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".
Probability density function | |||
Cumulative distribution function | |||
Parameters |
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Support | |||
CDF | |||
Mean | where Β() is the beta function | ||
Median | |||
Mode | |||
Variance | |||
Skewness | |||
Excess kurtosis | where moments (see) | ||
CF |
where is the Gamma function and is the Fox H-function.[1] |
The Burr (Type XII) distribution has probability density function:[4][5]
The 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 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.