# Log-logistic distribution

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Continuous probability distribution for a non-negative random variable

Log-logistic Probability density function α = 1 , {\displaystyle \alpha =1,} values of β {\displaystyle \beta } as shown in legend Cumulative distribution function α = 1 , {\displaystyle \alpha =1,} values of β {\displaystyle \beta } as shown in legend Parameters α > 0 {\displaystyle \alpha >0} scale β > 0 {\displaystyle \beta >0} shape Support x ∈ [ 0 , ∞ ) {\displaystyle x\in [0,\infty )} PDF ( β / α ) ( x / α ) β − 1 ( 1 + ( x / α ) β ) 2 {\displaystyle {\frac {(\beta /\alpha )(x/\alpha )^{\beta -1}}{\left(1+(x/\alpha )^{\beta }\right)^{2}}}} CDF 1 1 + ( x / α ) − β {\displaystyle {1 \over 1+(x/\alpha )^{-\beta }}} Quantile α ( p 1 − p ) 1 / β . {\displaystyle \alpha \left({\frac {p}{1-p}}\right)^{1/\beta }.} Mean α π / β sin ⁡ ( π / β ) {\displaystyle {\alpha \,\pi /\beta \over \sin(\pi /\beta )}} if β > 1 {\displaystyle \beta >1} , else undefined Median α {\displaystyle \alpha \,} Mode α ( β − 1 β + 1 ) 1 / β {\displaystyle \alpha \left({\frac {\beta -1}{\beta +1}}\right)^{1/\beta }} if β > 1 {\displaystyle \beta >1} , 0 otherwise Variance See main text Entropy ln ⁡ α − ln ⁡ β + 2 {\displaystyle \ln \alpha \ -\ln \beta \ +2} MGF β α − β ∫ 0 ∞ e t x x β − 1 ( 1 + ( x / α ) β ) 2 d x {\displaystyle \beta \alpha ^{-\beta }\int _{0}^{\infty }{\frac {e^{tx}x^{\beta -1}}{(1+(x/\alpha )^{\beta })^{2}}}dx} [1] = ∑ n = 0 ∞ ( α t ) n n ! B ( 1 + n β , 1 − n β ) {\displaystyle =\sum _{n=0}^{\infty }{\frac {(\alpha t)^{n}}{n!}}\mathrm {B} (1+{\frac {n}{\beta }},1-{\frac {n}{\beta }})} where B {\displaystyle \mathrm {B} } is the Beta function.[2] CF β α − β ∫ 0 ∞ e i t x x β − 1 ( 1 + ( x / α ) β ) 2 d x {\displaystyle \beta \alpha ^{-\beta }\int _{0}^{\infty }{\frac {e^{itx}x^{\beta -1}}{(1+(x/\alpha )^{\beta })^{2}}}dx} [1] = ∑ n = 0 ∞ ( i α t ) n n ! B ( 1 + n β , 1 − n β ) {\displaystyle =\sum _{n=0}^{\infty }{\frac {(i\alpha t)^{n}}{n!}}\mathrm {B} (1+{\frac {n}{\beta }},1-{\frac {n}{\beta }})} where B {\displaystyle \mathrm {B} } is the Beta function.[2]

In [probability](/source/Probability) and [statistics](/source/Statistics), the **log-logistic distribution** (known as the **Fisk distribution** in [economics](/source/Economics)) is a [continuous probability distribution](/source/Continuous_probability_distribution) for a non-negative [random variable](/source/Random_variable). It is used in [survival analysis](/source/Survival_analysis) as a [parametric model](/source/Parametric_model) for events whose rate increases initially and decreases later, as, for example, [mortality rate](/source/Mortality_rate) from cancer following diagnosis or treatment. It has also been used in [hydrology](/source/Hydrology) to model stream flow and [precipitation](/source/Precipitation), in [economics](/source/Economics) as a simple model of the [distribution of wealth](/source/Distribution_of_wealth) or [income](/source/Income_distribution), and in [networking](/source/Computer_network) to model the transmission times of data considering both the network and the software.

The log-logistic distribution is the probability distribution of a [random variable](/source/Random_variable) whose [logarithm](/source/Logarithm) has a [logistic](/source/Logistic_distribution) distribution.↵ It is similar in shape to the [log-normal distribution](/source/Log-normal_distribution) but has [heavier tails](/source/Heavy-tailed_distribution) . Unlike the log-normal, its [cumulative distribution function](/source/Cumulative_distribution_function) can be written in [closed form](/source/Closed_form_expression).

## Characterization

There are several different parameterizations of the distribution in use. The one shown here gives reasonably interpretable parameters and a simple form for the [cumulative distribution](/source/Cumulative_distribution_function) function [1].[4][5] The parameter α > 0 {\displaystyle \alpha >0} is a [scale parameter](/source/Scale_parameter) and is also the [median](/source/Median) of the distribution. The parameter β > 0 {\displaystyle \beta >0} is a [shape parameter](/source/Shape_parameter). The distribution is [unimodal](/source/Unimodal) when β > 1 {\displaystyle \beta >1} its [dispersion](/source/Statistical_dispersion) decreases as it β {\displaystyle \beta } increases.

The [cumulative distribution function](/source/Cumulative_distribution_function) is

- F ( x ; α , β ) = 1 1 + ( x / α ) − β = ( x / α ) β 1 + ( x / α ) β = x β α β + x β {\displaystyle {\begin{aligned}F(x;\alpha ,\beta )&={1 \over 1+(x/\alpha )^{-\beta }}\\[5pt]&={(x/\alpha )^{\beta } \over 1+(x/\alpha )^{\beta }}\\[5pt]&={x^{\beta } \over \alpha ^{\beta }+x^{\beta }}\end{aligned}}}

where x > 0 {\displaystyle x>0} , α > 0 {\displaystyle \alpha >0} , β > 0. {\displaystyle \beta >0.}

The [probability density function](/source/Probability_density_function) is

- f ( x ; α , β ) = ( β / α ) ( x / α ) β − 1 ( 1 + ( x / α ) β ) 2 {\displaystyle f(x;\alpha ,\beta )={\frac {(\beta /\alpha )(x/\alpha )^{\beta -1}}{\left(1+(x/\alpha )^{\beta }\right)^{2}}}}

### Alternative parameterization

An alternative parametrization is given by the pair μ , s {\displaystyle \mu ,s} in analogy with the logistic distribution:

- μ = ln ⁡ ( α ) {\displaystyle \mu =\ln(\alpha )}

- s = 1 / β {\displaystyle s=1/\beta }

## Properties

### Moments

The k {\displaystyle k} throw [moment](/source/Moment_(mathematics)) exists only when k < β , {\displaystyle k<\beta ,} when it is given by [3][4].[6]

- E ⁡ ( X k ) = α k B ⁡ ( 1 − k / β , 1 + k / β ) = α k k π / β sin ⁡ ( k π / β ) {\displaystyle {\begin{aligned}\operatorname {E} (X^{k})&=\alpha ^{k}\operatorname {B} (1-k/\beta ,1+k/\beta )\\[5pt]&=\alpha ^{k}\,{k\pi /\beta \over \sin(k\pi /\beta )}\end{aligned}}}

where B is the [beta](/source/Beta_function) function. Expressions for the [mean](/source/Expected_value), [variance](/source/Variance), [skewness](/source/Skewness), and [kurtosis](/source/Kurtosis) can be derived from this. Writing b = π / β {\displaystyle b=\pi /\beta } for convenience, the mean is

- E ⁡ ( X ) = α b / sin ⁡ b , β > 1 , {\displaystyle \operatorname {E} (X)=\alpha b/\sin b,\quad \beta >1,}

and the variance is

- Var ⁡ ( X ) = α 2 ( 2 b / sin ⁡ 2 b − b 2 / sin 2 ⁡ b ) , β > 2. {\displaystyle \operatorname {Var} (X)=\alpha ^{2}\left(2b/\sin 2b-b^{2}/\sin ^{2}b\right),\quad \beta >2.}

Explicit expressions for the skewness and kurtosis are lengthy.[7] As β {\displaystyle \beta } tends to infinity, the mean tends to the variance and skewness tend to zero, and the excess kurtosis tends to 6/5 (see also [related distributions](#Related_distributions) below).

### Quantiles

The [quantile function](/source/Quantile_function) (inverse cumulative distribution function) is

- F − 1 ( p ; α , β ) = α ( p 1 − p ) 1 / β . {\displaystyle F^{-1}(p;\alpha ,\beta )=\alpha \left({\frac {p}{1-p}}\right)^{1/\beta }.}

It follows that the [median](/source/Median) is, the lower [quartile](/source/Quartile) is, and the upper quartile is 3 1 / β α {\displaystyle 3^{1/\beta }\alpha } .

## Applications

[Hazard function](/source/Hazard_function).

        α
        =
        1
        ,

    {\displaystyle \alpha =1,}

 values of

        β

    {\displaystyle \beta }

 as shown in legend

### Survival analysis

The log-logistic distribution provides one [parametric model](/source/Parametric_model) for [survival analysis](/source/Survival_analysis). Unlike the more commonly used [Weibull distribution](/source/Weibull_distribution), it can have a non-[monotonic](/source/Monotonic) [hazard function](/source/Hazard_function): when β > 1 , {\displaystyle \beta >1,} the hazard function is [unimodal](/source/Unimodal) (when β {\displaystyle \beta } ≤ 1, the hazard decreases monotonically). The fact that the cumulative distribution function can be written in closed form is particularly useful for analysis of survival data with censoring.[8] The log-logistic distribution can be used as the basis of an [accelerated failure time model](/source/Accelerated_failure_time_model) by allowing it α {\displaystyle \alpha } to differ between groups, or more generally by introducing covariates that affect it α {\displaystyle \alpha } but not β {\displaystyle \beta } by modelling it log ⁡ ( α ) {\displaystyle \log(\alpha )} as a linear function of the covariates.[9]

The [survival function](/source/Survival_function) is

- S ( t ) = 1 − F ( t ) = [ 1 + ( t / α ) β ] − 1 , {\displaystyle S(t)=1-F(t)=[1+(t/\alpha )^{\beta }]^{-1},\,}

and so the [hazard function](/source/Hazard_function) is

- h ( t ) = f ( t ) S ( t ) = ( β / α ) ( t / α ) β − 1 1 + ( t / α ) β . {\displaystyle h(t)={\frac {f(t)}{S(t)}}={\frac {(\beta /\alpha )(t/\alpha )^{\beta -1}}{1+(t/\alpha )^{\beta }}}.}

The log-logistic distribution with shape parameter β = 1 {\displaystyle \beta =1} is the marginal distribution of the inter-times in a geometric-distributed [counting](/source/Counting_process) process [8].[10]

### Hydrology

Fitted cumulative log-logistic distribution to maximum one-day October rainfalls

The log-logistic distribution has been used in hydrology for modeling stream flow rates and precipitation.[4][5]

Extreme values like maximum one-day rainfall and river discharge per month or per year often follow a [log-normal](/source/Log-normal_distribution) distribution [9]. The log-normal distribution, however, needs a numeric approximation. As the log-logistic distribution, which can be solved analytically, is similar to the log-normal distribution, it can be used instead.

The blue picture illustrates an example of fitting the log-logistic distribution to ranked maximum one-day October rainfalls, and it shows the 90% [confidence belt](/source/Confidence_belt) based on the [binomial distribution](/source/Binomial_distribution). The rainfall data are represented by the [plotting position](/source/Plotting_position) *r*/(*n*+1) as part of the [cumulative frequency analysis](/source/Cumulative_frequency_analysis) .

### Economics

The log-logistic has been used as a simple model of the [distribution of wealth](/source/Distribution_of_wealth) or [income](/source/Income_distribution) in [economics](/source/Economics), where it is known as the Fisk distribution.[11] Its [Gini coefficient](/source/Gini_coefficient) is [11].[12]

Derivation of Gini coefficient The Gini coefficient for a continuous probability distribution takes the form: G = 1 μ ∫ 0 ∞ F ( 1 − F ) d x {\displaystyle G={1 \over {\mu }}\int _{0}^{\infty }F(1-F)dx} where F {\displaystyle F} is the CDF of the distribution and μ {\displaystyle \mu } is the expected value. For the log-logistic distribution, the formula for the Gini coefficient becomes: G = sin ⁡ ( π / β ) α π / β ∫ 0 ∞ d x [ 1 + ( x / α ) − β ] [ 1 + ( x / α ) β ] {\displaystyle G={\sin(\pi /\beta ) \over {\alpha \pi /\beta }}\int _{0}^{\infty }{dx \over {[1+(x/\alpha )^{-\beta }][1+(x/\alpha )^{\beta }]}}} Defining the substitution z = x / α {\displaystyle z=x/\alpha } leads to the simpler equation: G = sin ⁡ ( π / β ) π / β ∫ 0 ∞ d z ( 1 + z − β ) ( 1 + z β ) {\displaystyle G={\sin(\pi /\beta ) \over {\pi /\beta }}\int _{0}^{\infty }{dz \over {(1+z^{-\beta })(1+z^{\beta })}}} And making the substitution u = 1 / ( 1 + z β ) {\displaystyle u=1/(1+z^{\beta })} further simplifies the Gini coefficient formula to: G = sin ⁡ ( π / β ) π ∫ 0 1 u − 1 / β ( 1 − u ) 1 / β d u {\displaystyle G={\sin(\pi /\beta ) \over {\pi }}\int _{0}^{1}u^{-1/\beta }(1-u)^{1/\beta }du} The integral component is equivalent to the standard beta function B ( 1 − 1 / β , 1 + 1 / β ) {\displaystyle {\text{B}}(1-1/\beta ,1+1/\beta )} . The beta function may also be written as: B ( x , y ) = Γ ( x ) Γ ( y ) Γ ( x + y ) {\displaystyle {\text{B}}(x,y)={\Gamma (x)\Gamma (y) \over {\Gamma (x+y)}}} where Γ ( ⋅ ) {\displaystyle \Gamma (\cdot )} is the gamma function. Using the properties of the gamma function, it can be shown that: B ( 1 − 1 / β , 1 + 1 / β ) = 1 β Γ ( 1 − 1 / β ) Γ ( 1 / β ) {\displaystyle {\text{B}}(1-1/\beta ,1+1/\beta )={1 \over {\beta }}\Gamma (1-1/\beta )\Gamma (1/\beta )} From Euler's reflection formula, the expression can be simplified further: B ( 1 − 1 / β , 1 + 1 / β ) = 1 β π sin ⁡ ( π / β ) {\displaystyle {\text{B}}(1-1/\beta ,1+1/\beta )={1 \over {\beta }}{\pi \over {\sin(\pi /\beta )}}} Finally, we may conclude that the Gini coefficient for the log-logistic distribution G = 1 / β {\displaystyle G=1/\beta } .

### Networking

The log-logistic has been used as a model for the period of time beginning when some data leaves a software user application in a computer and the response is received by the same application after traveling through and being processed by other computers, applications, and network segments, most or all of them without hard [real-time](/source/Real-time_computing) guarantees (for example, when an application is displaying data coming from a remote [sensor](/source/Sensor) connected to the Internet). It has been shown to be a more accurate probabilistic model for that than the [log-normal distribution](/source/Log-normal_distribution) or others, as long as abrupt changes of regime in the sequences of those times are properly detected.[13]

## Related distributions

- If X ∼ LL ⁡ ( α , β ) {\displaystyle X\sim \operatorname {LL} (\alpha ,\beta )} then k X ∼ LL ⁡ ( k α , β ) . {\displaystyle kX\sim \operatorname {LL} (k\alpha ,\beta ).}

- If X ∼ LL ⁡ ( α , β ) {\displaystyle X\sim \operatorname {LL} (\alpha ,\beta )} then X k ∼ LL ⁡ ( α k , β / | k | ) . {\displaystyle X^{k}\sim \operatorname {LL} (\alpha ^{k},\beta /|k|).}

- LL ⁡ ( α , β ) ∼ Dagum ( 1 , α , β ) {\displaystyle \operatorname {LL} (\alpha ,\beta )\sim {\textrm {Dagum}}(1,\alpha ,\beta )} ([Dagum distribution](/source/Dagum_distribution)).

- LL ⁡ ( α , β ) ∼ SinghMaddala ( 1 , α , β ) {\displaystyle \operatorname {LL} (\alpha ,\beta )\sim {\textrm {SinghMaddala}}(1,\alpha ,\beta )} ([Singh–Maddala distribution](/source/Singh%E2%80%93Maddala_distribution) ).

- LL ( γ , σ ) ∼ β ′ ( 1 , 1 , γ , σ ) {\displaystyle {\textrm {LL}}(\gamma ,\sigma )\sim \beta '(1,1,\gamma ,\sigma )} ([Beta prime distribution](/source/Beta_prime_distribution)).

- If *X* has a log-logistic distribution with scale parameter α {\displaystyle \alpha } and shape parameter, β {\displaystyle \beta } then *Y* = log(*X*) has a [logistic distribution](/source/Logistic_distribution) with location parameter log ⁡ ( α ) {\displaystyle \log(\alpha )} and scale parameter 1 / β . {\displaystyle 1/\beta .}

- As the shape parameter β {\displaystyle \beta } of the log-logistic distribution increases, its shape increasingly resembles that of a (very narrow) [logistic distribution](/source/Logistic_distribution). Informally:

- - LL ⁡ ( α , β ) → L ( α , α / β ) as β → ∞ . {\displaystyle \operatorname {LL} (\alpha ,\beta )\to L(\alpha ,\alpha /\beta )\quad {\text{as}}\quad \beta \to \infty .}

- The log-logistic distribution with shape parameter β = 1 {\displaystyle \beta =1} and scale parameter α {\displaystyle \alpha } is the same as the [generalized Pareto distribution](/source/Pareto_distribution#Generalized_Pareto_distributions) with location parameter μ = 0 {\displaystyle \mu =0} , shape parameter ξ = 1 {\displaystyle \xi =1} and scale parameter α : {\displaystyle \alpha :}

- - LL ⁡ ( α , 1 ) = GPD ⁡ ( 1 , α , 1 ) . {\displaystyle \operatorname {LL} (\alpha ,1)=\operatorname {GPD} (1,\alpha ,1).}

- The addition of another parameter (a shift parameter) formally results in a [shifted log-logistic distribution](/source/Shifted_log-logistic_distribution), but this is usually considered in a different parameterization so that the distribution can be bounded above or bounded below.

### Generalizations

Several different distributions are sometimes referred to as the **generalized log-logistic distribution**, as they contain the log-logistic as a special case.[12] These include the [Burr Type XII distribution](/source/Burr_Type_XII_distribution) (also known as the *Singh–Maddala distribution*) and the [Dagum distribution](/source/Dagum_distribution), both of which include a second shape parameter. Both are in turn special cases of the even more general *generalized beta distribution of the second kind*. Another more straightforward generalization of the log-logistic is the [shifted log-logistic distribution](/source/Shifted_log-logistic_distribution) .

Another **generalized log-logistic distribution** is the log-transform of the [metalog distribution](/source/Metalog_distribution), in which power series expansions in terms of p {\displaystyle p} are substituted for [logistic distribution](/source/Logistic_distribution) parameters μ {\displaystyle \mu } . The resulting [log-metalog distribution](/source/Metalog_distribution#/Unbounded,_semibounded,_and_bounded_metalog_distributions) is highly shape flexible, has a simple closed-form [PDF](/source/Probability_density_function) and [quantile function](/source/Quantile_function), can be fit to data with linear least squares, and subsumes the log-logistic distribution as a special case.

## See also

- [Probability distributions: List of important distributions supported on semi-infinite intervals](/source/List_of_probability_distributions#Supported_on_semi-infinite_intervals,_usually_.5B0.2C.E2.88.9E.29)

## References

1. ^ [***a***](#cite_ref-mgf_cite_1-0) [***b***](#cite_ref-mgf_cite_1-1) Leemis, Larry. ["Log-Logistic distribution"](http://www.math.wm.edu/~leemis/chart/UDR/PDFs/Loglogistic.pdf) (PDF). College of William & Mary.

1. ^ [***a***](#cite_ref-mgf_cite2_2-0) [***b***](#cite_ref-mgf_cite2_2-1) Ekawati, D.; Warsono; Kurniasari, D. (2014). "On the Moments, Cumulants, and Characteristic Function of the Log-Logistic Distribution". *IPTEK, the Journal for Technology and Science*. **25** (3): 78–82.

1. **[^](#cite_ref-norton_3-0)** Norton, Matthew; Khokhlov, Valentyn; Uryasev, Stan (2019). ["Calculating CVaR and bPOE for common probability distributions with application to portfolio optimization and density estimation"](https://web.archive.org/web/20230331230821/http://uryasev.ams.stonybrook.edu/wp-content/uploads/2019/10/Norton2019_CVaR_bPOE.pdf) (PDF). *Annals of Operations Research*. **299** (1–2). Springer: 1281–1315. [arXiv](/source/ArXiv_(identifier)):[1811.11301](https://arxiv.org/abs/1811.11301). [doi](/source/Doi_(identifier)):[10.1007/s10479-019-03373-1](https://doi.org/10.1007%2Fs10479-019-03373-1). [S2CID](/source/S2CID_(identifier)) [254231768](https://api.semanticscholar.org/CorpusID:254231768). Archived from [the original](http://uryasev.ams.stonybrook.edu/wp-content/uploads/2019/10/Norton2019_CVaR_bPOE.pdf) (PDF) on 2023-03-31. Retrieved 2023-02-27.

1. ^ [***a***](#cite_ref-Shoukri_4-0) [***b***](#cite_ref-Shoukri_4-1) Shoukri, M.M.; Mian, I.U.M.; Tracy, D.S. (1988), "Sampling Properties of Estimators of the Log-Logistic Distribution with Application to Canadian Precipitation Data", *[The Canadian Journal of Statistics](/source/The_Canadian_Journal_of_Statistics)*, **16** (3): 223–236, [doi](/source/Doi_(identifier)):[10.2307/3314729](https://doi.org/10.2307%2F3314729), [JSTOR](/source/JSTOR_(identifier)) [3314729](https://www.jstor.org/stable/3314729)

1. ^ [***a***](#cite_ref-Ashkar06_5-0) [***b***](#cite_ref-Ashkar06_5-1) Ashkar, Fahim; Mahdi, Smail (2006), "Fitting the log-logistic distribution by generalized moments", *[Journal of Hydrology](/source/Journal_of_Hydrology)*, **328** (3–4): 694–703, [Bibcode](/source/Bibcode_(identifier)):[2006JHyd..328..694A](https://ui.adsabs.harvard.edu/abs/2006JHyd..328..694A), [doi](/source/Doi_(identifier)):[10.1016/j.jhydrol.2006.01.014](https://doi.org/10.1016%2Fj.jhydrol.2006.01.014)

1. **[^](#cite_ref-TadiBurr80_6-0)** Tadikamalla, Pandu R. (1980), "A Look at the Burr and Related Distributions", *International Statistical Review*, **48** (3): 337–344, [doi](/source/Doi_(identifier)):[10.2307/1402945](https://doi.org/10.2307%2F1402945), [JSTOR](/source/JSTOR_(identifier)) [1402945](https://www.jstor.org/stable/1402945)

1. **[^](#cite_ref-7)** McLaughlin, Michael P. (2001), [*A Compendium of Common Probability Distributions*](http://www.causascientia.org/math_stat/Dists/Compendium.pdf) (PDF), p. A–37, retrieved 2008-02-15

1. **[^](#cite_ref-8)** Bennett, Steve (1983), "Log-Logistic Regression Models for Survival Data", *Journal of the Royal Statistical Society, Series C*, **32** (2): 165–171, [doi](/source/Doi_(identifier)):[10.2307/2347295](https://doi.org/10.2307%2F2347295), [JSTOR](/source/JSTOR_(identifier)) [2347295](https://www.jstor.org/stable/2347295)

1. **[^](#cite_ref-9)** Collett, Dave (2003), *Modelling Survival Data in Medical Research* (2nd ed.), CRC press, [ISBN](/source/ISBN_(identifier)) [978-1-58488-325-8](https://en.wikipedia.org/wiki/Special:BookSources/978-1-58488-325-8)

1. **[^](#cite_ref-10)** Di Crescenzo, Antonio; Pellerey, Franco (2019), "Some results and applications of geometric counting processes", *Methodology and Computing in Applied Probability*, **21** (1): 203–233, [doi](/source/Doi_(identifier)):[10.1007/s11009-018-9649-9](https://doi.org/10.1007%2Fs11009-018-9649-9), [S2CID](/source/S2CID_(identifier)) [254793416](https://api.semanticscholar.org/CorpusID:254793416)

1. **[^](#cite_ref-11)** Fisk, P.R. (1961), "The Graduation of Income Distributions", *Econometrica*, **29** (2): 171–185, [doi](/source/Doi_(identifier)):[10.2307/1909287](https://doi.org/10.2307%2F1909287), [JSTOR](/source/JSTOR_(identifier)) [1909287](https://www.jstor.org/stable/1909287)

1. ^ [***a***](#cite_ref-KK_12-0) [***b***](#cite_ref-KK_12-1) Kleiber, C.; Kotz, S (2003), *Statistical Size Distributions in Economics and Actuarial Sciences*, Wiley, [ISBN](/source/ISBN_(identifier)) [978-0-471-15064-0](https://en.wikipedia.org/wiki/Special:BookSources/978-0-471-15064-0)

1. **[^](#cite_ref-13)** Gago-Benítez, A.; Fernández-Madrigal J.-A., Cruz-Martín, A. (2013), "Log-Logistic Modeling of Sensory Flow Delays in Networked Telerobots", *IEEE Sensors Journal*, **13** (8), IEEE Sensors 13(8): 2944–2953, [Bibcode](/source/Bibcode_(identifier)):[2013ISenJ..13.2944G](https://ui.adsabs.harvard.edu/abs/2013ISenJ..13.2944G), [doi](/source/Doi_(identifier)):[10.1109/JSEN.2013.2263381](https://doi.org/10.1109%2FJSEN.2013.2263381), [S2CID](/source/S2CID_(identifier)) [47511693](https://api.semanticscholar.org/CorpusID:47511693){{[citation](https://en.wikipedia.org/wiki/Template:Citation)}}: CS1 maint: multiple names: authors list ([link](https://en.wikipedia.org/wiki/Category:CS1_maint:_multiple_names:_authors_list))

v t e Probability distributions (list) Discrete univariate with finite support Benford Bernoulli Beta-binomial Binomial Categorical Hypergeometric Negative Poisson binomial Rademacher Soliton Discrete uniform Zipf Zipf–Mandelbrot with infinite support Beta negative binomial Borel Conway–Maxwell–Poisson Discrete phase-type Delaporte Extended negative binomial Flory–Schulz Gauss–Kuzmin Geometric Logarithmic Mixed Poisson Negative binomial Panjer Parabolic fractal Poisson Skellam Yule–Simon Zeta Continuous univariate supported on a bounded interval Arcsine ARGUS Balding–Nichols Bates Beta Generalized Beta rectangular Continuous Bernoulli Continuous binomial Irwin–Hall Kumaraswamy Logit-normal Noncentral beta PERT Power function Raised cosine Reciprocal Triangular U-quadratic Uniform Wigner semicircle supported on a semi-infinite interval Benini Benktander 1st kind Benktander 2nd kind Beta prime Burr Chi Chi-squared Noncentral Inverse Scaled Dagum Davis Erlang Hyper Exponential Hyperexponential Hypoexponential Logarithmic F Noncentral Folded normal Fréchet Gamma Generalized Inverse gamma/Gompertz Gompertz Shifted Half-logistic Half-normal Hotelling's T-squared Hartman–Watson Inverse Gaussian Generalized Kolmogorov Lévy Log-Cauchy Log-Laplace Log-logistic Log-normal Log-t Lomax Matrix-exponential Maxwell–Boltzmann Maxwell–Jüttner Mittag-Leffler Nakagami Pareto Phase-type Poly-Weibull Rayleigh Relativistic Breit–Wigner Rice Truncated normal type-2 Gumbel Weibull Discrete Wilks's lambda supported on the whole real line Cauchy Exponential power Fisher's z Kaniadakis κ-Gaussian Gaussian q Generalized hyperbolic Generalized logistic (logistic-beta) Generalized normal Geometric stable Gumbel Holtsmark Hyperbolic secant Johnson's SU Landau Laplace Asymmetric Logistic Noncentral t Normal (Gaussian) Normal-inverse Gaussian Skew normal Slash Stable Student's t Tracy–Widom Variance-gamma Voigt with support whose type varies Generalized chi-squared Generalized extreme value Generalized Pareto Marchenko–Pastur Kaniadakis κ-exponential Kaniadakis κ-Gamma Kaniadakis κ-Weibull Kaniadakis κ-Logistic Kaniadakis κ-Erlang q-exponential q-Gaussian q-Weibull Shifted log-logistic Tukey lambda Mixed univariate continuous- discrete Rectified Gaussian Multivariate (joint) Discrete: Ewens Multinomial Dirichlet Negative Continuous: Dirichlet Generalized Multivariate Laplace Multivariate normal Multivariate stable Multivariate t Normal-gamma Inverse Matrix-valued: LKJ Matrix beta Matrix F Matrix normal Matrix t Matrix gamma Inverse Wishart Normal Inverse Normal-inverse Complex Uniform distribution on a Stiefel manifold Directional Univariate (circular) directional Circular uniform Univariate von Mises Wrapped normal Wrapped Cauchy Wrapped exponential Wrapped asymmetric Laplace Wrapped Lévy Bivariate (spherical) Kent Bivariate (toroidal) Bivariate von Mises Multivariate von Mises–Fisher Bingham Degenerate and singular Degenerate Dirac delta function Singular Cantor Families Circular Compound Poisson Elliptical Exponential Natural exponential Location–scale Maximum entropy Mixture Pearson Tweedie Wrapped Category Commons

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