# 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[1] is a continuous probability distribution for a non-negative random variable. It is also known as the Singh–Maddala distribution[2] 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:[3][4]

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

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.[5][6]

The Burr Type XII distribution is a member of a system of continuous distributions introduced by Irving W. Burr (1942), which comprises 12 distributions.[7]