Burr distribution
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". Definitions

Probability density function 

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

 

The  parameter scales the underlying variate and is a positive real. Cumulative distribution function 

The cumulative distribution function is:

Applications 

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. Random variate generation 

Given a random variable  drawn from the uniform distribution in the interval , the random variable

has a Burr Type XII distribution with parameters , and . This follows from the inverse cumulative distribution function given above. Related distributions 

 When c = 1, the Burr distribution becomes the Pareto Type II (Lomax) distribution. When k = 1, the Burr distribution is a log-logistic distribution sometimes referred to as the Fisk distribution, a special case of the Champernowne 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. The Dagum distribution, also known as the inverse Burr distribution, is the distribution of 1 / X, where X has the Burr distribution

References

Further reading

External links 

 

Continuous distributions
Systems of probability distributions