BorelTannerDistribution

BorelTannerDistribution[α,n]

represents a Borel–Tanner distribution with shape parameters α and n.

Details

Background & Context

  • BorelTannerDistribution[α,n] represents a discrete statistical distribution defined for integer values and determined by the parameters α and n called "shape parameters." Here, , n is any positive integer, and together these two parameters determine the overall shape, height, and horizontal placement of the probability density function (PDF) within the plane. The Borel–Tanner distribution has a discrete unimodal PDF. The Borel–Tanner distribution is sometimes referred to as the Tanner–Borel distribution and (in finance) as the herd size distribution.
  • The history of the Borel–Tanner distribution dates back to the 1940s, when French mathematician Félix Borel investigated the behavior of the PDF corresponding to the value . A decade later, Borel's methods were adapted by J. C. Tanner for the case of general positive integers n, thus marking the genesis of the distribution in its current form. Traditionally, the Borel–Tanner distribution is rooted in queueing theory, where its PDF returns for a given x the probability that exactly x members of a queue having n starting members and traffic intensity α will be served before the queue first vanishes, assuming Poisson arrivals and constant service time. More recently, the distribution has been used to model a variety of real-world phenomena, including highway traffic flows, online server traffic, and various investment behaviors relative to existing financial portfolios.
  • RandomVariate can be used to give one or more machine- or arbitrary-precision (the latter via the WorkingPrecision option) pseudorandom variates from a Borel–Tanner distribution. Distributed[x,BorelTannerDistribution[α,n]], written more concisely as xBorelTannerDistribution[α,n], can be used to assert that a random variable x is distributed according to a Borel–Tanner 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[BorelTannerDistribution[α,n],x] and CDF[BorelTannerDistribution[α,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 Borel–Tanner distribution, EstimatedDistribution to estimate a Borel–Tanner parametric distribution from given data, and FindDistributionParameters to fit data to a Borel–Tanner distribution. ProbabilityPlot can be used to generate a plot of the CDF of given data against the CDF of a symbolic Borel–Tanner distribution and QuantilePlot to generate a plot of the quantiles of given data against the quantiles of a symbolic Borel–Tanner distribution.
  • TransformedDistribution can be used to represent a transformed Borel–Tanner 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 Borel–Tanner distribution, and ProductDistribution can be used to compute a joint distribution with independent component distributions involving Borel–Tanner distributions.
  • BorelTannerDistribution is related to a number of other statistical distributions. BorelTannerDistribution is related to PoissonConsulDistribution in the sense that PoissonConsulDistribution[μ,α] can be obtained from BorelTannerDistribution[α,n] whenever nPoissonDistribution[μ]. BorelTannerDistribution is also related to BinomialDistribution, NegativeBinomialDistribution, and LogSeriesDistribution.

Examples

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Basic Examples  (3)

Probability mass function:

Cumulative distribution function:

Mean and variance:

Scope  (8)

Generate a sample of pseudorandom numbers from a Borel–Tanner distribution:

Compare sample histogram to the PDF of the Borel-Tanner distribution:

Distribution parameters estimation:

Estimate the distribution parameters from sample data:

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

Skewness:

Skewness attains its minimum independently of n:

Limiting values:

For large n, distribution becomes more symmetric:

Kurtosis:

Limiting values:

For large values of n, kurtosis converges to the kurtosis of the standard NormalDistribution:

Different moments with closed forms as functions of parameters:

Moment:

CentralMoment:

FactorialMoment:

Cumulant:

Hazard function:

Quantile function:

Use dimensionless Quantity to define BorelTannerDistribution:

Applications  (2)

The CDF of BorelTannerDistribution is an example of a right-continuous function:

Customers arrive at a service desk at unit rate per unit time. Serving each customer takes constant time . At the time of the opening of the service desk, the line contained people. The total number of customers served before there is no one in line follows a BorelTannerDistribution:

Note that the time must be less than 1 for the line to eventually disappear:

Show the distribution mass function for specific parameters:

Expected number of customers served before there is no one in line:

Compute for the specific parameters used above:

Properties & Relations  (2)

Sum of random variates from BorelTannerDistribution with common α parameter also follows BorelTannerDistribution:

PoissonConsulDistribution is a parameter mixture of BorelTannerDistribution and PoissonDistribution:

Possible Issues  (1)

Estimating via method of moments may yield invalid parameters:

Use the definition of the second parameter to specify its value:

Wolfram Research (2010), BorelTannerDistribution, Wolfram Language function, https://reference.wolfram.com/language/ref/BorelTannerDistribution.html (updated 2016).

Text

Wolfram Research (2010), BorelTannerDistribution, Wolfram Language function, https://reference.wolfram.com/language/ref/BorelTannerDistribution.html (updated 2016).

CMS

Wolfram Language. 2010. "BorelTannerDistribution." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2016. https://reference.wolfram.com/language/ref/BorelTannerDistribution.html.

APA

Wolfram Language. (2010). BorelTannerDistribution. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/BorelTannerDistribution.html

BibTeX

@misc{reference.wolfram_2024_boreltannerdistribution, author="Wolfram Research", title="{BorelTannerDistribution}", year="2016", howpublished="\url{https://reference.wolfram.com/language/ref/BorelTannerDistribution.html}", note=[Accessed: 18-November-2024 ]}

BibLaTeX

@online{reference.wolfram_2024_boreltannerdistribution, organization={Wolfram Research}, title={BorelTannerDistribution}, year={2016}, url={https://reference.wolfram.com/language/ref/BorelTannerDistribution.html}, note=[Accessed: 18-November-2024 ]}