Burr distribution

Burr Type XII

Probability density function
Cumulative distribution function
Parameters	$c > 0\!$ $k > 0\!$
Support	$x > 0\!$
PDF	$ck\frac{x^{c-1}}{(1+x^c)^{k+1}}\!$
CDF	$1-\left(1+x^c\right)^{-k}$
Mean	$\mu_1=k\operatorname{\Beta}(k-1/c,\, 1+1/c)$ where Β() is the beta function
Median	$\left(2^{\frac{1}{k}}-1\right)^\frac{1}{c}$
Mode	$\left(\frac{c-1}{kc+1}\right)^\frac{1}{c}$
Variance	$-\mu_1^2+\mu_2$
Skewness	$2\mu_1^3-3\mu_1\mu_2+\mu_3$
Ex. kurtosis	$-3\mu_1^4+6\mu_1^2\mu_2-4\mu_1\mu_3+\mu_4$ where moments (see) $\mu_r =k\operatorname{\Beta}\left(\frac{ck-r}{c},\, \frac{c+r}{c}\right)$

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:

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

and cumulative distribution function:

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

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.

See also

• Dagum distribution, also known as the inverse Burr Distribution.