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# Burr distribution

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 Title: Burr distribution Author: World Heritage Encyclopedia Language: English Subject: Collection: Publisher: World Heritage Encyclopedia Publication Date:

### Burr distribution

 Parameters Probability density function Cumulative distribution function c > 0\! k > 0\! x > 0\! ck\frac{x^{c-1}}{(1+x^c)^{k+1}}\! 1-\left(1+x^c\right)^{-k} k\operatorname{\Beta}(k-1/c,\, 1+1/c) where Β() is the beta function \left(2^{\frac{1}{k}}-1\right)^\frac{1}{c} \left(\frac{c-1}{kc+1}\right)^\frac{1}{c}

In probability theory, statistics and econometrics, the Burr Type XII distribution or simply the Burr distribution is a continuous probability distribution for a non-negative random variable. It is also known as the Singh–Maddala distribution and is one of a number of different distributions sometimes called the "generalized log-logistic distribution". It is most commonly used to model household income (See: Household income in the U.S. and compare to magenta graph at right).

The Burr (Type XII) distribution has probability density function:

f(x;c,k) = ck\frac{x^{c-1}}{(1+x^c)^{k+1}}\!
F(x;c,k) = 1-\left(1+x^c\right)^{-k} .

Note when c=1, the Burr distribution becomes the Pareto Type II distribution. When k=1, the Burr distribution is a special case of the Champernowne distribution, often referred to as the Fisk distribution.

The Burr Type XII distribution is a member of a system of continuous distributions introduced by Irving W. Burr (1942), which comprises 12 distributions.