LogLogisticDistribution

LogLogisticDistribution[γ,σ]

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

Details

Background & Context

Examples

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Basic Examples  (4)

Probability density function:

Cumulative distribution function:

Mean and variance:

Median:

Scope  (8)

Generate a sample of pseudorandom numbers from a log-logistic 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 the shape parameter γ:

For large values of γ, log-logistic distribution becomes symmetric:

Kurtosis depends only on the shape parameter γ:

Kurtosis has horizontal asymptote as γ gets large:

Different moments with closed forms as functions of parameters:

Moment:

Closed form for symbolic order:

CentralMoment:

FactorialMoment:

Cumulant:

Hazard function:

Quantile function:

Consistent use of Quantity in parameters yields QuantityDistribution:

Find the interquartile range:

Applications  (2)

LogLogisticDistribution can be used to model incomes:

Adjust part-time to full-time and select nonzero values, attach currency unit:

Fit log-logistic distribution to 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 $15,000:

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:

BetaPrimeDistribution can be used to model state per capita incomes:

Attach currency units:

Fit log-logistic distribution to 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:

Properties & Relations  (6)

Log-logistic distribution is closed under scaling by a positive factor:

Relationships to other distributions:

LogLogisticDistribution is a special case of DagumDistribution:

LogLogisticDistribution is a special case of SinghMaddalaDistribution:

LogLogisticDistribution is a special case of BetaPrimeDistribution:

LogLogisticDistribution is related to LogisticDistribution:

Neat Examples  (1)

PDFs for different γ values with CDF contours:

Introduced in 2010
 (8.0)
 |
Updated in 2016
 (10.4)