Burr distribution

Probability distribution
name =Burr| type =density
pdf_

cdf_

parameters =$c\; >\; 0!$
$k\; >\; 0!$
support =$x\; >\; 0!$
pdf =$ckfrac\{x^\{c-1${(1+x^c)^{k+1!
cdf =$1-left(1+x^c\; ight)^\{-k\}$
mean =$koperatorname\{B\}(k-1/c,,\; 1+1/c)$ where B() is the beta function
median =$left(2^\{frac\{1\}\{k-1\; ight)^frac\{1\}\{c\}$
mode =$left(frac\{c-1\}\{kc+1\}\; ight)^frac\{1\}\{c\}$
variance =
skewness =
kurtosis =
entropy =
mgf =
char =

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 distribution has probability density function: [Maddala, G.S.. 1983, 1996. "Limited-Dependent and Qualitative Variables in Econometrics". Cambridge University Press.] [Citation|title=A Look at the Burr and Related Distributions| first=Pandu R.| last=Tadikamalla| journal=International Statistical Review| volume=48| number=3| year=1980| pages=337-344| url=http://links.jstor.org/sici?sici=0306-7734%28198012%2948%3A3%3C337%3AALATBA%3E2.0.CO%3B2-Z]

:$p(x,c,k)\; =\; ckfrac\{x^\{c-1\{(1+x^c)^\{k+1!$

and cumulative distribution function:

:$P(x,c,k)\; =\; 1-left(1+x^c\; ight)^\{-k\}$

References

ee also

Log-logistic distribution