Burr distribution

Burr

Probability density function
Cumulative distribution function
Parameters	<math>c > 0\!</math> <math>k > 0\!</math>
Support	<math>x > 0\!</math>
Probability density function (pdf)	<math>ck\frac{x^{c-1
Cumulative distribution function (cdf)	{{{cdf}}}
Mean	{{{mean}}}
Median	{{{median}}}
Mode	{{{mode}}}
Variance	{{{variance}}}
Skewness	{{{skewness}}}
Excess kurtosis	{{{kurtosis}}}
Entropy	{{{entropy}}}
Moment-generating function (mgf)	{{{mgf}}}
Characteristic function	{{{char}}}

{(1+x^c)^{k+1}}\!</math>|

 cdf        =<math>1-\left(1+x^c\right)^{-k}</math>|
 mean       =<math>\left(-\frac{k}{2k+1}\right)^\frac{1}{c}</math>|
 median     =<math>\left(2^{-\frac{1}{k}}-1\right)^\frac{1}{c}</math>|
 mode       =<math>\left(-\frac{ck-1}{2ck+c+1}\right)^\frac{1}{c}</math>|
 variance   =|
 skewness   =|
 kurtosis   =|
 entropy    =|
 mgf        =|
 char       =|

}} The Burr distribution is a probability distribution used in econometrics. 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]

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

and continuous distribution function:

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

References

1. ^ Maddala, G.S.. 1983, 1996. Limited-Dependent and Qualitative Variables in Econometrics. Cambridge University Press.