WOLFRAM

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

Details

Background & Context

Examples

open allclose all

Basic Examples  (4)Summary of the most common use cases

Probability density function:

Out[1]=1
Out[2]=2
Out[3]=3

Cumulative distribution function:

Out[1]=1
Out[2]=2
Out[3]=3

Mean and variance:

Out[1]=1
Out[2]=2

Median:

Out[1]=1

Scope  (8)Survey of the scope of standard use cases

Generate a sample of pseudorandom numbers from a log-logistic distribution:

Compare its histogram to the PDF:

Out[2]=2

Distribution parameters estimation:

Estimate the distribution parameters from sample data:

Out[2]=2

Compare the density histogram of the sample with the PDF of the estimated distribution:

Out[3]=3

Skewness depends only on the shape parameter γ:

Out[1]=1
Out[2]=2

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

Out[3]=3

Kurtosis depends only on the shape parameter γ:

Out[1]=1
Out[2]=2

Kurtosis has horizontal asymptote as γ gets large:

Out[3]=3

Different moments with closed forms as functions of parameters:

Moment:

Out[2]=2

Closed form for symbolic order:

Out[3]=3

CentralMoment:

Out[4]=4

FactorialMoment:

Out[5]=5

Cumulant:

Out[6]=6

Hazard function:

Out[1]=1
Out[2]=2
Out[3]=3

Quantile function:

Out[1]=1
Out[2]=2
Out[3]=3

Consistent use of Quantity in parameters yields QuantityDistribution:

Out[1]=1

Find the interquartile range:

Out[2]=2

Applications  (2)Sample problems that can be solved with this function

LogLogisticDistribution can be used to model incomes:

Out[1]=1

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

Fit log-logistic distribution to the data:

Out[4]=4

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

Out[5]=5

Find the average income at a large state university:

Out[6]=6

Find the probability that a salary is at most $15,000:

Out[7]=7

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

Out[8]=8

Find the median salary:

Out[9]=9

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

Out[10]=10

BetaPrimeDistribution can be used to model state per capita incomes:

Out[1]=1

Attach currency units:

Fit log-logistic distribution to the data:

Out[3]=3

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

Out[4]=4

Find the average income per capita:

Out[5]=5

Find states with income close to the average:

Out[6]=6

Find the median income per capita:

Out[7]=7

Find states with income close to the median:

Out[8]=8

Find log-likelihood value:

Out[9]=9

Properties & Relations  (6)Properties of the function, and connections to other functions

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

Out[1]=1

Relationships to other distributions:

LogLogisticDistribution is a special case of DagumDistribution:

Out[1]=1
Out[2]=2
Out[3]=3

LogLogisticDistribution is a special case of SinghMaddalaDistribution:

Out[1]=1
Out[2]=2
Out[3]=3

LogLogisticDistribution is a special case of BetaPrimeDistribution:

Out[1]=1
Out[2]=2
Out[3]=3

LogLogisticDistribution is related to LogisticDistribution:

Out[1]=1
Out[2]=2

Neat Examples  (1)Surprising or curious use cases

PDFs for different γ values with CDF contours:

Out[4]=4
Wolfram Research (2010), LogLogisticDistribution, Wolfram Language function, https://reference.wolfram.com/language/ref/LogLogisticDistribution.html (updated 2016).
Wolfram Research (2010), LogLogisticDistribution, Wolfram Language function, https://reference.wolfram.com/language/ref/LogLogisticDistribution.html (updated 2016).

Text

Wolfram Research (2010), LogLogisticDistribution, Wolfram Language function, https://reference.wolfram.com/language/ref/LogLogisticDistribution.html (updated 2016).

Wolfram Research (2010), LogLogisticDistribution, Wolfram Language function, https://reference.wolfram.com/language/ref/LogLogisticDistribution.html (updated 2016).

CMS

Wolfram Language. 2010. "LogLogisticDistribution." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2016. https://reference.wolfram.com/language/ref/LogLogisticDistribution.html.

Wolfram Language. 2010. "LogLogisticDistribution." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2016. https://reference.wolfram.com/language/ref/LogLogisticDistribution.html.

APA

Wolfram Language. (2010). LogLogisticDistribution. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/LogLogisticDistribution.html

Wolfram Language. (2010). LogLogisticDistribution. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/LogLogisticDistribution.html

BibTeX

@misc{reference.wolfram_2025_loglogisticdistribution, author="Wolfram Research", title="{LogLogisticDistribution}", year="2016", howpublished="\url{https://reference.wolfram.com/language/ref/LogLogisticDistribution.html}", note=[Accessed: 17-May-2025 ]}

@misc{reference.wolfram_2025_loglogisticdistribution, author="Wolfram Research", title="{LogLogisticDistribution}", year="2016", howpublished="\url{https://reference.wolfram.com/language/ref/LogLogisticDistribution.html}", note=[Accessed: 17-May-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_loglogisticdistribution, organization={Wolfram Research}, title={LogLogisticDistribution}, year={2016}, url={https://reference.wolfram.com/language/ref/LogLogisticDistribution.html}, note=[Accessed: 17-May-2025 ]}

@online{reference.wolfram_2025_loglogisticdistribution, organization={Wolfram Research}, title={LogLogisticDistribution}, year={2016}, url={https://reference.wolfram.com/language/ref/LogLogisticDistribution.html}, note=[Accessed: 17-May-2025 ]}