BetaDistribution

BetaDistribution[α,β]
represents a continuous beta distribution with shape parameters α and β.

DetailsDetails

Background
Background

  • BetaDistribution[α,β] represents a statistical distribution defined over the interval and parametrized by two positive values α, β known as "shape parameters", which, roughly speaking, determine the "fatness" of the left and right tails in the probability distribution function (PDF). Depending on the values of α and β, the PDF of the beta distribution may be monotonic increasing, monotonic decreasing, or unimodal with potential singularities approaching the boundaries of its domain. In addition, the tails of the PDF are "fat" in the sense that the PDF decreases algebraically rather than decreasing exponentially for large values . (This behavior can be made quantitatively precise by analyzing the SurvivalFunction of the distribution.)
  • The beta distribution arises as a prior distribution for binomial proportions in Bayesian analysis. It is also commonly used to model random variables limited to a finite interval. For example, the distribution of the ^(th) smallest element in a continuous, independent, and uniformly distributed sample of size of can be computed using OrderDistribution[{UniformDistribution[],n},k] and is precisely equal to BetaDistribution[k,n-k+1]. In addition to its statistical significance, the beta distribution also plays a fundamental role in a number of scientific fields, including phenomena related to allele frequency distribution, soil property variability, geological mineral-to-rock ratios, and HIV transmission behavior.
  • RandomVariate can be used to give one or more machine- or arbitrary-precision (the latter via the WorkingPrecision option) pseudorandom variates from a beta distribution. Distributed[x,BetaDistribution[α,β]], written more concisely as , can be used to assert that a random variable x is distributed according to a beta distribution. Such an assertion can then be used in functions such as Probability, NProbability, Expectation, and NExpectation.
  • The probability distribution and cumulative density functions may be given using PDF[BetaDistribution[α,β],x] and CDF[BetaDistribution[α,β],x]. The mean, median, variance, raw moments, and central moments may be computed using Mean, Median, Variance, Moment, and CentralMoment, respectively.
  • DistributionFitTest can be used to test if a given dataset is consistent with a beta distribution, EstimatedDistribution to estimate a beta parametric distribution from given data, and FindDistributionParameters to fit data to a beta distribution. ProbabilityPlot can be used to generate a plot of the CDF of given data against the CDF of a symbolic beta distribution and QuantilePlot to generate a plot of the quantiles of given data against the quantiles of a symbolic beta distribution.
  • TransformedDistribution can be used to represent a transformed beta distribution, CensoredDistribution to represent the distribution of values censored between upper and lower values, and TruncatedDistribution to represent the distribution of values truncated between upper and lower values. CopulaDistribution can be used to build higher-dimensional distributions that contain a beta distribution, and ProductDistribution can be used to compute a joint distribution with independent component distributions involving beta distributions.
  • The beta distribution is related to a number of other distributions. For example, BetaDistribution is the so-called "conjugate prior" for the parameters of a number of other distributions, including BernoulliDistribution, BinomialDistribution, NegativeBinomialDistribution, and GeometricDistribution. Moreover, BetaDistribution generalizes both UniformDistribution and PowerDistribution in the sense that (modulo inclusion of the endpoints and ), PDF[BetaDistribution[1,1],x] is equal to both PDF[UniformDistribution[],x] and PDF[PowerDistribution[1,1],x]. BetaDistribution can also be obtained as transformations of KumuraswamyDistribution and NoncentralBetaDistribution and is closely related to PERTDistribution, PearsonDistribution, ChiSquareDistribution, GammaDistribution, FRatioDistribution, and BetaPrimeDistribution.
Introduced in 2007
(6.0)