[{"@context":"http:\/\/schema.org\/","@type":"BlogPosting","@id":"https:\/\/wiki.edu.vn\/en\/wiki5\/burr-distribution-wikipedia\/#BlogPosting","mainEntityOfPage":"https:\/\/wiki.edu.vn\/en\/wiki5\/burr-distribution-wikipedia\/","headline":"Burr distribution – Wikipedia","name":"Burr distribution – Wikipedia","description":"before-content-x4 From Wikipedia, the free encyclopedia Burr Type XII after-content-x4 Probability density function Cumulative distribution function after-content-x4 Parameters 0!”\/> 0!”\/>","datePublished":"2015-02-23","dateModified":"2015-02-23","author":{"@type":"Person","@id":"https:\/\/wiki.edu.vn\/en\/wiki5\/author\/lordneo\/#Person","name":"lordneo","url":"https:\/\/wiki.edu.vn\/en\/wiki5\/author\/lordneo\/","image":{"@type":"ImageObject","@id":"https:\/\/secure.gravatar.com\/avatar\/c9645c498c9701c88b89b8537773dd7c?s=96&d=mm&r=g","url":"https:\/\/secure.gravatar.com\/avatar\/c9645c498c9701c88b89b8537773dd7c?s=96&d=mm&r=g","height":96,"width":96}},"publisher":{"@type":"Organization","name":"Enzyklop\u00e4die","logo":{"@type":"ImageObject","@id":"https:\/\/wiki.edu.vn\/wiki4\/wp-content\/uploads\/2023\/08\/download.jpg","url":"https:\/\/wiki.edu.vn\/wiki4\/wp-content\/uploads\/2023\/08\/download.jpg","width":600,"height":60}},"image":{"@type":"ImageObject","@id":"https:\/\/upload.wikimedia.org\/wikipedia\/commons\/thumb\/d\/d4\/Burr_pdf.svg\/325px-Burr_pdf.svg.png","url":"https:\/\/upload.wikimedia.org\/wikipedia\/commons\/thumb\/d\/d4\/Burr_pdf.svg\/325px-Burr_pdf.svg.png","height":"252","width":"325"},"url":"https:\/\/wiki.edu.vn\/en\/wiki5\/burr-distribution-wikipedia\/","wordCount":5276,"articleBody":"     (adsbygoogle = window.adsbygoogle || []).push({});before-content-x4From Wikipedia, the free encyclopediaBurr Type XII       (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4Probability density functionCumulative distribution function       (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4Parameters0!”\/> 0!”\/>Support0!”\/>PDF       (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4ckxc\u22121(1+xc)k+1{displaystyle ck{frac {x^{c-1}}{(1+x^{c})^{k+1}}}!}CDF1\u2212(1+xc)\u2212k{displaystyle 1-left(1+x^{c}right)^{-k}}Mean\u03bc1=kB\u2061(k\u22121\/c,1+1\/c){displaystyle mu _{1}=koperatorname {mathrm {B} } (k-1\/c,,1+1\/c)} where \u0392() is the beta functionMedian(21k\u22121)1c{displaystyle left(2^{frac {1}{k}}-1right)^{frac {1}{c}}}Mode(c\u22121kc+1)1c{displaystyle left({frac {c-1}{kc+1}}right)^{frac {1}{c}}}Variance\u2212\u03bc12+\u03bc2{displaystyle -mu _{1}^{2}+mu _{2}}Skewness2\u03bc13\u22123\u03bc1\u03bc2+\u03bc3(\u2212\u03bc12+\u03bc2)3\/2{displaystyle {frac {2mu _{1}^{3}-3mu _{1}mu _{2}+mu _{3}}{left(-mu _{1}^{2}+mu _{2}right)^{3\/2}}}}Ex. kurtosis\u22123\u03bc14+6\u03bc12\u03bc2\u22124\u03bc1\u03bc3+\u03bc4(\u2212\u03bc12+\u03bc2)2\u22123{displaystyle {frac {-3mu _{1}^{4}+6mu _{1}^{2}mu _{2}-4mu _{1}mu _{3}+mu _{4}}{left(-mu _{1}^{2}+mu _{2}right)^{2}}}-3} where moments (see) \u03bcr=kB\u2061(ck\u2212rc,c+rc){displaystyle mu _{r}=koperatorname {mathrm {B} } left({frac {ck-r}{c}},,{frac {c+r}{c}}right)}CF=c(\u2212it)kc\u0393(k)H1,22,1[(\u2212it)c|(\u2212k,1)(0,1),(\u2212kc,c)],t\u22600{displaystyle ={frac {c(-it)^{kc}}{Gamma (k)}}H_{1,2}^{2,1}!left[(-it)^{c}left|{begin{matrix}(-k,1)\\(0,1),(-kc,c)end{matrix}}right.right],tneq 0}=1,t=0{displaystyle =1,t=0}where \u0393{displaystyle Gamma } is the Gamma function and H{displaystyle H} is the Fox H-function.[1]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\u2013Maddala distribution[3] 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 for example: Household income in the U.S. and compare to magenta graph at right.The Burr (Type XII) distribution has probability density function:[4][5]f(x;c,k)=ckxc\u22121(1+xc)k+1f(x;c,k,\u03bb)=ck\u03bb(x\u03bb)c\u22121[1+(x\u03bb)c]\u2212k\u22121{displaystyle {begin{aligned}f(x;c,k)&=ck{frac {x^{c-1}}{(1+x^{c})^{k+1}}}\\[6pt]f(x;c,k,lambda )&={frac {ck}{lambda }}left({frac {x}{lambda }}right)^{c-1}left[1+left({frac {x}{lambda }}right)^{c}right]^{-k-1}end{aligned}}}and cumulative distribution function:F(x;c,k)=1\u2212(1+xc)\u2212k{displaystyle F(x;c,k)=1-left(1+x^{c}right)^{-k}}F(x;c,k,\u03bb)=1\u2212[1+(x\u03bb)c]\u2212k{displaystyle F(x;c,k,lambda )=1-left[1+left({frac {x}{lambda }}right)^{c}right]^{-k}}Table of ContentsRelated distributions[edit]References[edit]Further reading[edit]External links[edit]Related distributions[edit]The Burr Type XII distribution is a member of a system of continuous distributions introduced by Irving W. Burr (1942), which comprises 12 distributions.[8]The Dagum distribution, also known as the inverse Burr distribution, is the distribution of 1 \/ X, where X has the Burr distributionReferences[edit]^ Nadarajah, S.; Pog\u00e1ny, T. K.; Saxena, R. K. (2012). “On the characteristic function for Burr distributions”. Statistics. 46 (3): 419\u2013428. doi:10.1080\/02331888.2010.513442.^ Burr, I. W. (1942). “Cumulative frequency functions”. Annals of Mathematical Statistics. 13 (2): 215\u2013232. doi:10.1214\/aoms\/1177731607. JSTOR\u00a02235756.^ Singh, S.; Maddala, G. (1976). “A Function for the Size Distribution of Incomes”. Econometrica. 44 (5): 963\u2013970. doi:10.2307\/1911538. JSTOR\u00a01911538.^ Maddala, G. S. (1996) [1983]. Limited-Dependent and Qualitative Variables in Econometrics. Cambridge University Press. ISBN\u00a00-521-33825-5.^ Tadikamalla, Pandu R. (1980), “A Look at the Burr and Related Distributions”, International Statistical Review, 48 (3): 337\u2013344, doi:10.2307\/1402945, JSTOR\u00a01402945^ C. Kleiber and S. Kotz (2003). Statistical Size Distributions in Economics and Actuarial Sciences. New York: Wiley. See Sections 7.3 “Champernowne Distribution” and 6.4.1 “Fisk Distribution.”^ Champernowne, D. G. (1952). “The graduation of income distributions”. Econometrica. 20 (4): 591\u2013614. doi:10.2307\/1907644. JSTOR\u00a01907644.^ See Kleiber and Kotz (2003), Table 2.4, p. 51, “The Burr Distributions.”Further reading[edit]External links[edit]     (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4"},{"@context":"http:\/\/schema.org\/","@type":"BreadcrumbList","itemListElement":[{"@type":"ListItem","position":1,"item":{"@id":"https:\/\/wiki.edu.vn\/en\/wiki5\/#breadcrumbitem","name":"Enzyklop\u00e4die"}},{"@type":"ListItem","position":2,"item":{"@id":"https:\/\/wiki.edu.vn\/en\/wiki5\/burr-distribution-wikipedia\/#breadcrumbitem","name":"Burr distribution – Wikipedia"}}]}]