# DavisDistribution

DavisDistribution[b,n,μ]

represents a Davis distribution with scale parameter b, shape parameter n, and location parameter μ.

# Details • The probability density for value in a Davis distribution is proportional to for .
• DavisDistribution allows b to be any positive real number, μ to be any non-negative real number, and .
• DavisDistribution allows μ and b to be any quantities of any unit dimensions, and n to be a dimensionless quantity. »
• DavisDistribution can be used with such functions as Mean, CDF, and RandomVariate.

# Background & Context

• DavisDistribution[b,n,μ] represents a continuous statistical distribution defined over the interval and parametrized by the values b, n, and μ. Together, the parameters b (a positive real number called the "scale parameter") and n (a positive real number called the "shape parameter") determine the overall height and steepness of the probability density function (PDF) of the Davis distribution. The non-negative real "location parameter" μ determines the horizontal location of the PDF. The PDF of the Davis distribution is unimodal and has "fat" tails, in the sense that the PDF decreases algebraically rather than exponentially for large values of .
• The Davis distribution originated with American mathematician and statistician Harold Davis, who in the 1940s proposed it as an alternative model for income sizes and distributions. The Davis distribution is also a generalization of Planck's law from statistical physics.
• RandomVariate can be used to give one or more machine- or arbitrary-precision (the latter via the WorkingPrecision option) pseudorandom variates from a Davis distribution. Distributed[x,DavisDistribution[b,n,μ]], written more concisely as xDavisDistribution[b,n,μ], can be used to assert that a random variable x is distributed according to a Davis 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[DavisDistribution[b,n,μ],x] and CDF[DavisDistribution[b,n,μ],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 Davis distribution, EstimatedDistribution to estimate a Davis parametric distribution from given data, and FindDistributionParameters to fit data to a Davis distribution. ProbabilityPlot can be used to generate a plot of the CDF of given data against the CDF of a symbolic Davis distribution and QuantilePlot to generate a plot of the quantiles of given data against the quantiles of a symbolic Davis distribution.
• TransformedDistribution can be used to represent a transformed Davis 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 Davis distribution, and ProductDistribution can be used to compute a joint distribution with independent component distributions involving Davis distributions.
• DavisDistribution is related to a number of other distributions. For example, the long-term decay of DavisDistribution is asymptotic to that of ParetoDistribution, consistent with it's having fat tails. Qualitatively, DavisDistribution is related to a number of other distributions that are also used to model income, including BetaPrimeDistribution, DagumDistribution, LogLogisticDistribution, and BeniniDistribution. DavisDistribution is also related to BenktanderGibratDistribution and BenktanderWeibullDistribution.

# Examples

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## Basic Examples(3)

Probability density function:

Cumulative distribution function:

Mean and variance:

## Scope(8)

Generate a sample of pseudorandom numbers from a Davis distribution:

Compare its histogram to the PDF:

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 depends only on shape parameter n:

Kurtosis depends only on shape parameter n:

Different moments with closed forms as functions of parameters:

Hazard function:

Quantile function:

Consistent use of Quantity in parameters yields QuantityDistribution:

Find the median salary:

## Applications(2)

DavisDistribution can be used to model incomes:

Adjust part-time to full-time and select nonzero values:

Fit Davis distribution to the data:

Compare the histogram of the data to the PDF of the estimated distribution:

Find average income at a large state university:

Find the probability that a salary is at most \$30,000:

Find the probability that a salary is at least \$150,000:

Find the median salary:

Simulate the incomes for 100 randomly selected employees of such a university:

DavisDistribution can be used to model state per capita incomes:

Fit Davis distribution to the data:

Compare the histogram of the data to the PDF of the estimated distribution:

Find average income per capita:

Find states with income close to the average:

Find the median income per capita:

Find states with income close to the median:

Find log-likelihood value:

## Properties & Relations(1)

Davis distribution is closed under translation and scaling by a positive factor:

## Neat Examples(1)

PDFs for different b values with CDF contours: