com/articles/gamma-function).\nThe [beta function](https://bliptext.com/articles/beta-function), \\Beta, is a [normalization constant](https://bliptext.com/articles/normalization-constant) to ensure that the total probability is 1.\nIn the above equations x is a [realization](https://bliptext.com/articles/realization-probability)—an observed value that actually occurred—of a [random variable](https://bliptext.com/articles/random-variable) X.\nSeveral authors, including [N. L. Johnson](https://bliptext.com/articles/norman-lloyd-johnson) and [S. Kotz](https://bliptext.com/articles/samuel-kotz), use the symbols p and q (instead of \\alpha and \\beta) for the shape parameters of the beta distribution, reminiscent of the symbols traditionally used for the parameters of the [Bernoulli distribution](https://bliptext.com/articles/bernoulli-distribution), because the beta distribution approaches the [Bernoulli distribution](https://bliptext.com/articles/bernoulli-distribution) in the limit when both shape parameters \\alpha and \\beta approach the value of zero.\nIn the following, a random variable X beta-distributed with parameters \\alpha and \\beta will be denoted by:\nOther notations for beta-distributed random variables used in the statistical literature are and.\n\n## Cumulative distribution function\nThe [cumulative distribution function](https://bliptext.com/articles/cumulative-distribution-function) is\nwhere is the [incomplete beta function](https://bliptext.com/articles/beta-function) and is the regularized [incomplete beta function](https://bliptext.com/articles/beta-function).\n# [beta distribution | upload.wikimedia.org/wikipedia/commons/d/d3/CDF///for///symmetric///Beta///distribution///vs.///x///and///alpha%3Dbeta///-///J.///Rodal.jpg]\n# [beta distribution | upload.wikimedia.org/wikipedia/commons/9/9f/CDF///for///skewed///Beta///distribution///vs.///x///and///beta%3D///5///alpha///-///J.///Rodal.jpg]\n\n## Alternative parameterizations\n\n### Two parameters\n\n#### Mean and sample size\nThe beta distribution may also be reparameterized in terms of its mean μ (0 0( p. 83).\nDenoting by αPosterior and βPosterior the shape parameters of the posterior beta distribution resulting from applying Bayes theorem to a binomial likelihood function and a prior probability, the interpretation of the addition of both shape parameters to be sample size = ν = α·Posterior + β·Posterior is only correct for the Haldane prior probability Beta(0,0).\nSpecifically, for the Bayes (uniform) prior Beta(1,1) the correct interpretation would be sample size = α·Posterior + β Posterior − 2, or ν = (sample size) + 2.\nFor sample size much larger than 2, the difference between these two priors becomes negligible.\n(See section [Bayesian inference](https://bliptext.com/articles/) for further details.) ν = α + β is referred to as the \"sample size\" of a beta distribution, but one should remember that it is, strictly speaking, the \"sample size\" of a binomial likelihood function only when using a Haldane Beta(0,0) prior in Bayes theorem.\nThis parametrization may be useful in Bayesian parameter estimation.\nFor example, one may administer a test to a number of individuals.\nIf it is assumed that each person's score (0 ≤ θ ≤ 1) is drawn from a population-level beta distribution, then an important statistic is the mean of this population-level distribution.\nThe mean and sample size parameters are related to the shape parameters α and β via\nUnder this [parametrization](https://bliptext.com/articles/statistical-parameter), one may place an [uninformative prior](https://bliptext.com/articles/uninformative-prior) probability over the mean, and a vague prior probability (such as an [exponential](https://bliptext.com/articles/exponential-distribution) or [gamma distribution](https://bliptext.com/articles/gamma-distribution)) over the positive reals for the sample size, if they are independent, and prior data and/or beliefs justify it.\n\n#### Mode and concentration\n[Concave](https://bliptext.com/articles/concave-function) beta distributions, which have, can be parametrized in terms of mode and \"concentration\".\nThe mode,, and concentration, , can be used to define the usual shape parameters as follows:\nFor the mode, 0 2.\nIf instead we define the concentration as, the condition simplifies to c>0 and the beta density at and can be written as:\nwhere c directly scales the [sufficient statistics](https://bliptext.com/articles/sufficient-statistics), \\log(x) and \\log(1-x).\nNote also that in the limit, c\\to0, the distribution becomes flat.\n\n#### Mean and variance\nSolving the system of (coupled) equations given in the above sections as the equations for the mean and the variance of the beta distribution in terms of the original parameters α and β, one can express the α and β parameters in terms of the mean (μ) and the variance (var):\nThis [parametrization](https://bliptext.com/articles/statistical-parameter) of the beta distribution may lead to a more intuitive understanding than the one based on the original parameters α and β.\nFor example, by expressing the mode, skewness, excess kurtosis and differential entropy in terms of the mean and the variance:\n# [325px | upload.wikimedia.org/wikipedia/commons/6/68/Mode///Beta///Distribution///for///both///alpha///and///beta///greater///than///1///-///J.///Rodal.jpg]\n# [325px | upload.wikimedia.org/wikipedia/commons/c/c2/Mode///Beta///Distribution///for///both///alpha///and///beta///greater///than///1///-///another///view///-///J.///Rodal.jpg]\n# [325px | upload.wikimedia.org/wikipedia/commons/a/a4/Skewness///Beta///Distribution///for///mean///full///range///and///variance///between///0.05///and///0.25///-///Dr.///J.///Rodal.jpg]\n# [325px | upload.wikimedia.org/wikipedia/commons/9/95/Skewness///Beta///Distribution///for///mean///and///variance///both///full///range///-///J.///Rodal.jpg]\n# [325px | upload.wikimedia.org/wikipedia/commons/0/01/Excess///Kurtosis///Beta///Distribution///with///mean///for///full///range///and///variance///from///0.05///to///0.25///-///J.///Rodal.jpg]\n# [325px | upload.wikimedia.org/wikipedia/commons/6/66/Excess///Kurtosis///Beta///Distribution///with///mean///and///variance///for///full///range///-///J.///Rodal.jpg]\n# [325px | upload.wikimedia.org/wikipedia/commons/0/02/Differential///Entropy///Beta///Distribution///with///mean///from///0.2///to///0.8///and///variance///from///0.01///to///0.09///-///J.///Rodal.jpg]\n# [325px | upload.wikimedia.org/wikipedia/commons/8/8f/Differential///Entropy///Beta///Distribution///with///mean///from///0.3///to///0.7///and///variance///from///0///to///0.2///-///J.///Rodal.jpg]\n\n### Four parameters\nA beta distribution with the two shape parameters α and β is supported on the range [0,1] or (0,1).\nIt is possible to alter the location and scale of the distribution by introducing two further parameters representing the minimum, a, and maximum c (c > a), values of the distribution, by a linear transformation substituting the non-dimensional variable x in terms of the new variable y (with support [a,c] or (a,c)) and the parameters a and c:\nThe [probability density function](https://bliptext.com/articles/probability-density-function) of the four parameter beta distribution is equal to the two parameter distribution, scaled by the range (c − a), (so that the total area under the density curve equals a probability of one), and with the \"y\" variable shifted and scaled as follows:\nThat a random variable Y is beta-distributed with four parameters α, β, a, and c will be denoted by:\nSome measures of central location are scaled (by (c − a)) and shifted (by a), as follows:\nNote: the geometric mean and harmonic mean cannot be transformed by a linear transformation in the way that the mean, median and mode can.\nThe shape parameters of Y can be written in term of its mean and variance as\nThe statistical dispersion measures are scaled (they do not need to be shifted because they are already centered on the mean) by the range (c − a), linearly for the mean deviation and nonlinearly for the variance:\nSince the [skewness](https://bliptext.com/articles/skewness) and [excess kurtosis](https://bliptext.com/articles/excess-kurtosis) are non-dimensional quantities (as [moments](https://bliptext.com/articles/moment-mathematics) centered on the mean and normalized by the [standard deviation](https://bliptext.com/articles/standard-deviation)), they are independent of the parameters a and c, and therefore equal to the expressions given above in terms of X (with support [0,1] or (0,1)):\n\n# Properties\n\n## Measures of central tendency\n\n### Mode\nThe [mode](https://bliptext.com/articles/mode-statistics) of a beta distributed [random variable](https://bliptext.com/articles/random-variable) X with α, β > 1 is the most likely value of the distribution (corresponding to the peak in the PDF), and is given by the following expression:\nWhen both parameters are less than one (α, β 1 the mode (resp. anti-mode when α, β 2) it is known that the sample mean (as an estimate of location) is not as [robust](https://bliptext.com/articles/robust-statistics) as the sample median, the opposite is the case for uniform or \"U-shaped\" bimodal distributions (with Beta(α, β) such that α, β ≤ 1), with the modes located at the ends of the distribution.\nAs Mosteller and Tukey remark ( p. 207) \"the average of the two extreme observations uses all the sample information. This illustrates how, for short-tailed distributions, the extreme observations should get more weight.\"\nBy contrast, it follows that the median of \"U-shaped\" bimodal distributions with modes at the edge of the distribution (with Beta(α, β) such that α, β ≤ 1) is not robust, as the sample median drops the extreme sample observations from consideration.\nA practical application of this occurs for example for [random walks](https://bliptext.com/articles/random-walk), since the probability for the time of the last visit to the origin in a random walk is distributed as the [arcsine distribution](https://bliptext.com/articles/arcsine-distribution) Beta(1/2, 1/2): the mean of a number of [realizations](https://bliptext.com/articles/realization-probability) of a random walk is a much more robust estimator than the median (which is an inappropriate sample measure estimate in this case).\n# [Mean for beta distribution for 0 ≤ α ≤ 5 and 0 ≤ β ≤ 5 | upload.wikimedia.org/wikipedia/commons/1/12/Mean///Beta///Distribution///for///alpha///and///beta///from///0///to///5///-///J.///Rodal.jpg]\n\n### Geometric mean\nThe logarithm of the [geometric mean](https://bliptext.com/articles/geometric-mean) GX of a distribution with [random variable](https://bliptext.com/articles/random-variable) X is the arithmetic mean of ln(X), or, equivalently, its expected value:\nFor a beta distribution, the expected value integral gives:\nwhere ψ is the [digamma function](https://bliptext.com/articles/digamma-function).\nTherefore, the geometric mean of a beta distribution with shape parameters α and β is the exponential of the digamma functions of α and β as follows:\nWhile for a beta distribution with equal shape parameters α = β, it follows that skewness = 0 and mode = mean = median = 1/2, the geometric mean is less than 1/2: 0 1), the value of α required tends towards β + 1/2 as β → ∞.\nFor example, all these couples have the same geometric mean of 1/2: [β = 1, α = 1.4427], [β = 2, α = 2.46958], [β = 3, α = 3.47943], [β = 4, α = 4.48449], [β = 5, α = 5.48756], [β = 10, α = 10.4938], [β = 100, α = 100.499].\nThe fundamental property of the geometric mean, which can be proven to be false for any other mean, is\nThis makes the geometric mean the only correct mean when averaging normalized results, that is results that are presented as ratios to reference values.\nThis is relevant because the beta distribution is a suitable model for the random behavior of percentages and it is particularly suitable to the statistical modelling of proportions.\nThe geometric mean plays a central role in maximum likelihood estimation, see section \"Parameter estimation, maximum likelihood.\"\nActually, when performing maximum likelihood estimation, besides the [[geometric mean](https://bliptext.com/articles/geometric-mean)](https://bliptext.com/articles/geometric-mean) GX based on the random variable X, also another [[geometric mean](https://bliptext.com/articles/geometric-mean)](https://bliptext.com/articles/geometric-mean) appears naturally: the [[geometric mean](https://bliptext.com/articles/geometric-mean)](https://bliptext.com/articles/geometric-mean) based on the linear transformation ––(1 − X), the mirror-image of X, denoted by G(1−X):\nAlong a line α = β, the following limits apply:\nFollowing are the limits with one parameter finite (non-zero) and the other approaching these limits:\nIt has the following approximate value:\nAlthough both GX and G(1−X) are asymmetric, in the case that both shape parameters are equal α = β, the geometric means are equal: GX = G(1−X).\nThis equality follows from the following symmetry displayed between both geometric means:\n# [(Mean − GeometricMean) for beta distribution versus α and β from 0 to 2, showing the asymmetry between α and β for the geometric mean | upload.wikimedia.org/wikipedia/commons/9/9f/%28Mean///-///GeometricMean%29///for///Beta///Distribution///versus///alpha///and///beta///from///0///to///2///-///J.///Rodal.jpg]\n# [beta distribution | upload.wikimedia.org/wikipedia/commons/1/1e/Geometric///Means///for///Beta///distribution///Purple%3DG%28X%29%2C///Yellow%3DG%281-X%29%2C///smaller///values///alpha///and///beta///in///front///-///J./ //Rodal.jpg]\n# [beta distribution | upload.wikimedia.org/wikipedia/commons/1/1e/Geometric///Means///for///Beta///distribution///Purple%3DG%28X%29%2C///Yellow%3DG%281-X%29%2C///larger///values///alpha///and///beta///in///front///-///J.// /Rodal.jpg]\n\n### Harmonic mean\nThe inverse of the [harmonic mean](https://bliptext.com/articles/harmonic-mean) (HX) of a distribution with [random variable](https://bliptext.com/articles/random-variable) X is the arithmetic mean of 1/X, or, equivalently, its expected value.\nTherefore, the [harmonic mean](https://bliptext.com/articles/harmonic-mean) (HX) of a beta distribution with shape parameters α and β is:\nThe [harmonic mean](https://bliptext.com/articles/harmonic-mean) (HX) of a beta distribution with α 0) (see subsection [Mean and sample size](https://bliptext.com/articles/)):\nUsing this [parametrization](https://bliptext.com/articles/statistical-parameter), one can express the variance in terms of the mean μ and the sample size ν as follows:\nSince ν = α + β > 0, it follows that var(X) 2, as it depends on the linear (absolute) deviations rather than the square deviations from the mean.\nTherefore, the effect of very large deviations from the mean are not as overly weighted.\nUsing [Stirling's approximation](https://bliptext.com/articles/stirling-s-approximation) to the [Gamma function](https://bliptext.com/articles/gamma-function), [N.L.Johnson](https://bliptext.com/articles/norman-lloyd-johnson) and [S.Kotz](https://bliptext.com/articles/samuel-kotz) derived the following approximation for values of the shape parameters greater than unity (the relative error for this approximation is only −3.5% for α = β = 1, and it decreases to zero as α → ∞, β → ∞):\nAt the limit α → ∞, β → ∞, the ratio of the mean absolute deviation to the standard deviation (for the beta distribution) becomes equal to the ratio of the same measures for the normal distribution:.\nFor α = β = 1 this ratio equals, so that from α = β = 1 to α, β → ∞ the ratio decreases by 8.5%.\nFor α = β = 0 the standard deviation is exactly equal to the mean absolute deviation around the mean.\nTherefore, this ratio decreases by 15% from α = β = 0 to α = β = 1, and by 25% from α = β = 0 to α, β → ∞.\nHowever, for skewed beta distributions such that α → 0 or β → 0, the ratio of the standard deviation to the mean absolute deviation approaches infinity (although each of them, individually, approaches zero) because the mean absolute deviation approaches zero faster than the standard deviation.\nUsing the [parametrization](https://bliptext.com/articles/statistical-parameter) in terms of mean μ and sample size ν = α + β > 0:\none can express the mean [absolute deviation](https://bliptext.com/articles/absolute-deviation) around the mean in terms of the mean μ and the sample size ν as follows:\nFor a symmetric distribution, the mean is at the middle of the distribution, μ = 1/2, and therefore:\nAlso, the following limits (with only the noted variable approaching the limit) can be obtained from the above expressions:\n# [Ratio of ,ean abs.dev. to std.dev. for beta distribution with α and β ranging from 0 to 5 | upload.wikimedia.org/wikipedia/commons/6/60/Ratio///of///Mean///Abs.///Dev.///to///Std.Dev.///Beta///distribution///with///alpha///and///beta///from///0///to///5///-///J.///Rodal.jpg]\n# [Ratio of mean abs.dev. to std.dev. for beta distribution with mean 0 ≤ μ ≤ 1 and sample size 0 β.\nUsing the [parametrization](https://bliptext.com/articles/statistical-parameter) in terms of mean μ and sample size ν = α + β:\none can express the skewness in terms of the mean μ and the sample size ν as follows:\nThe skewness can also be expressed just in terms of the variance var and the mean μ as follows:\nThe accompanying plot of skewness as a function of variance and mean shows that maximum variance (1/4) is coupled with zero skewness and the symmetry condition (μ = 1/2), and that maximum skewness (positive or negative infinity) occurs when the mean is located at one end or the other, so that the \"mass\" of the probability distribution is concentrated at the ends (minimum variance).\nThe following expression for the square of the skewness, in terms of the sample size ν = α + β and the variance var, is useful for the method of moments estimation of four parameters:\nThis expression correctly gives a skewness of zero for α = β, since in that case (see ):.\nFor the symmetric case (α = β), skewness = 0 over the whole range, and the following limits apply:\nFor the asymmetric cases (α ≠ β) the following limits (with only the noted variable approaching the limit) can be obtained from the above expressions:\n# [Skewness for Beta Distribution as a function of variance and mean | upload.wikimedia.org/wikipedia/commons/1/1b/Skewness///for///Beta///Distribution///as///a///function///of///the///variance///and///the///mean///-///J.///Rodal.jpg]\n# [325px | upload.wikimedia.org/wikipedia/commons/4/43/Skewness///Beta///Distribution///for///alpha///and///beta///from///1///to///5///-///J.///Rodal.jpg]\n# [325px | upload.wikimedia.org/wikipedia/commons/4/48/Skewness///Beta///Distribution///for///alpha///and///beta///from///.1///to///5///-///J.///Rodal.jpg]\n\n## Kurtosis\nThe beta distribution has been applied in acoustic analysis to assess damage to gears, as the kurtosis of the beta distribution has been reported to be a good indicator of the condition of a gear.\nKurtosis has also been used to distinguish the seismic signal generated by a person's footsteps from other signals.\nAs persons or other targets moving on the ground generate continuous signals in the form of seismic waves, one can separate different targets based on the seismic waves they generate.\nKurtosis is sensitive to impulsive signals, so it's much more sensitive to the signal generated by human footsteps than other signals generated by vehicles, winds, noise, etc.\nUnfortunately, the notation for kurtosis has not been standardized.\nKenney and Keeping use the symbol γ2 for the [excess kurtosis](https://bliptext.com/articles/excess-kurtosis), but [Abramowitz and Stegun](https://bliptext.com/articles/abramowitz-and-stegun) use different terminology.\nTo prevent confusion between kurtosis (the fourth moment centered on the mean, normalized by the square of the variance) and excess kurtosis, when using symbols, they will be spelled out as follows:\nLetting α = β in the above expression one obtains\nTherefore, for symmetric beta distributions, the excess kurtosis is negative, increasing from a minimum value of −2 at the limit as {α = β} → 0, and approaching a maximum value of zero as {α = β} → ∞.\nThe value of −2 is the minimum value of excess kurtosis that any distribution (not just beta distributions, but any distribution of any possible kind) can ever achieve.\nThis minimum value is reached when all the probability density is entirely concentrated at each end x = 0 and x = 1, with nothing in between: a 2-point [Bernoulli distribution](https://bliptext.com/articles/bernoulli-distribution) with equal probability 1/2 at each end (a coin toss: see section below \"Kurtosis bounded by the square of the skewness\" for further discussion).\nThe description of [kurtosis](https://bliptext.com/articles/kurtosis) as a measure of the \"potential outliers\" (or \"potential rare, extreme values\") of the probability distribution, is correct for all distributions including the beta distribution.\nWhen rare, extreme values can occur in the beta distribution, the higher its kurtosis; otherwise, the kurtosis is lower.\nFor α ≠ β, skewed beta distributions, the excess kurtosis can reach unlimited positive values (particularly for α → 0 for finite β, or for β → 0 for finite α) because the side away from the mode will produce occasional extreme values.\nMinimum kurtosis takes place when the mass density is concentrated equally at each end (and therefore the mean is at the center), and there is no probability mass density in between the ends.\nUsing the [parametrization](https://bliptext.com/articles/statistical-parameter) in terms of mean μ and sample size ν = α + β:\none can express the excess kurtosis in terms of the mean μ and the sample size ν as follows:\nThe excess kurtosis can also be expressed in terms of just the following two parameters: the variance var, and the sample size ν as follows:\nand, in terms of the variance var and the mean μ as follows:\nThe plot of excess kurtosis as a function of the variance and the mean shows that the minimum value of the excess kurtosis (−2, which is the minimum possible value for excess kurtosis for any distribution) is intimately coupled with the maximum value of variance (1/4) and the symmetry condition: the mean occurring at the midpoint (μ = 1/2).\nThis occurs for the symmetric case of α = β = 0, with zero skewness.\nAt the limit, this is the 2 point [Bernoulli distribution](https://bliptext.com/articles/bernoulli-distribution) with equal probability 1/2 at each [Dirac delta function](https://bliptext.com/articles/dirac-delta-function) end x = 0 and x = 1 and zero probability everywhere else.\n(A coin toss: one face of the coin being x = 0 and the other face being x = 1.) Variance is maximum because the distribution is bimodal with nothing in between the two modes (spikes) at each end.\nExcess kurtosis is minimum: the probability density \"mass\" is zero at the mean and it is concentrated at the two peaks at each end.\nExcess kurtosis reaches the minimum possible value (for any distribution) when the probability density function has two spikes at each end: it is bi-\"peaky\" with nothing in between them.\nOn the other hand, the plot shows that for extreme skewed cases, where the mean is located near one or the other end (μ = 0 or μ = 1), the variance is close to zero, and the excess kurtosis rapidly approaches infinity when the mean of the distribution approaches either end.\nAlternatively, the excess kurtosis can also be expressed in terms of just the following two parameters: the square of the skewness, and the sample size ν as follows:\nFrom this last expression, one can obtain the same limits published over a century ago by [Karl Pearson](https://bliptext.com/articles/karl-pearson) for the beta distribution (see section below titled \"Kurtosis bounded by the square of the skewness\").\nSetting α + β = ν = 0 in the above expression, one obtains Pearson's lower boundary (values for the skewness and excess kurtosis below the boundary (excess kurtosis + 2 − skewness2 = 0) cannot occur for any distribution, and hence [Karl Pearson](https://bliptext.com/articles/karl-pearson) appropriately called the region below this boundary the \"impossible region\").\nThe limit of α + β = ν → ∞ determines Pearson's upper boundary.\ntherefore:\nValues of ν = α + β such that ν ranges from zero to infinity, 0 1), and the mean in terms of α and β:\nIf 1 1 the absolute distance between the mean and the median is less than 5% of the distance between the maximum and minimum values of x.\nOn the other hand, the absolute distance between the mean and the mode can reach 50% of the distance between the [maximum](https://bliptext.com/articles/maxima-and-minima) and minimum values of x, for the ([pathological](https://bliptext.com/articles/pathological-mathematics)) case of α = 1 and β = 1, for which values the beta distribution approaches the uniform distribution and the [differential entropy](https://bliptext.com/articles/information-entropy) approaches its [maximum](https://bliptext.com/articles/maxima-and-minima) value, and hence [maximum](https://bliptext.com/articles/maxima-and-minima) \"disorder\".\nFor example, for α = 1.0001 and β = 1.00000001:\nwhere PDF stands for the value of the [probability density function](https://bliptext.com/articles/probability-density-function).\n# [325px | upload.wikimedia.org/wikipedia/commons/f/ff/Mean///Median///Difference///-///Beta///Distribution///for///alpha///and///beta///from///1///to///5///-///J.///Rodal.jpg]\n# [325px | upload.wikimedia.org/wikipedia/commons/8/88/Mean///Mode///Difference///-///Beta///Distribution///for///alpha///and///beta///from///1///to///5///-///J.///Rodal.jpg]\n\n### Mean, geometric mean and harmonic mean relationship\nIt is known from the [inequality of arithmetic and geometric means](https://bliptext.com/articles/inequality-of-arithmetic-and-geometric-means) that the geometric mean is lower than the mean.\nSimilarly, the harmonic mean is lower than the geometric mean.\nThe accompanying plot shows that for α = β, both the mean and the median are exactly equal to 1/2, regardless of the value of α = β, and the mode is also equal to 1/2 for α = β > 1, however the geometric and harmonic means are lower than 1/2 and they only approach this value asymptotically as α = β → ∞.\n# [:Mean, median, geometric mean and harmonic mean for beta distribution with 0 0:\n\n## Geometry of the probability density function\n\n### Inflection points\nFor certain values of the shape parameters α and β, the [probability density function](https://bliptext.com/articles/probability-density-function) has [inflection points](https://bliptext.com/articles/inflection-points), at which the [curvature](https://bliptext.com/articles/curvature) changes sign.\nThe position of these inflection points can be useful as a measure of the [dispersion](https://bliptext.com/articles/statistical-dispersion) or spread of the distribution.\nDefining the following quantity:\nPoints of inflection occur, depending on the value of the shape parameters α and β, as follows:\nThere are no inflection points in the remaining (symmetric and skewed) regions: U-shaped: (α, β 2) or J-shaped: (α > 2, β 0.\nThe accompanying plot shows these two solutions as surfaces in a space with horizontal axes of (sample excess kurtosis) and (sample squared skewness) and the shape parameters as the vertical axis.\nThe surfaces are constrained by the condition that the sample excess kurtosis must be bounded by the sample squared skewness as stipulated in the above equation.\nThe two surfaces meet at the right edge defined by zero skewness.\nAlong this right edge, both parameters are equal and the distribution is symmetric U-shaped for α = β 2.\nThe surfaces also meet at the front (lower) edge defined by \"the impossible boundary\" line (excess kurtosis + 2 - skewness2 = 0).\nAlong this front (lower) boundary both shape parameters approach zero, and the probability density is concentrated more at one end than the other end (with practically nothing in between), with probabilities at the left end x = 0 and at the right end x = 1.\nThe two surfaces become further apart towards the rear edge.\nAt this rear edge the surface parameters are quite different from each other.\nAs remarked, for example, by Bowman and Shenton, sampling in the neighborhood of the line (sample excess kurtosis - (3/2)(sample skewness)2 = 0) (the just-J-shaped portion of the rear edge where blue meets beige), \"is dangerously near to chaos\", because at that line the denominator of the expression above for the estimate ν = α + β becomes zero and hence ν approaches infinity as that line is approached.\nBowman and Shenton write that \"the higher moment parameters (kurtosis and skewness) are extremely fragile (near that line). However, the mean and standard deviation are fairly reliable.\"\nTherefore, the problem is for the case of four parameter estimation for very skewed distributions such that the excess kurtosis approaches (3/2) times the square of the skewness.\nThis boundary line is produced by extremely skewed distributions with very large values of one of the parameters and very small values of the other parameter.\nSee for a numerical example and further comments about this rear edge boundary line (sample excess kurtosis - (3/2)(sample skewness)2 = 0).\nAs remarked by Karl Pearson himself this issue may not be of much practical importance as this trouble arises only for very skewed J-shaped (or mirror-image J-shaped) distributions with very different values of shape parameters that are unlikely to occur much in practice). The usual skewed-bell-shape distributions that occur in practice do not have this parameter estimation problem.\nThe remaining two parameters can be determined using the sample mean and the sample variance using a variety of equations.\nOne alternative is to calculate the support interval range based on the sample variance and the sample kurtosis.\nFor this purpose one can solve, in terms of the range, the equation expressing the excess kurtosis in terms of the sample variance, and the sample size ν (see and ):\nto obtain:\nAnother alternative is to calculate the support interval range based on the sample variance and the sample skewness.\nFor this purpose one can solve, in terms of the range, the equation expressing the squared skewness in terms of the sample variance, and the sample size ν (see section titled \"Skewness\" and \"Alternative parametrizations, four parameters\"):\nto obtain:\nThe remaining parameter can be determined from the sample mean and the previously obtained parameters: :\nand finally,.\nIn the above formulas one may take, for example, as estimates of the sample moments:\nThe estimators G1 for [sample skewness](https://bliptext.com/articles/skewness) and G2 for [sample kurtosis](https://bliptext.com/articles/kurtosis) are used by [DAP](https://bliptext.com/articles/dap-software)/[SAS](https://bliptext.com/articles/sas-system), [PSPP](https://bliptext.com/articles/pspp)/[SPSS](https://bliptext.com/articles/spss), and [Excel](https://bliptext.com/articles/microsoft-excel).\nHowever, they are not used by [BMDP](https://bliptext.com/articles/bmdp) and (according to ) they were not used by [MINITAB](https://bliptext.com/articles/minitab) in 1998.\nActually, Joanes and Gill in their 1998 study concluded that the skewness and kurtosis estimators used in [BMDP](https://bliptext.com/articles/bmdp) and in [MINITAB](https://bliptext.com/articles/minitab) (at that time) had smaller variance and mean-squared error in normal samples, but the skewness and kurtosis estimators used in [DAP](https://bliptext.com/articles/dap-software)/[SAS](https://bliptext.com/articles/sas-system), [PSPP](https://bliptext.com/articles/pspp)/[SPSS](https://bliptext.com/articles/spss), namely G1 and G2, had smaller mean-squared error in samples from a very skewed distribution.\nIt is for this reason that we have spelled out \"sample skewness\", etc., in the above formulas, to make it explicit that the user should choose the best estimator according to the problem at hand, as the best estimator for skewness and kurtosis depends on the amount of skewness (as shown by Joanes and Gill ).\n# [Solutions for parameter estimates vs. (sample) excess Kurtosis and (sample) squared Skewness Beta distribution | upload.wikimedia.org/wikipedia/commons/0/06/%28alpha///and///beta%29///Parameter///estimates///vs.///excess///Kurtosis///and///%28squared%29///Skewness///Beta///distribution///-///J.///Rodal.png]\n\n### Maximum likelihood\n\n#### Two unknown parameters\nAs is also the case for [maximum likelihood](https://bliptext.com/articles/maximum-likelihood) estimates for the [gamma distribution](https://bliptext.com/articles/gamma-distribution), the [maximum likelihood](https://bliptext.com/articles/maximum-likelihood) estimates for the beta distribution do not have a general closed form solution for arbitrary values of the shape parameters.\nIf X1, ..., XN are independent random variables each having a beta distribution, the joint log likelihood function for N [iid](https://bliptext.com/articles/independent-and-identically-distributed-random-variables) observations is:\nFinding the maximum with respect to a shape parameter involves taking the partial derivative with respect to the shape parameter and setting the expression equal to zero yielding the [maximum likelihood](https://bliptext.com/articles/maximum-likelihood) estimator of the shape parameters:\nwhere:\nsince the **[di[gamma function](https://bliptext.com/articles/gamma-function)](https://bliptext.com/articles/digamma-function)** denoted ψ(α) is defined as the [logarithmic derivative](https://bliptext.com/articles/logarithmic-derivative) of the [gamma function](https://bliptext.com/articles/gamma-function):\nTo ensure that the values with zero tangent slope are indeed a maximum (instead of a saddle-point or a minimum) one has to also satisfy the condition that the curvature is negative.\nThis amounts to satisfying that the second partial derivative with respect to the shape parameters is negative\nusing the previous equations, this is equivalent to:\nwhere the **[trigamma function](https://bliptext.com/articles/trigamma-function)**, denoted ψ1(α), is the second of the [polygamma functions](https://bliptext.com/articles/polygamma-function), and is defined as the derivative of the [digamma](https://bliptext.com/articles/digamma) function:\nThese conditions are equivalent to stating that the variances of the logarithmically transformed variables are positive, since:\nTherefore, the condition of negative curvature at a maximum is equivalent to the statements:\nAlternatively, the condition of negative curvature at a maximum is also equivalent to stating that the following [logarithmic derivatives](https://bliptext.com/articles/logarithmic-derivative) of the [geometric means](https://bliptext.com/articles/geometric-mean) GX and G(1−X) are positive, since:\nWhile these slopes are indeed positive, the other slopes are negative:\nThe slopes of the mean and the median with respect to α and β display similar sign behavior.\nFrom the condition that at a maximum, the partial derivative with respect to the shape parameter equals zero, we obtain the following system of coupled [maximum likelihood estimate](https://bliptext.com/articles/maximum-likelihood-estimate) equations (for the average log-likelihoods) that needs to be inverted to obtain the (unknown) shape parameter estimates in terms of the (known) average of logarithms of the samples X1, ..., XN:\nwhere we recognize as the logarithm of the sample [[geometric mean](https://bliptext.com/articles/geometric-mean)](https://bliptext.com/articles/geometric-mean) and as the logarithm of the sample [[geometric mean](https://bliptext.com/articles/geometric-mean)](https://bliptext.com/articles/geometric-mean) based on (1 − X), the mirror-image of X.\nFor, it follows that.\nThese coupled equations containing [digamma functions](https://bliptext.com/articles/digamma-function) of the shape parameter estimates must be solved by numerical methods as done, for example, by Beckman et al. Gnanadesikan et al. give numerical solutions for a few cases.\n[N.L.Johnson](https://bliptext.com/articles/norman-lloyd-johnson) and [S.Kotz](https://bliptext.com/articles/samuel-kotz) suggest that for \"not too small\" shape parameter estimates, the logarithmic approximation to the digamma function may be used to obtain initial values for an iterative solution, since the equations resulting from this approximation can be solved exactly:\nwhich leads to the following solution for the initial values (of the estimate shape parameters in terms of the sample geometric means) for an iterative solution:\nAlternatively, the estimates provided by the method of moments can instead be used as initial values for an iterative solution of the maximum likelihood coupled equations in terms of the digamma functions.\nWhen the distribution is required over a known interval other than [0, 1] with random variable X, say [a, c] with random variable Y, then replace ln(Xi) in the first equation with\nand replace ln(1−Xi) in the second equation with\n(see \"Alternative parametrizations, four parameters\" section below).\nIf one of the shape parameters is known, the problem is considerably simplified.\nThe following [logit](https://bliptext.com/articles/logit) transformation can be used to solve for the unknown shape parameter (for skewed cases such that, otherwise, if symmetric, both -equal- parameters are known when one is known):\nThis [logit](https://bliptext.com/articles/logit) transformation is the logarithm of the transformation that divides the variable X by its mirror-image (X/(1 - X) resulting in the \"inverted beta distribution\" or [beta prime distribution](https://bliptext.com/articles/beta-prime-distribution) (also known as beta distribution of the second kind or [Pearson's Type VI](https://bliptext.com/articles/pearson-distribution)) with support [0, +∞).\nAs previously discussed in the section \"Moments of logarithmically transformed random variables,\" the [logit](https://bliptext.com/articles/logit) transformation, studied by Johnson, extends the finite support [0, 1] based on the original variable X to infinite support in both directions of the real line (−∞, +∞).\nIf, for example, \\hat{\\beta} is known, the unknown parameter can be obtained in terms of the inverse digamma function of the right hand side of this equation:\nIn particular, if one of the shape parameters has a value of unity, for example for (the power function distribution with bounded support [0,1]), using the identity ψ(x + 1) = ψ(x) + 1/x in the equation, the maximum likelihood estimator for the unknown parameter is, exactly:\nThe beta has support [0, 1], therefore, and hence , and therefore\nIn conclusion, the maximum likelihood estimates of the shape parameters of a beta distribution are (in general) a complicated function of the sample [[geometric mean](https://bliptext.com/articles/geometric-mean)](https://bliptext.com/articles/geometric-mean), and of the sample [[geometric mean](https://bliptext.com/articles/geometric-mean)](https://bliptext.com/articles/geometric-mean) based on (1−X), the mirror-image of X.\nOne may ask, if the variance (in addition to the mean) is necessary to estimate two shape parameters with the method of moments, why is the (logarithmic or geometric) variance not necessary to estimate two shape parameters with the maximum likelihood method, for which only the geometric means suffice?\nThe answer is because the mean does not provide as much information as the geometric mean.\nFor a beta distribution with equal shape parameters α = β, the mean is exactly 1/2, regardless of the value of the shape parameters, and therefore regardless of the value of the statistical dispersion (the variance).\nOn the other hand, the geometric mean of a beta distribution with equal shape parameters α = β, depends on the value of the shape parameters, and therefore it contains more information.\nAlso, the geometric mean of a beta distribution does not satisfy the symmetry conditions satisfied by the mean, therefore, by employing both the geometric mean based on X and geometric mean based on (1 − X), the maximum likelihood method is able to provide best estimates for both parameters α = β, without need of employing the variance.\nOne can express the joint log likelihood per N [iid](https://bliptext.com/articles/independent-and-identically-distributed-random-variables) observations in terms of the [sufficient statistics](https://bliptext.com/articles/sufficient-statistic) (the sample geometric means) as follows:\nWe can plot the joint log likelihood per N observations for fixed values of the sample geometric means to see the behavior of the likelihood function as a function of the shape parameters α and β.\nIn such a plot, the shape parameter estimators correspond to the maxima of the likelihood function.\nSee the accompanying graph that shows that all the likelihood functions intersect at α = β = 1, which corresponds to the values of the shape parameters that give the maximum entropy (the maximum entropy occurs for shape parameters equal to unity: the uniform distribution).\nIt is evident from the plot that the likelihood function gives sharp peaks for values of the shape parameter estimators close to zero, but that for values of the shape parameters estimators greater than one, the likelihood function becomes quite flat, with less defined peaks.\nObviously, the maximum likelihood parameter estimation method for the beta distribution becomes less acceptable for larger values of the shape parameter estimators, as the uncertainty in the peak definition increases with the value of the shape parameter estimators.\nOne can arrive at the same conclusion by noticing that the expression for the curvature of the likelihood function is in terms of the geometric variances\nThese variances (and therefore the curvatures) are much larger for small values of the shape parameter α and β.\nHowever, for shape parameter values α, β > 1, the variances (and therefore the curvatures) flatten out.\nEquivalently, this result follows from the [Cramér–Rao bound](https://bliptext.com/articles/cram-r-rao-bound), since the [Fisher information](https://bliptext.com/articles/fisher-information) matrix components for the beta distribution are these logarithmic variances.\nThe [Cramér–Rao bound](https://bliptext.com/articles/cram-r-rao-bound) states that the [variance](https://bliptext.com/articles/variance) of any unbiased estimator of α is bounded by the [reciprocal](https://bliptext.com/articles/multiplicative-inverse) of the [Fisher information](https://bliptext.com/articles/fisher-information):\nso the variance of the estimators increases with increasing α and β, as the logarithmic variances decrease.\nAlso one can express the joint log likelihood per N [iid](https://bliptext.com/articles/independent-and-identically-distributed-random-variables) observations in terms of the [digamma function](https://bliptext.com/articles/digamma-function) expressions for the logarithms of the sample geometric means as follows:\nthis expression is identical to the negative of the cross-entropy (see section on \"Quantities of information (entropy)\").\nTherefore, finding the maximum of the joint log likelihood of the shape parameters, per N [iid](https://bliptext.com/articles/independent-and-identically-distributed-random-variables) observations, is identical to finding the minimum of the cross-entropy for the beta distribution, as a function of the shape parameters.\nwith the cross-entropy defined as follows:\n# [beta distribution | upload.wikimedia.org/wikipedia/commons/5/58/Max///%28Joint///Log///Likelihood///per///N%29///for///Beta///distribution///Maxima///at///alpha%3Dbeta%3D2///-///J.///Rodal.png]\n# [beta distribution | upload.wikimedia.org/wikipedia/commons/1/1d/Max///%28Joint///Log///Likelihood///per///N%29///for///Beta///distribution///Maxima///at///alpha%3Dbeta%3D///0.25%2C0.5%2C1%2C2%2C4%2C6%2C8///-///J.///Rodal .png]\n\n#### Four unknown parameters\nThe procedure is similar to the one followed in the two unknown parameter case.\nIf Y1, ..., YN are independent random variables each having a beta distribution with four parameters, the joint log likelihood function for N [iid](https://bliptext.com/articles/independent-and-identically-distributed-random-variables) observations is:\nFinding the maximum with respect to a shape parameter involves taking the partial derivative with respect to the shape parameter and setting the expression equal to zero yielding the [maximum likelihood](https://bliptext.com/articles/maximum-likelihood) estimator of the shape parameters:\nthese equations can be re-arranged as the following system of four coupled equations (the first two equations are geometric means and the second two equations are the harmonic means) in terms of the maximum likelihood estimates for the four parameters :\nwith sample geometric means:\nThe parameters are embedded inside the geometric mean expressions in a nonlinear way (to the power 1/N).\nThis precludes, in general, a closed form solution, even for an initial value approximation for iteration purposes.\nOne alternative is to use as initial values for iteration the values obtained from the method of moments solution for the four parameter case.\nFurthermore, the expressions for the harmonic means are well-defined only for, which precludes a maximum likelihood solution for shape parameters less than unity in the four-parameter case.\nFisher's information matrix for the four parameter case is [positive-definite](https://bliptext.com/articles/positive-definite-matrix) only for α, β > 2 (for further discussion, see section on Fisher information matrix, four parameter case), for bell-shaped (symmetric or unsymmetric) beta distributions, with inflection points located to either side of the mode.\nThe following Fisher information components (that represent the expectations of the curvature of the log likelihood function) have [singularities](https://bliptext.com/articles/mathematical-singularity) at the following values:\n(for further discussion see section on Fisher information matrix).\nThus, it is not possible to strictly carry on the maximum likelihood estimation for some well known distributions belonging to the four-parameter beta distribution family, like the [uniform distribution](https://bliptext.com/articles/continuous-uniform-distribution) (Beta(1, 1, a, c)), and the [arcsine distribution](https://bliptext.com/articles/arcsine-distribution) (Beta(1/2, 1/2, a, c)).\n[N.L.Johnson](https://bliptext.com/articles/norman-lloyd-johnson) and [S.Kotz](https://bliptext.com/articles/samuel-kotz) ignore the equations for the harmonic means and instead suggest \"If a and c are unknown, and maximum likelihood estimators of a, c, α and β are required, the above procedure (for the two unknown parameter case, with X transformed as X = (Y − a)/(c − a)) can be repeated using a succession of trial values of a and c, until the pair (a, c) for which maximum likelihood (given a and c) is as great as possible, is attained\" (where, for the purpose of clarity, their notation for the parameters has been translated into the present notation).\n\n### Fisher information matrix\nLet a random variable X have a probability density f(x;α).\nThe partial derivative with respect to the (unknown, and to be estimated) parameter α of the log [likelihood function](https://bliptext.com/articles/likelihood-function) is called the [score](https://bliptext.com/articles/score-statistics).\nThe second moment of the score is called the [Fisher information](https://bliptext.com/articles/fisher-information):\nThe [expectation](https://bliptext.com/articles/expected-value) of the [score](https://bliptext.com/articles/score-statistics) is zero, therefore the Fisher information is also the second moment centered on the mean of the [score](https://bliptext.com/articles/score-statistics): the [variance](https://bliptext.com/articles/variance) of the [score](https://bliptext.com/articles/score-statistics).\nIf the log [likelihood function](https://bliptext.com/articles/likelihood-function) is twice differentiable with respect to the parameter α, and under certain regularity conditions, then the Fisher information may also be written as follows (which is often a more convenient form for calculation purposes):\nThus, the Fisher information is the negative of the expectation of the second [derivative](https://bliptext.com/articles/derivative) with respect to the parameter α of the log [likelihood function](https://bliptext.com/articles/likelihood-function).\nTherefore, Fisher information is a measure of the [curvature](https://bliptext.com/articles/curvature) of the log likelihood function of α.\nA low [[curvature](https://bliptext.com/articles/curvature)](https://bliptext.com/articles/[curvature](https://bliptext.com/articles/curvature)) (and therefore high radius of [[curvature](https://bliptext.com/articles/curvature)](https://bliptext.com/articles/[curvature](https://bliptext.com/articles/curvature))), flatter log likelihood function curve has low Fisher information; while a log likelihood function curve with large [[curvature](https://bliptext.com/articles/curvature)](https://bliptext.com/articles/[curvature](https://bliptext.com/articles/curvature)) (and therefore low radius of [[curvature](https://bliptext.com/articles/curvature)](https://bliptext.com/articles/[curvature](https://bliptext.com/articles/curvature))) has high Fisher information.\nWhen the Fisher information matrix is computed at the evaluates of the parameters (\"the observed Fisher information matrix\") it is equivalent to the replacement of the true log likelihood surface by a Taylor's series approximation, taken as far as the quadratic terms.\nThe word information, in the context of Fisher information, refers to information about the parameters.\nInformation such as: estimation, sufficiency and properties of variances of estimators.\nThe [Cramér–Rao bound](https://bliptext.com/articles/cram-r-rao-bound) states that the inverse of the Fisher information is a lower bound on the variance of any [estimator](https://bliptext.com/articles/estimator) of a parameter α:\nThe precision to which one can estimate the estimator of a parameter α is limited by the Fisher Information of the log likelihood function.\nThe Fisher information is a measure of the minimum error involved in estimating a parameter of a distribution and it can be viewed as a measure of the resolving power of an experiment needed to discriminate between two alternative hypothesis of a parameter.\nWhen there are N parameters\nthen the Fisher information takes the form of an N×N [positive semidefinite](https://bliptext.com/articles/positive-semidefinite-matrix) [symmetric matrix](https://bliptext.com/articles/symmetric-matrix), the Fisher information matrix, with typical element:\nUnder certain regularity conditions, the Fisher Information Matrix may also be written in the following form, which is often more convenient for computation:\nWith X1, ..., XN [iid](https://bliptext.com/articles/iid) random variables, an N-dimensional \"box\" can be constructed with sides X1, ..., XN.\nCosta and Cover show that the (Shannon) differential entropy h(X) is related to the volume of the typical set (having the sample entropy close to the true entropy), while the Fisher information is related to the surface of this typical set.\n\n#### Two parameters\nFor X1, ..., XN independent random variables each having a beta distribution parametrized with shape parameters α and β, the joint log likelihood function for N [iid](https://bliptext.com/articles/independent-and-identically-distributed-random-variables) observations is:\ntherefore the joint log likelihood function per N [iid](https://bliptext.com/articles/independent-and-identically-distributed-random-variables) observations is\nFor the two parameter case, the Fisher information has 4 components: 2 diagonal and 2 off-diagonal.\nSince the Fisher information matrix is symmetric, one of these off diagonal components is independent.\nTherefore, the Fisher information matrix has 3 independent components (2 diagonal and 1 off diagonal).\nAryal and Nadarajah calculated Fisher's information matrix for the four-parameter case, from which the two parameter case can be obtained as follows:\nSince the Fisher information matrix is symmetric\nThe Fisher information components are equal to the log geometric variances and log geometric covariance.\nTherefore, they can be expressed as **[trigamma functions](https://bliptext.com/articles/trigamma-function)**, denoted ψ1(α), the second of the [polygamma functions](https://bliptext.com/articles/polygamma-function), defined as the derivative of the [digamma](https://bliptext.com/articles/digamma) function:\nThese derivatives are also derived in the and plots of the log likelihood function are also shown in that section.\ncontains plots and further discussion of the Fisher information matrix components: the log geometric variances and log geometric covariance as a function of the shape parameters α and β.\ncontains formulas for moments of logarithmically transformed random variables.\nImages for the Fisher information components and are shown in.\nThe determinant of Fisher's information matrix is of interest (for example for the calculation of [Jeffreys prior](https://bliptext.com/articles/jeffreys-prior) probability).\nFrom the expressions for the individual components of the Fisher information matrix, it follows that the determinant of Fisher's (symmetric) information matrix for the beta distribution is:\nFrom [Sylvester's criterion](https://bliptext.com/articles/sylvester-s-criterion) (checking whether the diagonal elements are all positive), it follows that the Fisher information matrix for the two parameter case is [positive-definite](https://bliptext.com/articles/positive-definite-matrix) (under the standard condition that the shape parameters are positive α > 0 and β > 0).\n\n#### Four parameters\nIf Y1, ..., YN are independent random variables each having a beta distribution with four parameters: the exponents α and β, and also a (the minimum of the distribution range), and c (the maximum of the distribution range) (section titled \"Alternative parametrizations\", \"Four parameters\"), with [probability density function](https://bliptext.com/articles/probability-density-function):\nthe joint log likelihood function per N [iid](https://bliptext.com/articles/independent-and-identically-distributed-random-variables) observations is:\nFor the four parameter case, the Fisher information has 4*4=16 components.\nIt has 12 off-diagonal components = (4×4 total − 4 diagonal).\nSince the Fisher information matrix is symmetric, half of these components (12/2=6) are independent.\nTherefore, the Fisher information matrix has 6 independent off-diagonal + 4 diagonal = 10 independent components.\nAryal and Nadarajah calculated Fisher's information matrix for the four parameter case as follows:\nIn the above expressions, the use of X instead of Y in the expressions var[ln(X)] = ln(varGX) is not an error.\nThe expressions in terms of the log geometric variances and log geometric covariance occur as functions of the two parameter X ~ Beta(α, β) parametrization because when taking the partial derivatives with respect to the exponents (α, β) in the four parameter case, one obtains the identical expressions as for the two parameter case: these terms of the four parameter Fisher information matrix are independent of the minimum a and maximum c of the distribution's range.\nThe only non-zero term upon double differentiation of the log likelihood function with respect to the exponents α and β is the second derivative of the log of the beta function: ln(B(α, β)).\nThis term is independent of the minimum a and maximum c of the distribution's range.\nDouble differentiation of this term results in trigamma functions.\nThe sections titled \"Maximum likelihood\", \"Two unknown parameters\" and \"Four unknown parameters\" also show this fact.\nThe Fisher information for N [i.i.d.](https://bliptext.com/articles/i-i-d) samples is N times the individual Fisher information (eq. 11.279, page 394 of Cover and Thomas ).\n(Aryal and Nadarajah take a single observation, N = 1, to calculate the following components of the Fisher information, which leads to the same result as considering the derivatives of the log likelihood per N observations. Moreover, below the erroneous expression for in Aryal and Nadarajah has been corrected.)\nThe lower two diagonal entries of the Fisher information matrix, with respect to the parameter a (the minimum of the distribution's range):, and with respect to the parameter c (the maximum of the distribution's range): are only defined for exponents α > 2 and β > 2 respectively.\nThe Fisher information matrix component for the minimum a approaches infinity for exponent α approaching 2 from above, and the Fisher information matrix component for the maximum c approaches infinity for exponent β approaching 2 from above.\nThe Fisher information matrix for the four parameter case does not depend on the individual values of the minimum a and the maximum c, but only on the total range (c − a).\nMoreover, the components of the Fisher information matrix that depend on the range (c − a), depend only through its inverse (or the square of the inverse), such that the Fisher information decreases for increasing range (c − a).\nThe accompanying images show the Fisher information components and.\nImages for the Fisher information components and are shown in.\nAll these Fisher information components look like a basin, with the \"walls\" of the basin being located at low values of the parameters.\nThe following four-parameter-beta-distribution Fisher information components can be expressed in terms of the two-parameter: X ~ Beta(α, β) expectations of the transformed ratio ((1 − X)/X) and of its mirror image (X/(1 − X)), scaled by the range (c − a), which may be helpful for interpretation:\nThese are also the expected values of the \"inverted beta distribution\" or [beta prime distribution](https://bliptext.com/articles/beta-prime-distribution) (also known as beta distribution of the second kind or [Pearson's Type VI](https://bliptext.com/articles/pearson-distribution)) and its mirror image, scaled by the range (c − a).\nAlso, the following Fisher information components can be expressed in terms of the harmonic (1/X) variances or of variances based on the ratio transformed variables ((1-X)/X) as follows:\nSee section \"Moments of linearly transformed, product and inverted random variables\" for these expectations.\nThe determinant of Fisher's information matrix is of interest (for example for the calculation of [Jeffreys prior](https://bliptext.com/articles/jeffreys-prior) probability).\nFrom the expressions for the individual components, it follows that the determinant of Fisher's (symmetric) information matrix for the beta distribution with four parameters is:\nUsing [Sylvester's criterion](https://bliptext.com/articles/sylvester-s-criterion) (checking whether the diagonal elements are all positive), and since diagonal components and have [singularities](https://bliptext.com/articles/mathematical-singularity) at α=2 and β=2 it follows that the Fisher information matrix for the four parameter case is [positive-definite](https://bliptext.com/articles/positive-definite-matrix) for α>2 and β>2.\nSince for α > 2 and β > 2 the beta distribution is (symmetric or unsymmetric) bell shaped, it follows that the Fisher information matrix is positive-definite only for bell-shaped (symmetric or unsymmetric) beta distributions, with inflection points located to either side of the mode.\nThus, important well known distributions belonging to the four-parameter beta distribution family, like the parabolic distribution (Beta(2,2,a,c)) and the [uniform distribution](https://bliptext.com/articles/continuous-uniform-distribution) (Beta(1,1,a,c)) have Fisher information components that blow up (approach infinity) in the four-parameter case (although their Fisher information components are all defined for the two parameter case).\nThe four-parameter [Wigner semicircle distribution](https://bliptext.com/articles/wigner-semicircle-distribution) (Beta(3/2,3/2,a,c)) and [arcsine distribution](https://bliptext.com/articles/arcsine-distribution) (Beta(1/2,1/2,a,c)) have negative Fisher information determinants for the four-parameter case.\n# [Fisher Information I(a,a) for α = β vs range (c − a) and exponent α = β | upload.wikimedia.org/wikipedia/commons/0/08/Fisher///Information///I%28a%2Ca%29///for///alpha%3Dbeta///vs///range///%28c-a%29///and///exponent///alpha%3Dbeta///-///J.///Rodal.png]\n# [Fisher Information I(α,a) for α = β, vs. range (c − a) and exponent α = β | upload.wikimedia.org/wikipedia/commons/6/64/Fisher///Information///I%28alpha%2Ca%29///for///alpha%3Dbeta%2C///vs.///range///%28c///-///a%29///and///exponent///alpha%3Dbeta///-///J.///Rodal.png]\n\n## Bayesian inference\nThe use of Beta distributions in [Bayesian inference](https://bliptext.com/articles/bayesian-inference) is due to the fact that they provide a family of [conjugate prior probability distributions](https://bliptext.com/articles/conjugate-prior-distribution) for [binomial](https://bliptext.com/articles/binomial-distribution) (including [Bernoulli](https://bliptext.com/articles/bernoulli-distribution)) and [geometric distributions](https://bliptext.com/articles/geometric-distribution).\nThe domain of the beta distribution can be viewed as a probability, and in fact the beta distribution is often used to describe the distribution of a probability value p:\nExamples of beta distributions used as prior probabilities to represent ignorance of prior parameter values in Bayesian inference are Beta(1,1), Beta(0,0) and Beta(1/2,1/2).\n# [Beta(1,1): The uniform distribution probability density was proposed by Thomas Bayes to represent ignorance of prior probabilities in Bayesian inference. | upload.wikimedia.org/wikipedia/commons/0/09/Beta%281%2C1%29///Uniform///distribution///-///J.///Rodal.png]\n\n### Rule of succession\nA classic application of the beta distribution is the [rule of succession](https://bliptext.com/articles/rule-of-succession), introduced in the 18th century by [Pierre-Simon Laplace](https://bliptext.com/articles/pierre-simon-laplace) in the course of treating the [sunrise problem](https://bliptext.com/articles/sunrise-problem).\nIt states that, given s successes in n [conditionally independent](https://bliptext.com/articles/conditional-independence) [Bernoulli trials](https://bliptext.com/articles/bernoulli-trial) with probability p, that the estimate of the expected value in the next trial is.\nThis estimate is the expected value of the posterior distribution over p, namely Beta(s+1, n−s+1), which is given by [Bayes' rule](https://bliptext.com/articles/bayes-rule) if one assumes a uniform prior probability over p (i.e., Beta(1, 1)) and then observes that p generated s successes in n trials.\nLaplace's rule of succession has been criticized by prominent scientists.\nR. T. Cox described Laplace's application of the rule of succession to the [sunrise problem](https://bliptext.com/articles/sunrise-problem) ( p. 89) as \"a travesty of the proper use of the principle\".\nKeynes remarks ( Ch.XXX, p. 382) \"indeed this is so foolish a theorem that to entertain it is discreditable\".\nKarl Pearson showed that the probability that the next (n + 1) trials will be successes, after n successes in n trials, is only 50%, which has been considered too low by scientists like Jeffreys and unacceptable as a representation of the scientific process of experimentation to test a proposed scientific law.\nAs pointed out by Jeffreys ( p. 128) (crediting [C. D. Broad](https://bliptext.com/articles/c-d-broad) ) Laplace's rule of succession establishes a high probability of success ((n+1)/(n+2)) in the next trial, but only a moderate probability (50%) that a further sample (n+1) comparable in size will be equally successful.\nAs pointed out by Perks, \"The rule of succession itself is hard to accept. It assigns a probability to the next trial which implies the assumption that the actual run observed is an average run and that we are always at the end of an average run. It would, one would think, be more reasonable to assume that we were in the middle of an average run. Clearly a higher value for both probabilities is necessary if they are to accord with reasonable belief.\"\nThese problems with Laplace's rule of succession motivated Haldane, Perks, Jeffreys and others to search for other forms of prior probability (see the next ).\nAccording to Jaynes, the main problem with the [rule of succession](https://bliptext.com/articles/rule-of-succession) is that it is not valid when s=0 or s=n (see [rule of succession](https://bliptext.com/articles/rule-of-succession), for an analysis of its validity).\n\n### Bayes–Laplace prior probability (Beta(1,1))\nThe beta distribution achieves maximum differential entropy for Beta(1,1): the [uniform](https://bliptext.com/articles/uniform-density) probability density, for which all values in the domain of the distribution have equal density.\nThis uniform distribution Beta(1,1) was suggested (\"with a great deal of doubt\") by [Thomas Bayes](https://bliptext.com/articles/thomas-bayes) as the prior probability distribution to express ignorance about the correct prior distribution.\nThis prior distribution was adopted (apparently, from his writings, with little sign of doubt ) by [Pierre-Simon Laplace](https://bliptext.com/articles/pierre-simon-laplace), and hence it was also known as the \"Bayes–Laplace rule\" or the \"Laplace rule\" of \"[inverse probability](https://bliptext.com/articles/inverse-probability)\" in publications of the first half of the 20th century.\nIn the later part of the 19th century and early part of the 20th century, scientists realized that the assumption of uniform \"equal\" probability density depended on the actual functions (for example whether a linear or a logarithmic scale was most appropriate) and parametrizations used.\nIn particular, the behavior near the ends of distributions with finite support (for example near x = 0, for a distribution with initial support at x = 0) required particular attention.\nKeynes ( Ch.XXX, p. 381) criticized the use of Bayes's uniform prior probability (Beta(1,1)) that all values between zero and one are equiprobable, as follows: \"Thus experience, if it shows anything, shows that there is a very marked clustering of statistical ratios in the neighborhoods of zero and unity, of those for positive theories and for correlations between positive qualities in the neighborhood of zero, and of those for negative theories and for correlations between negative qualities in the neighborhood of unity. \"\n\n### Haldane's prior probability (Beta(0,0))\nThe Beta(0,0) distribution was proposed by [J.B.S. Haldane](https://bliptext.com/articles/j-b-s-haldane), who suggested that the prior probability representing complete uncertainty should be proportional to p−1(1−p)−1.\nThe function p−1(1−p)−1 can be viewed as the limit of the numerator of the beta distribution as both shape parameters approach zero: α, β → 0.\nThe Beta function (in the denominator of the beta distribution) approaches infinity, for both parameters approaching zero, α, β → 0.\nTherefore, p−1(1−p)−1 divided by the Beta function approaches a 2-point [Bernoulli distribution](https://bliptext.com/articles/bernoulli-distribution) with equal probability 1/2 at each end, at 0 and 1, and nothing in between, as α, β → 0.\nA coin-toss: one face of the coin being at 0 and the other face being at 1.\nThe Haldane prior probability distribution Beta(0,0) is an \"[improper prior](https://bliptext.com/articles/improper-prior)\" because its integration (from 0 to 1) fails to strictly converge to 1 due to the singularities at each end.\nHowever, this is not an issue for computing posterior probabilities unless the sample size is very small.\nFurthermore, Zellner points out that on the [log-odds](https://bliptext.com/articles/log-odds) scale, (the [logit](https://bliptext.com/articles/logit) transformation ln(p/1 − p)), the Haldane prior is the uniformly flat prior.\nThe fact that a uniform prior probability on the [logit](https://bliptext.com/articles/logit) transformed variable ln(p/1 − p) (with domain (−∞, ∞)) is equivalent to the Haldane prior on the domain [0, 1] was pointed out by [Harold Jeffreys](https://bliptext.com/articles/harold-jeffreys) in the first edition (1939) of his book Theory of Probability ( p. 123).\nJeffreys writes \"Certainly if we take the Bayes–Laplace rule right up to the extremes we are led to results that do not correspond to anybody's way of thinking. The (Haldane) rule dx/(x(1 − x)) goes too far the other way. It would lead to the conclusion that if a sample is of one type with respect to some property there is a probability 1 that the whole population is of that type.\"\nThe fact that \"uniform\" depends on the parametrization, led Jeffreys to seek a form of prior that would be invariant under different parametrizations.\n# [Beta(0,0): The Haldane prior probability expressing total ignorance about prior information, where we are not even sure whether it is physically possible for an experiment to yield either a success or a failure. As α, β → 0, the beta distribution approaches a two-point Bernoulli distribution with all probability density concentrated at each end, at 0 and 1, and nothing in between. A coin-toss: one face of the coin being at 0 and the other face being at 1. | upload.wikimedia.org/wikipedia/commons/1/1b/Beta///distribution///for///alpha///and///beta///approaching///zero///-///J.///Rodal.png]\n\n### Jeffreys' prior probability (Beta(1/2,1/2) for a Bernoulli or for a binomial distribution)\n[Harold Jeffreys](https://bliptext.com/articles/harold-jeffreys) proposed to use an [uninformative prior](https://bliptext.com/articles/uninformative-prior) probability measure that should be [invariant under reparameterization](https://bliptext.com/articles/parametrization-invariance): proportional to the square root of the [determinant](https://bliptext.com/articles/determinant) of [Fisher's information](https://bliptext.com/articles/fisher-s-information) matrix.\nFor the [Bernoulli distribution](https://bliptext.com/articles/bernoulli-distribution), this can be shown as follows: for a coin that is \"heads\" with probability p ∈ [0, 1] and is \"tails\" with probability 1 − p, for a given (H,T) ∈ {(0,1), (1,0)} the probability is pH(1 − p)T.\nSince T = 1 − H, the [Bernoulli distribution](https://bliptext.com/articles/bernoulli-distribution) is pH(1 − p)1 − H. Considering p as the only parameter, it follows that the log likelihood for the [Bernoulli distribution](https://bliptext.com/articles/bernoulli-distribution) is\nThe Fisher information matrix has only one component (it is a scalar, because there is only one parameter: p), therefore:\nSimilarly, for the [Binomial distribution](https://bliptext.com/articles/binomial-distribution) with n [Bernoulli trials](https://bliptext.com/articles/bernoulli-trials), it can be shown that\nThus, for the [Bernoulli](https://bliptext.com/articles/bernoulli-distribution), and [Binomial distributions](https://bliptext.com/articles/binomial-distribution), [Jeffreys prior](https://bliptext.com/articles/jeffreys-prior) is proportional to, which happens to be proportional to a beta distribution with domain variable x = p, and shape parameters α = β = 1/2, the [arcsine distribution](https://bliptext.com/articles/arcsine-distribution):\nIt will be shown in the next section that the normalizing constant for Jeffreys prior is immaterial to the final result because the normalizing constant cancels out in Bayes theorem for the posterior probability.\nHence Beta(1/2,1/2) is used as the Jeffreys prior for both Bernoulli and binomial distributions.\nAs shown in the next section, when using this expression as a prior probability times the likelihood in [Bayes theorem](https://bliptext.com/articles/bayes-theorem), the posterior probability turns out to be a beta distribution.\nIt is important to realize, however, that Jeffreys prior is proportional to for the Bernoulli and binomial distribution, but not for the beta distribution.\nJeffreys prior for the beta distribution is given by the determinant of Fisher's information for the beta distribution, which, as shown in the is a function of the [trigamma function](https://bliptext.com/articles/trigamma-function) ψ1 of shape parameters α and β as follows:\nAs previously discussed, Jeffreys prior for the Bernoulli and binomial distributions is proportional to the [arcsine distribution](https://bliptext.com/articles/arcsine-distribution) Beta(1/2,1/2), a one-dimensional curve that looks like a basin as a function of the parameter p of the Bernoulli and binomial distributions.\nThe walls of the basin are formed by p approaching the singularities at the ends p → 0 and p → 1, where Beta(1/2,1/2) approaches infinity.\nJeffreys prior for the beta distribution is a 2-dimensional surface (embedded in a three-dimensional space) that looks like a basin with only two of its walls meeting at the corner α = β = 0 (and missing the other two walls) as a function of the shape parameters α and β of the beta distribution.\nThe two adjoining walls of this 2-dimensional surface are formed by the shape parameters α and β approaching the singularities (of the trigamma function) at α, β → 0.\nIt has no walls for α, β → ∞ because in this case the determinant of Fisher's information matrix for the beta distribution approaches zero.\nIt will be shown in the next section that Jeffreys prior probability results in posterior probabilities (when multiplied by the binomial likelihood function) that are intermediate between the posterior probability results of the Haldane and Bayes prior probabilities.\nJeffreys prior may be difficult to obtain analytically, and for some cases it just doesn't exist (even for simple distribution functions like the asymmetric [triangular distribution](https://bliptext.com/articles/triangular-distribution)).\nBerger, Bernardo and Sun, in a 2009 paper defined a reference prior probability distribution that (unlike Jeffreys prior) exists for the asymmetric [triangular distribution](https://bliptext.com/articles/triangular-distribution).\nThey cannot obtain a closed-form expression for their reference prior, but numerical calculations show it to be nearly perfectly fitted by the (proper) prior\nwhere θ is the vertex variable for the asymmetric [triangular distribution](https://bliptext.com/articles/triangular-distribution) with support [0, 1] (corresponding to the following parameter values in Wikipedia's article on the [triangular distribution](https://bliptext.com/articles/triangular-distribution): vertex c = θ, left end a = 0,and right end b = 1).\nBerger et al. also give a heuristic argument that Beta(1/2,1/2) could indeed be the exact Berger–Bernardo–Sun reference prior for the asymmetric triangular distribution.\nTherefore, Beta(1/2,1/2) not only is Jeffreys prior for the Bernoulli and binomial distributions, but also seems to be the Berger–Bernardo–Sun reference prior for the asymmetric triangular distribution (for which the Jeffreys prior does not exist), a distribution used in project management and [PERT](https://bliptext.com/articles/pert) analysis to describe the cost and duration of project tasks.\nClarke and Barron prove that, among continuous positive priors, Jeffreys prior (when it exists) asymptotically maximizes Shannon's [mutual information](https://bliptext.com/articles/mutual-information) between a sample of size n and the parameter, and therefore Jeffreys prior is the most uninformative prior (measuring information as Shannon information).\nThe proof rests on an examination of the [Kullback–Leibler divergence](https://bliptext.com/articles/kullback-leibler-divergence) between probability density functions for [iid](https://bliptext.com/articles/iid) random variables.\n# [Jeffreys prior probability for the beta distribution: the square root of the determinant of Fisher's information matrix: is a function of the trigamma function ψ1 of shape parameters α, β | upload.wikimedia.org/wikipedia/commons/3/30/Jeffreys///prior///probability///for///the///beta///distribution///-///J.///Rodal.png]\n# [beta distribution | upload.wikimedia.org/wikipedia/commons/3/33/Beta///distribution///for///3///different///prior///probability///functions///-///J.///Rodal.png]\n# [beta distribution | upload.wikimedia.org/wikipedia/commons/4/49/Beta///distribution///for///3///different///prior///probability///functions%2C///skewed///case///-///J.///Rodal.png]\n# [beta distribution | upload.wikimedia.org/wikipedia/commons/a/a5/Beta///distribution///for///3///different///prior///probability///functions%2C///skewed///case///sample///size///%3D///%284%2C12%2C40%29///-///J.///Rodal.pn g]\n\n### Effect of different prior probability choices on the posterior beta distribution\nIf samples are drawn from the population of a random variable X that result in s successes and f failures in n [Bernoulli trials](https://bliptext.com/articles/bernoulli-trial) n = s + f, then the [likelihood function](https://bliptext.com/articles/likelihood-function) for parameters s and f given x = p (the notation x = p in the expressions below will emphasize that the domain x stands for the value of the parameter p in the [binomial distribution](https://bliptext.com/articles/binomial-distribution)), is the following [binomial distribution](https://bliptext.com/articles/binomial-distribution):\nIf beliefs about [prior probability](https://bliptext.com/articles/prior-probability) information are reasonably well approximated by a beta distribution with parameters α Prior and β Prior, then:\nAccording to [Bayes' theorem](https://bliptext.com/articles/bayes-theorem) for a continuous event space, the [posterior probability](https://bliptext.com/articles/posterior-probability) density is given by the product of the [prior probability](https://bliptext.com/articles/prior-probability) and the likelihood function (given the evidence s and f = n − s), normalized so that the area under the curve equals one, as follows:\nThe [binomial coefficient](https://bliptext.com/articles/binomial-coefficient)\nappears both in the numerator and the denominator of the posterior probability, and it does not depend on the integration variable x, hence it cancels out, and it is irrelevant to the final result.\nSimilarly the normalizing factor for the prior probability, the beta function B(αPrior,βPrior) cancels out and it is immaterial to the final result.\nThe same posterior probability result can be obtained if one uses an un-normalized prior\nbecause the normalizing factors all cancel out.\nSeveral authors (including Jeffreys himself) thus use an un-normalized prior formula since the normalization constant cancels out.\nThe numerator of the posterior probability ends up being just the (un-normalized) product of the prior probability and the likelihood function, and the denominator is its integral from zero to one.\nThe beta function in the denominator, B(s + α Prior, n − s + β Prior), appears as a normalization constant to ensure that the total posterior probability integrates to unity.\nThe ratio s/n of the number of successes to the total number of trials is a [sufficient statistic](https://bliptext.com/articles/sufficient-statistic) in the binomial case, which is relevant for the following results.\nFor the **Bayes**' prior probability (Beta(1,1)), the posterior probability is:\nFor the **Jeffreys**' prior probability (Beta(1/2,1/2)), the posterior probability is:\nand for the **Haldane** prior probability (Beta(0,0)), the posterior probability is:\nFrom the above expressions it follows that for s/n = 1/2) all the above three prior probabilities result in the identical location for the posterior probability mean = mode = 1/2. For s/n mean for Jeffreys prior > mean for Haldane prior. For s/n > 1/2 the order of these inequalities is reversed such that the Haldane prior probability results in the largest posterior mean. The Haldane prior probability Beta(0,0) results in a posterior probability density with mean (the expected value for the probability of success in the \"next\" trial) identical to the ratio s/n of the number of successes to the total number of trials. Therefore, the Haldane prior results in a posterior probability with expected value in the next trial equal to the maximum likelihood.\nThe Bayes prior probability Beta(1,1) results in a posterior probability density with mode identical to the ratio s/n (the maximum likelihood).\nIn the case that 100% of the trials have been successful s = n, the Bayes prior probability Beta(1,1) results in a posterior expected value equal to the rule of succession (n + 1)/(n + 2), while the Haldane prior Beta(0,0) results in a posterior expected value of 1 (absolute certainty of success in the next trial).\nJeffreys prior probability results in a posterior expected value equal to (n + 1/2)/(n + 1).\nPerks (p. 303) points out: \"This provides a new rule of succession and expresses a 'reasonable' position to take up, namely, that after an unbroken run of n successes we assume a probability for the next trial equivalent to the assumption that we are about half-way through an average run, i.e. that we expect a failure once in (2n + 2) trials. The Bayes–Laplace rule implies that we are about at the end of an average run or that we expect a failure once in (n + 2) trials. The comparison clearly favours the new result (what is now called Jeffreys prior) from the point of view of 'reasonableness'.\"\nConversely, in the case that 100% of the trials have resulted in failure (s = 0), the Bayes prior probability Beta(1,1) results in a posterior expected value for success in the next trial equal to 1/(n + 2), while the Haldane prior Beta(0,0) results in a posterior expected value of success in the next trial of 0 (absolute certainty of failure in the next trial).\nJeffreys prior probability results in a posterior expected value for success in the next trial equal to (1/2)/(n + 1), which Perks (p. 303) points out: \"is a much more reasonably remote result than the Bayes–Laplace result 1/(n + 2)\".\nJaynes questions (for the uniform prior Beta(1,1)) the use of these formulas for the cases s = 0 or s = n because the integrals do not converge (Beta(1,1) is an improper prior for s = 0 or s = n).\nIn practice, the conditions 0 1 and β > 1).\nThe above estimate for the [mean](https://bliptext.com/articles/mean) is known as the [PERT](https://bliptext.com/articles/pert) [three-point estimation](https://bliptext.com/articles/three-point-estimation) and it is exact for either of the following values of β (for arbitrary α within these ranges):\nor\n[skewness](https://bliptext.com/articles/skewness), and [excess kurtosis](https://bliptext.com/articles/excess-kurtosis)\nThe above estimate for the [standard deviation](https://bliptext.com/articles/standard-deviation) σ(X) = (c − a)/6 is exact for either of the following values of α and β:\nOtherwise, these can be poor approximations for beta distributions with other values of α and β, exhibiting average errors of 40% in the mean and 549% in the variance.\n# [beta distribution | upload.wikimedia.org/wikipedia/commons/0/0b/Beta///Distribution///beta%3Dalpha///from///1.05///to///4.95.svg]\n# [beta distribution | upload.wikimedia.org/wikipedia/commons/b/b9/Beta///Distribution///beta%3D6-alpha///from///1.05///to///4.95.svg]\n# [beta distribution | upload.wikimedia.org/wikipedia/commons/f/fc/Beta///Distribution///for///conjugate///alpha///beta.svg]\n\n# Random variate generation\nIf X and Y are independent, with and then\nSo one algorithm for generating beta variates is to generate, where X is a [gamma variate](https://bliptext.com/articles/gamma-distribution) with parameters (α, 1) and Y is an independent [gamma variate](https://bliptext.com/articles/gamma-distribution) with parameters (β, 1).\nIn fact, here and X+Y are independent, and.\nIf and Z is independent of X and Y, then and is independent of.\nThis shows that the product of independent and random variables is a random variable.\nAlso, the kth [order statistic](https://bliptext.com/articles/order-statistic) of n [uniformly distributed](https://bliptext.com/articles/uniform-distribution-continuous) variates is, so an alternative if α and β are small integers is to generate α + β − 1 uniform variates and choose the α-th smallest.\nAnother way to generate the Beta distribution is by [Pólya urn model](https://bliptext.com/articles/p-lya-urn-model).\nAccording to this method, one start with an \"urn\" with α \"black\" balls and β \"white\" balls and draw uniformly with replacement.\nEvery trial an additional ball is added according to the color of the last ball which was drawn.\nAsymptotically, the proportion of black and white balls will be distributed according to the Beta distribution, where each repetition of the experiment will produce a different value.\nIt is also possible to use the [inverse transform sampling](https://bliptext.com/articles/inverse-transform-sampling).\n\n# Normal approximation to the Beta distribution\nA beta distribution with α ~ β and α and β >> 1 is approximately normal with mean 1/2 and variance 1/(4(2α + 1)).\nIf α ≥ β the normal approximation can be improved by taking the cube-root of the logarithm of the reciprocal of\n\n# History\n[Thomas Bayes](https://bliptext.com/articles/thomas-bayes), in a posthumous paper published in 1763 by [Richard Price](https://bliptext.com/articles/richard-price), obtained a beta distribution as the density of the probability of success in Bernoulli trials (see ), but the paper does not analyze any of the moments of the beta distribution or discuss any of its properties.\nThe first systematic modern discussion of the beta distribution is probably due to [Karl Pearson](https://bliptext.com/articles/karl-pearson).\nIn Pearson's papers the beta distribution is couched as a solution of a differential equation: [Pearson's Type I distribution](https://bliptext.com/articles/pearson-distribution) which it is essentially identical to except for arbitrary shifting and re-scaling (the beta and Pearson Type I distributions can always be equalized by proper choice of parameters).\nIn fact, in several English books and journal articles in the few decades prior to World War II, it was common to refer to the beta distribution as Pearson's Type I distribution.\n[William P. Elderton](https://bliptext.com/articles/william-palin-elderton) in his 1906 monograph \"Frequency curves and correlation\" further analyzes the beta distribution as Pearson's Type I distribution, including a full discussion of the method of moments for the four parameter case, and diagrams of (what Elderton describes as) U-shaped, J-shaped, twisted J-shaped, \"cocked-hat\" shapes, horizontal and angled straight-line cases.\nElderton wrote \"I am chiefly indebted to Professor Pearson, but the indebtedness is of a kind for which it is impossible to offer formal thanks.\"\n[Elderton](https://bliptext.com/articles/william-palin-elderton) in his 1906 monograph provides an impressive amount of information on the beta distribution, including equations for the origin of the distribution chosen to be the mode, as well as for other Pearson distributions: types I through VII.\nElderton also included a number of appendixes, including one appendix (\"II\") on the beta and gamma functions.\nIn later editions, Elderton added equations for the origin of the distribution chosen to be the mean, and analysis of Pearson distributions VIII through XII.\nAs remarked by Bowman and Shenton \"Fisher and Pearson had a difference of opinion in the approach to (parameter) estimation, in particular relating to (Pearson's method of) moments and (Fisher's method of) maximum likelihood in the case of the Beta distribution.\"\nAlso according to Bowman and Shenton, \"the case of a Type I (beta distribution) model being the center of the controversy was pure serendipity. A more difficult model of 4 parameters would have been hard to find.\"\nThe long running public conflict of Fisher with Karl Pearson can be followed in a number of articles in prestigious journals.\nFor example, concerning the estimation of the four parameters for the beta distribution, and Fisher's criticism of Pearson's method of moments as being arbitrary, see Pearson's article \"Method of moments and method of maximum likelihood\" (published three years after his retirement from University College, London, where his position had been divided between Fisher and Pearson's son Egon) in which Pearson writes \"I read (Koshai's paper in the Journal of the Royal Statistical Society, 1933) which as far as I am aware is the only case at present published of the application of Professor Fisher's method. To my astonishment that method depends on first working out the constants of the frequency curve by the (Pearson) Method of Moments and then superposing on it, by what Fisher terms \"the Method of Maximum Likelihood\" a further approximation to obtain, what he holds, he will thus get, 'more efficient values' of the curve constants\".\nDavid and Edwards's treatise on the history of statistics cites the first modern treatment of the beta distribution, in 1911, using the beta designation that has become standard, due to [Corrado Gini](https://bliptext.com/articles/corrado-gini), an Italian [statistician](https://bliptext.com/articles/statistician), [demographer](https://bliptext.com/articles/demography), and [sociologist](https://bliptext.com/articles/sociology), who developed the [Gini coefficient](https://bliptext.com/articles/gini-coefficient).\n[N.L.Johnson](https://bliptext.com/articles/norman-lloyd-johnson) and [S.Kotz](https://bliptext.com/articles/samuel-kotz), in their comprehensive and very informative monograph on leading historical personalities in statistical sciences credit [Corrado Gini](https://bliptext.com/articles/corrado-gini) as \"an early Bayesian...who dealt with the problem of eliciting the parameters of an initial Beta distribution, by singling out techniques which anticipated the advent of the so-called empirical Bayes approach.\"\n# [Karl Pearson analyzed the beta distribution as the solution Type I of Pearson distributions | upload.wikimedia.org/wikipedia/commons/2/21/Karl///Pearson///2.jpg]","revisionCount":0},"hasMore":false,"nextCursor":null}