A random variable X is said to have the beta distribution with parameters a and b if its probability density function is
where
denotes the beta function. The beta family, whose origin can be traced to 1676. in a letter from Sir Isaac Newton to Henry Oldenberg, has been utilized extensively in statistical theory and practice.
Originally defined on the unit interval, many generalizations of (1) have been proposed in the literature; see Karian and Dudewicz (2000) for a four parameter generalization defined over a finite interval, McDonald and Richards (1987a, b) for a generalization obtained by power transformation of X; Libby and Novick (1982) and Armero and Bayarri (1994) for generalizations obtained by dividing (1) by certain algebraic functions; Gordy (1998) for a generalization obtained by multiplying (1) by an exponential function; and,...
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References and Further Reading
Armero C, Bayarri MJ (1994) Prior assessments for prediction in queues. The Statistician 43:139–153
Gordy MB (1988) Computationally convenient distributional assumptions for commonvalue auctions. Comput Econ 12: 61–78
Gupta AK, Nadarajah S (2004) Handbook of beta distribution and its applications. Marcel Dekker, New York
Johnson NL, Kotz S, Balakrishnan N (1994) Continuous univariate distributions, vol 2, 2nd edn. Wiley, New York
Karian ZA, Dudewicz EJ (2000) Fitting statistical distributions: the generalized lambda distribution and generalized bootstrap methods. CRC, Boca Raton, Florida
Libby DL, Novick MR (1982) Multivariate generalized beta-distributions with applications to utility assessment. J Educ Stat 7:271–294
McDonald JB, Richards DO (1987a) Model selection: some generalized distributions. Commun Stat 16A:1049–1074
McDonald JB, Richards DO (1987b) Some generalized models with application to reliability. J Stat Plann Inf 16:365–376
Nadarajah S, Kotz S (2004) A generalized beta distribution II. InterStat
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Gupta, A.K. (2011). Beta Distribution. In: Lovric, M. (eds) International Encyclopedia of Statistical Science. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-04898-2_144
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