represents a Pareto distribution with minimum value parameter k and shape parameter α.
represents a Pareto type II distribution with location parameter μ.
represents a Pareto type IV distribution with shape parameter γ.
- The probability density for value in a Pareto distribution is proportional to for , and is zero for . »
- ParetoDistribution includes Pareto distributions of types I, II, III, and IV:
ParetoDistribution[k,α] Pareto type I distribution ParetoDistribution[k,α,μ] Pareto type II distribution ParetoDistribution[k,1,γ,μ] Pareto type III distribution ParetoDistribution[k,α,γ,μ] Pareto type IV distribution
- ParetoDistribution[k,α,0] is also known as Lomax distribution.
- The survival function for value in a Pareto distribution corresponds to:
ParetoDistribution[k,α] ParetoDistribution[k,α,μ] ParetoDistribution[k,α,γ,μ]
- ParetoDistribution allows k, α, and γ to be any positive real numbers and μ any real number.
- ParetoDistribution allows k and μ to be any quantities of the same unit dimensions, and α, γ to be dimensionless quantities. »
- ParetoDistribution can be used with such functions as Mean, CDF, and RandomVariate. »
Background & Context
- ParetoDistribution represents a statistical distribution belonging to one of four types—type I, II, III, or IV—as determined by its argument structure. The overall shape of the probability density function (PDF) of a Pareto distribution varies significantly based on its arguments. For example, the PDF of types I and II Pareto distributions are monotonically decreasing while type IV distributions may have a single peak. In addition, the PDF of all types of ParetoDistribution are defined over a half-infinite interval and the tails of the PDF are "fat" in the sense that the PDF decreases as a power law rather than decreasing exponentially for large values . (This behavior can be made quantitatively precise by analyzing the SurvivalFunction of the distribution.)
- Pareto distributions originate with Italian economist Vilfredo Pareto, who noticed that approximately 80% of the peas in his garden were produced by roughly 20% of the pea pods. Later, Pareto observed that wealth distribution among nations followed a similar distribution, a result that led him to devise the so-called 80-20 rule (also called the Pareto principle), the basis for which is a type-I distribution corresponding to ParetoDistribution[k,α] with . Pareto distributions also arise in a number of other mathematical and scientific contexts and are applicable to phenomena including hard disk error rates, price returns among stocks, and Bose–Einstein statistics.
- RandomVariate can be used to give one or more machine- or arbitrary-precision (the latter via the WorkingPrecision option) pseudorandom variates from a Pareto distribution. Distributed[x,ParetoDistribution[k,α]], written more concisely as xParetoDistribution[k,α], can be used to assert that a random variable x is distributed according to a type-I Pareto distribution. Here, the positive parameter α is known as the Pareto index. Such an assertion can then be used in functions such as Probability, NProbability, Expectation, and NExpectation.
- The probability density and cumulative distribution functions for type-I Pareto distributions may be given using PDF[ParetoDistribution[k,α]],x] and CDF[ParetoDistribution[k,α]],x], with similar expressions for type II, III, and IV distributions. In general, Pareto distributions have PDFs that are proportional to . 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 Pareto distribution, EstimatedDistribution to estimate a Pareto parametric distribution from given data, and FindDistributionParameters to fit data to a Pareto distribution. ProbabilityPlot can be used to generate a plot of the CDF of given data against the CDF of a symbolic Pareto distribution and QuantilePlot to generate a plot of the quantiles of given data against the quantiles of a symbolic Pareto distribution.
- TransformedDistribution can be used to represent a transformed Pareto 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 Pareto distribution and ProductDistribution can be used to compute a joint distribution with independent component distributions involving Pareto distributions.
- ParetoDistribution is closely related to a number of other distributions. For example, the Pareto distribution is the continuous analogue of ZipfDistribution. As is result of its definition, the reciprocal of a Pareto‐distributed random variable follows the PowerDistribution. Furthermore, the distribution of the appropriately centered and scaled total values of independent samples of a Pareto-distributed random variable approaches a StableDistribution, while an exponential function of a random variable that follows ExponentialDistribution follows a ParetoDistribution. ParetoDistribution is also closely related to LogNormalDistribution, BenktanderWeibullDistribution, BeniniDistribution, BenktanderGibratDistribution, ChiSquareDistribution, PearsonDistribution, and BetaPrimeDistribution.