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".
The Burr (Type XII) distribution has probability density function:[4][5]
The λ λ --> {\displaystyle \lambda } parameter scales the underlying variate and is a positive real.
The cumulative distribution function is:
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.
Given a random variable U {\displaystyle U} drawn from the uniform distribution in the interval ( 0 , 1 ) {\displaystyle \left(0,1\right)} , the random variable
has a Burr Type XII distribution with parameters c {\displaystyle c} , k {\displaystyle k} and λ λ --> {\displaystyle \lambda } . This follows from the inverse cumulative distribution function given above.
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