represents a beta negative binomial mixture distribution with beta distribution parameters α and β and n successful trials.


Background & Context

  • BetaNegativeBinomialDistribution[α,β,n] represents a discrete statistical distribution defined at integer values . The parameters α, β are positive real numbers known as shape parameters, and they determine the overall shape and behavior of the probability density function (PDF). The beta negative binomial distribution has a discrete PDF and, depending on the values of α and β, the PDF may have monotonically increasing or decreasing values or may be unimodal. The beta negative binomial distribution is sometimes referred to as the inverse MarkovPólya distribution, the beta-Pascal distribution, and the generalized Waring distribution.
  • The beta negative binomial distribution can be thought of as an abstraction of the Bernoulli (BernoulliDistribution) and negative binomial (NegativeBinomialDistribution) distributions in which the success probability p of a known number of Bernoulli trials is random, the associated binomial distribution has success probability p that follows the beta distribution (BetaDistribution), and the distribution of the failures is studied. In Bayesian terms, this means that the beta negative binomial distribution arises as a posterior predictive distribution of a negative binomial variable in which the prior distribution on the success probability p is a beta distribution.
  • The first documented mention of the beta negative binomial distribution is in the work of Kemp and Kemp from the 1950s and was obtained using methods analogous to those used by the authors to derive and study the beta binomial distribution (BetaBinomialDistribution). A number of real-world phenomena can be modeled by a beta binomial distribution. For example, the beta negative binomial distribution can be utilized in inverse sampling from a Pólya urn model for a specific set of drawing rules and with additional replacements. More recently, beta negative binomial distributions have been applied to model distributions of contagions and have been used in accident theory to describe the distribution of accidents in an accident-prone community that is exposed to variable risk.
  • RandomVariate can be used to give one or more machine- or arbitrary-precision (the latter via the WorkingPrecision option) pseudorandom variates from a beta negative binomial distribution. Distributed[x,BetaNegativeBinomialDistribution[α,β,n]], written more concisely as xBetaNegativeBinomialDistribution[α,β,n], can be used to assert that a random variable x is distributed according to a beta negative binomial distribution. Such an assertion can then be used in functions such as Probability, NProbability, Expectation, and NExpectation.
  • The probability density and cumulative distribution functions may be given using PDF[BetaNegativeBinomialDistribution[α,β,n],x] and CDF[BetaNegativeBinomialDistribution[α,β,n],x]. The mean, median, variance, raw moments, and central moments may be computed using Mean, Median, Variance, Moment, and CentralMoment, respectively. These quantities can be visualized using DiscretePlot.
  • DistributionFitTest can be used to test if a given dataset is consistent with a beta negative binomial distribution, EstimatedDistribution to estimate a beta negative binomial parametric distribution from given data, and FindDistributionParameters to fit data to a beta negative binomial distribution. ProbabilityPlot can be used to generate a plot of the CDF of given data against the CDF of a symbolic beta negative binomial distribution and QuantilePlot to generate a plot of the quantiles of given data against the quantiles of a symbolic beta negative binomial distribution.
  • TransformedDistribution can be used to represent a transformed beta negative binomial 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 negative binomial distribution, and ProductDistribution can be used to compute a joint distribution with independent component distributions involving beta negative binomial distributions.
  • BetaNegativeBinomialDistribution is related to a number of other statistical distributions. As previously noted, BetaNegativeBinomialDistribution combines features from both NegativeBinomialDistribution and BetaDistribution, a fact made precise by observing that ParameterMixtureDistribution[NegativeBinomialDistribution[n,p],pBetaDistribution[α,β]] evaluates to BetaNegativeBinomialDistribution[α,β,n]. Similarly, WaringYuleDistribution is a special case of BetaNegativeBinomialDistribution in the sense that WaringYuleDistribution[α,β] has the same PDF as BetaNegativeBinomialDistribution[α,β,1]. In a very natural way, BetaNegativeBinomialDistribution is related to BetaBinomialDistribution, and because of the fact that NegativeMultinomialDistribution and DirichletDistribution are higher-dimensional analogues of NegativeBinomialDistribution and BetaDistribution, respectively, BetaNegativeBinomialDistribution can be viewed as a one-dimensional analogue of the so-called Dirichlet negative multinomial distribution.


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Basic Examples  (3)

Probability mass function:

Cumulative distribution function:

Mean and variance:

Scope  (7)

Generate a sample of pseudorandom numbers from a beta negative binomial distribution:

Compare its histogram to the PDF:

Distribution parameters estimation:

Estimate the distribution parameters from sample data:

Compare a density histogram of the sample with the PDF of the estimated distribution:



Different moments with closed forms as functions of parameters:




Closed form for symbolic order:


Hazard function:

Quantile function:

Applications  (2)

CDF of BetaNegativeBinomialDistribution is an example of a right-continuous function:

The probability of at least 50 failures before 10 successes, assuming a beta distribution on :

Verify by computing the probability for a fixed value of and averaging:

Properties & Relations  (5)

The probability of getting negative integers, integers beyond n, or non-integer numbers is zero:

Relationships to other distributions:

WaringYuleDistribution is a special case of beta negative binomial distribution:

WaringYuleDistribution is a special case of beta negative binomial distribution:

Beta negative binomial distribution is a mixture of NegativeBinomialDistribution and BetaDistribution:

Possible Issues  (2)

BetaNegativeBinomialDistribution is not defined when α, β, or n is non-positive:

Substitution of invalid parameters into symbolic outputs gives results that are not meaningful:

Introduced in 2007