Burr distribution

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Burr
Probability density function
Cumulative distribution function
Parameters c > 0\!
k > 0\!
Support x > 0\!
Probability density function (pdf) ck\frac{x^{c-1}}{(1+x^c)^{k+1}}\!
Cumulative distribution function (cdf) 1-\left(1+x^c\right)^{-k}
Mean k\operatorname{B}(k-1/c,\, 1+1/c) where B() 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
Skewness
Excess kurtosis
Entropy
Moment-generating function (mgf)
Characteristic function

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:[1][2]

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

and cumulative distribution function:

P(x,c,k) = 1-\left(1+x^c\right)^{-k}

[edit] References

 Please expand this article using the suggested source(s) below.
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  1. ^ Maddala, G.S.. 1983, 1996. Limited-Dependent and Qualitative Variables in Econometrics. Cambridge University Press.
  2. ^ Tadikamalla, Pandu R. (1980), “A Look at the Burr and Related Distributions”, International Statistical Review 48 (3): 337-344, <http://links.jstor.org/sici?sici=0306-7734%28198012%2948%3A3%3C337%3AALATBA%3E2.0.CO%3B2-Z> 

[edit] See also

Log-logistic distribution

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Probability distributions
 
Continuous univariate supported on a semi-infinite interval