This is documentation for Mathematica 8, which was
based on an earlier version of the Wolfram Language.

# LogLogisticDistribution

 LogLogisticDistribution represents a log-logistic distribution with shape parameter and scale parameter .
• The probability density for value in a log-logistic distribution is proportional to for .
Probability density function:
Cumulative distribution function:
Mean and variance:
Median:
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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 Scope   (7)
Generate a set of pseudorandom numbers that are log-logistic distributed:
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 the shape parameter :
For large values of , log-logistic distribution becomes symmetric:
Kurtosis depends only on the shape parameter :
Kurtosis has horizontal asymptote as gets large:
Different moments with closed forms as functions of parameters:
Closed form for symbolic order:
Hazard function:
Quantile function:
 Applications   (2)
LogLogisticDistribution can be used to model incomes:
Adjust part-time to full-time and select nonzero values:
Fit log-logistic distribution into the data:
Compare the histogram of the data to the PDF of the estimated distribution:
Find the average income at a large state university:
Find the probability that a salary is at most \$150000:
Find the probability that a salary is at least \$150000:
Find the median salary:
Simulate the incomes for 100 randomly selected employees of such a university:
BetaPrimeDistribution can be used to model state per capita incomes:
Fit a log-logistic distribution into the data:
Compare the histogram of the data to the PDF of the estimated distribution:
Find the 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:
Compare with the fit using BetaPrimeDistribution:
Compare with the fit using DagumDistribution:
Compare with the fit using DavisDistribution:
Parameter influence on the CDF for each :
Log-logistic distribution is closed under scaling by a positive factor:
Relationships to other distributions:
New in 8