Burr Type XII
Probability density function  |
Cumulative distribution function  |
Parameters |
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Support |
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PDF |
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CDF |
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Mean |
where Β() is the beta function |
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Median |
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Mode |
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Variance |
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Skewness |
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Ex. kurtosis |
where moments (see)  |
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CF |
![{\displaystyle ={\frac {c(-it)^{kc)){\Gamma (k)))H_{1,2}^{2,1}\!\left[(-it)^{c}\left|{\begin{matrix}(-k,1)\\(0,1),(-kc,c)\end{matrix))\right.\right],t\neq 0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4d57864ffe50078a34292d9f949f09e0f47ab2b3)
 where is the Gamma function and is the Fox H-function.[1] |
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In probability theory, statistics and econometrics, the Burr Type XII distribution or simply the Burr distribution[2] is a continuous probability distribution for a non-negative random variable. It is also known as the Singh–Maddala distribution[3] and is one of a number of different distributions sometimes called the "generalized log-logistic distribution".
Definitions
Probability density function
The Burr (Type XII) distribution has probability density function:[4][5]
![{\displaystyle {\begin{aligned}f(x;c,k)&=ck{\frac {x^{c-1)){(1+x^{c})^{k+1))}\\[6pt]f(x;c,k,\lambda )&={\frac {ck}{\lambda ))\left({\frac {x}{\lambda ))\right)^{c-1}\left[1+\left({\frac {x}{\lambda ))\right)^{c}\right]^{-k-1}\end{aligned))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a3c758cb4945e4a43a37fb66ad628b53ca8de40c)
The
parameter scales the underlying variate and is a positive real.
Cumulative distribution function
The cumulative distribution function is:

![{\displaystyle F(x;c,k,\lambda )=1-\left[1+\left({\frac {x}{\lambda ))\right)^{c}\right]^{-k))](https://wikimedia.org/api/rest_v1/media/math/render/svg/00238e8a5ecdca4563db2344bfca37b93baee67f)
Applications
It is most commonly used to model household income, see for example: Household income in the U.S. and compare to magenta graph at right.