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

- LogLogisticDistribution is also known as Fisk distribution.
- The probability density for value
in a log-logistic distribution is proportional to
for
.
- LogLogisticDistribution allows γ and σ to be any positive real numbers.
- LogLogisticDistribution allows σ to be a quantity of any unit dimension, and γ to be a dimensionless quantity. »
- LogLogisticDistribution can be used with such functions as Mean, CDF, and RandomVariate.
Background & Context
- LogLogisticDistribution[γ,σ] 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 LogLogisticDistribution[γ,σ] 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
open allclose allBasic Examples (4)Summary of the most common use cases

https://wolfram.com/xid/0nx3ljk9ysmq9gk86-mm0u45


https://wolfram.com/xid/0nx3ljk9ysmq9gk86-lzvs1w


https://wolfram.com/xid/0nx3ljk9ysmq9gk86-ud24zr

Cumulative distribution function:

https://wolfram.com/xid/0nx3ljk9ysmq9gk86-k8opug


https://wolfram.com/xid/0nx3ljk9ysmq9gk86-sl6mbg


https://wolfram.com/xid/0nx3ljk9ysmq9gk86-t1dyqi


https://wolfram.com/xid/0nx3ljk9ysmq9gk86-pkjpxb


https://wolfram.com/xid/0nx3ljk9ysmq9gk86-4b9ibt


https://wolfram.com/xid/0nx3ljk9ysmq9gk86-6kedsx

Scope (8)Survey of the scope of standard use cases
Generate a sample of pseudorandom numbers from a log-logistic distribution:

https://wolfram.com/xid/0nx3ljk9ysmq9gk86-qhtk5j
Compare its histogram to the PDF:

https://wolfram.com/xid/0nx3ljk9ysmq9gk86-03mwaz

Distribution parameters estimation:

https://wolfram.com/xid/0nx3ljk9ysmq9gk86-45b7g2
Estimate the distribution parameters from sample data:

https://wolfram.com/xid/0nx3ljk9ysmq9gk86-epi747

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

https://wolfram.com/xid/0nx3ljk9ysmq9gk86-f8ui5o

Skewness depends only on the shape parameter γ:

https://wolfram.com/xid/0nx3ljk9ysmq9gk86-s2le6s


https://wolfram.com/xid/0nx3ljk9ysmq9gk86-h5v6ql

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

https://wolfram.com/xid/0nx3ljk9ysmq9gk86-8w817q

Kurtosis depends only on the shape parameter γ:

https://wolfram.com/xid/0nx3ljk9ysmq9gk86-o0xwfi


https://wolfram.com/xid/0nx3ljk9ysmq9gk86-0vcsns

Kurtosis has horizontal asymptote as γ gets large:

https://wolfram.com/xid/0nx3ljk9ysmq9gk86-tcrg9l

Different moments with closed forms as functions of parameters:

https://wolfram.com/xid/0nx3ljk9ysmq9gk86-js043h

https://wolfram.com/xid/0nx3ljk9ysmq9gk86-rx074o

Closed form for symbolic order:

https://wolfram.com/xid/0nx3ljk9ysmq9gk86-vdnhk1


https://wolfram.com/xid/0nx3ljk9ysmq9gk86-pknsqa


https://wolfram.com/xid/0nx3ljk9ysmq9gk86-zg9ct4


https://wolfram.com/xid/0nx3ljk9ysmq9gk86-9gzmth


https://wolfram.com/xid/0nx3ljk9ysmq9gk86-2iq3ge


https://wolfram.com/xid/0nx3ljk9ysmq9gk86-5hz33a


https://wolfram.com/xid/0nx3ljk9ysmq9gk86-ilrg2w


https://wolfram.com/xid/0nx3ljk9ysmq9gk86-x3ve33


https://wolfram.com/xid/0nx3ljk9ysmq9gk86-iml77m


https://wolfram.com/xid/0nx3ljk9ysmq9gk86-sdhku4

Consistent use of Quantity in parameters yields QuantityDistribution:

https://wolfram.com/xid/0nx3ljk9ysmq9gk86-bc5g3d


https://wolfram.com/xid/0nx3ljk9ysmq9gk86-ep3kcu

Applications (2)Sample problems that can be solved with this function
LogLogisticDistribution can be used to model incomes:

https://wolfram.com/xid/0nx3ljk9ysmq9gk86-5mvzlk


https://wolfram.com/xid/0nx3ljk9ysmq9gk86-87moqy
Adjust part-time to full-time and select nonzero values, attach currency unit:

https://wolfram.com/xid/0nx3ljk9ysmq9gk86-w8fgub
Fit log-logistic distribution to the data:

https://wolfram.com/xid/0nx3ljk9ysmq9gk86-83xl39

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

https://wolfram.com/xid/0nx3ljk9ysmq9gk86-wrn7k9

Find the average income at a large state university:

https://wolfram.com/xid/0nx3ljk9ysmq9gk86-for81j

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

https://wolfram.com/xid/0nx3ljk9ysmq9gk86-fl05eu

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

https://wolfram.com/xid/0nx3ljk9ysmq9gk86-mzibvb


https://wolfram.com/xid/0nx3ljk9ysmq9gk86-7hcsn8

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

https://wolfram.com/xid/0nx3ljk9ysmq9gk86-18pcw5

BetaPrimeDistribution can be used to model state per capita incomes:

https://wolfram.com/xid/0nx3ljk9ysmq9gk86-bdhhru


https://wolfram.com/xid/0nx3ljk9ysmq9gk86-zrq5bc
Fit log-logistic distribution to the data:

https://wolfram.com/xid/0nx3ljk9ysmq9gk86-24wzdp

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

https://wolfram.com/xid/0nx3ljk9ysmq9gk86-4l4py7

Find the average income per capita:

https://wolfram.com/xid/0nx3ljk9ysmq9gk86-belxf6

Find states with income close to the average:

https://wolfram.com/xid/0nx3ljk9ysmq9gk86-uen50d

Find the median income per capita:

https://wolfram.com/xid/0nx3ljk9ysmq9gk86-wbveuy

Find states with income close to the median:

https://wolfram.com/xid/0nx3ljk9ysmq9gk86-0mngys


https://wolfram.com/xid/0nx3ljk9ysmq9gk86-w0ag8d

Properties & Relations (6)Properties of the function, and connections to other functions
Log-logistic distribution is closed under scaling by a positive factor:

https://wolfram.com/xid/0nx3ljk9ysmq9gk86-u1rz2n

Relationships to other distributions:

LogLogisticDistribution is a special case of DagumDistribution:

https://wolfram.com/xid/0nx3ljk9ysmq9gk86-s5fo3i


https://wolfram.com/xid/0nx3ljk9ysmq9gk86-gzciyh


https://wolfram.com/xid/0nx3ljk9ysmq9gk86-bmd664

LogLogisticDistribution is a special case of SinghMaddalaDistribution:

https://wolfram.com/xid/0nx3ljk9ysmq9gk86-c1jcmn


https://wolfram.com/xid/0nx3ljk9ysmq9gk86-d26l8s


https://wolfram.com/xid/0nx3ljk9ysmq9gk86-38spiv

LogLogisticDistribution is a special case of BetaPrimeDistribution:

https://wolfram.com/xid/0nx3ljk9ysmq9gk86-h0l32b


https://wolfram.com/xid/0nx3ljk9ysmq9gk86-n117d9


https://wolfram.com/xid/0nx3ljk9ysmq9gk86-clrien

LogLogisticDistribution is related to LogisticDistribution:

https://wolfram.com/xid/0nx3ljk9ysmq9gk86-6cu6y0


https://wolfram.com/xid/0nx3ljk9ysmq9gk86-1p3yh7

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
]}
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
]}