Some One Parameter Models for Continuous Random Variables Defined on the Interval [0, 1]

Author(s)
John J. Wiorkowski

ABSTRACT

The Beta Distribution is almost exclusively used for situations, after range normalization, wherein a continuous random variable is defined on the closed range [0, 1]. Since the beta distribution is intrinsically a two parameter distribution, this creates problems in some applications where specification of more than one parameter is difficult. In this note, two new classes of single parameter continuous probability distributions on a closed interval are introduced. These distributions remove some of the theoretical and practical problems of using the Beta Distribution for applications. The Burr Type XI Distribution has desirable characteristics for many applications especially when there is ambiguity in the definition of the specified parameter.

Cite this paper

J. Wiorkowski, "Some One Parameter Models for Continuous Random Variables Defined on the Interval [0, 1],"*Applied Mathematics*, Vol. 4 No. 4, 2013, pp. 604-613. doi: 10.4236/am.2013.44085.

J. Wiorkowski, "Some One Parameter Models for Continuous Random Variables Defined on the Interval [0, 1],"

References

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[2] A. O’Hagan, C. Buck, A. Daneshkhah, J. Eiser, P. Garth waite, D. Jenkinson, J. Oakley and T. Rakow, “Uncertain Judgements, Eliciting Experts’ Probabilities,” John Wiley and Sons, Ltd., Chichester, 2006. doi:10.1002/0470033312

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[4] D. Johnson, “The Triangular Distribution as a Proxy for the Beta Distribution in Risk Analysis,” The Statistician, Vol. 46, No. 3, 1997, pp. 387-398. doi:10.1111/1467-9884.00091

[5] A. Lau, H. Lau and Y. Zhang, “A Simple and Logical Alternative for Making PERT Time Estimates,” IIE Transactions, Vol. 28, No. 3, 1996, pp. 183-192. doi:10.1080/07408179608966265

[6] H. Lau, A. Lau and C. Ho, “Improved Moment-Estimation Formulas Using More than Three Subjective Fractiles,” Management Science, Vol. 44, No. 3, 1998, pp. 346-351. doi:10.1287/mnsc.44.3.346

[7] S. Mohan, M. Gopalakrishnan, H. Balasubramanian and A. Chandrashekar, “A Lognormal Approximation of Activity Duration in PERT Using Two Time Estimates,” Journal of the Operational Research Society, Vol. 58, No. 6, 2007, pp. 827-831. doi:10.1057/palgrave.jors.2602204

[8] I. J. Premachandra, “An Approximation of the Activity Duration Distribution in PERT,” Computers and Operations Research, Vol. 28, No. 5, 2001, pp. 443-452. doi:10.1016/S0305-0548(99)00129-X

[9] W. Fazar, “Program Evaluation and Review Technique,” The American Statistician, Vol. 13, No. 1, 1959, p. 10.

[10] M. Trout, “On the Generality of the PERT Average Time Formula,” Decisions Sciences, Vol. 20, No. 2, 1989, pp. 410-412. doi:10.1111/j.1540-5915.1989.tb01888.x

[11] A. Davison, “Statistical Models,” Cambridge University Press, London, 2003. doi:10.1017/CBO9780511815850

[12] A. Edwards, “Gilbert’s Sine Distribution,” Teaching Statistics, Vol. 22, No. 3, 2000, pp. 70-71. doi:10.1111/1467-9639.00026

[13] J. Proakis, “Digital Communications,” 3rd Edition, McGraw Hill, Inc., New York, 1995.

[14] I. Burr, “Cumulative Frequency Functions,” Annals of Mathematical Statistics, Vol. 13, No. 2, 1942, pp. 215-232. doi:10.1214/aoms/1177731607

[15] S. Kotz and N. Johnson, “Encyclopedia of Statistical Sciences,” John Wiley and Sons, Inc., New York, 1982.

[16] J. Berny, “A New Distribution Function for Risk Analysis,” Journal of the Operational Research Society, Vol. 40, No. 2, 1989, pp. 1121-1127.

[17] C. W. Topp and F. C. Leone, “A Family of J-Shaped Distributions,” Journal of the American Statistical Association, Vol. 50, No. 269, 1955, pp. 209-219. doi:10.1080/01621459.1955.10501259

[18] N. Johnson, S. Kotz and N. Balakrishnan, “Continuous Univariate Distributions,” Vol. 2, 2nd Edition, John Wiley and Sons, Inc., New York, 1995.

[19] N. Johnson, S. Kotz and N. Balakrishnan, “Continuous Univariate Distributions,” Vol. 1, 2nd Edition, John Wiley and Sons, Inc. New York, 1994.