# LogLogisticDistribution

represents a log-logistic distribution with shape parameter γ and scale parameter σ.

# Background & Context

• represents a continuous statistical distribution supported over the interval and parametrized by positive real numbers γ (called a "shape parameter") and σ (called a "scale parameter"), which together determine the overall behavior of its probability density function (PDF). Depending on the values of γ and σ, the PDF of a log-logistic distribution may be either unimodal with a single "peak" (i.e. a global maximum) or monotone decreasing with a potential singularity approaching the lower boundary of its domain. In addition, the PDF of the log-logistic distribution has tails that are "fat" in the sense that its PDF decreases algebraically rather than exponentially for large values of . (This behavior can be made quantitatively precise by analyzing the SurvivalFunction of the distribution.) LogLogisticDistribution is sometimes referred to as the Fisk distribution, particularly in applications to economics.
• LogLogisticDistribution is the distribution followed by the logarithm of a logistic-distributed random variable. In other words, if is a random variable and (where denotes "is distributed as"), then . Qualitatively, the log-logistic distribution is similar to the log-normal distribution (LogNormalDistribution) and as such, both are commonly utilized tools for approximating lifetime data across various disciplines. In addition, the log-logistic distribution has been employed to model numerous phenomena including precipitation, wealth and income distribution, and data transmission and processing times.
• RandomVariate can be used to give one or more machine- or arbitrary-precision (the latter via the WorkingPrecision option) pseudorandom variates from a log-logistic distribution. Distributed[x,LogLogisticDistribution[γ,σ]], written more concisely as xLogLogisticDistribution[γ,σ], can be used to assert that a random variable x is distributed according to a log-logistic distribution. Such an assertion can then be used in functions such as Probability, NProbability, Expectation, and NExpectation.
• The probability density and cumulative distribution functions for log-logistic distributions may be given using PDF[LogLogisticDistribution[γ,σ],x] and CDF[LogLogisticDistribution[γ,σ],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 log-logistic distribution, EstimatedDistribution to estimate a log-logistic parametric distribution from given data, and FindDistributionParameters to fit data to a log-logistic distribution. ProbabilityPlot can be used to generate a plot of the CDF of given data against the CDF of a symbolic log-logistic distribution, and QuantilePlot to generate a plot of the quantiles of given data against the quantiles of a symbolic log-logistic distribution.
• TransformedDistribution can be used to represent a transformed log-logistic 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 log-logistic distribution, and ProductDistribution can be used to compute a joint distribution with independent component distributions involving log-logistic distributions.
• LogLogisticDistribution is related to a number of other distributions. LogLogisticDistribution is connected to LogisticDistribution and is qualitatively similar to LogNormalDistribution as discussed above. LogLogisticDistribution is a special case of several distributions including DagumDistribution, SinghMaddalaDistribution, and BetaPrimeDistribution in the sense that the PDF of is precisely the same as that of DagumDistribution[1,γ,σ], SinghMaddalaDistribution[1,γ,σ], and BetaPrimeDistribution[1,1,γ,σ]. Its logarithmic behavior is qualitatively similar to that of LogGammaDistribution, LogMultinormalDistribution, and LogNormalDistribution. LogLogisticDistribution is also related to DavisDistribution, NormalDistribution, ExponentialDistribution, WeibullDistribution, GompertzMakehamDistribution, and GammaDistribution.

# Examples

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## Basic Examples(4)

Probability density function:

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Cumulative distribution function:

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Mean and variance:

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Median:

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