Beta exponential distribution

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The beta exponential distribution is a generalization of the exponential distribution, generated from from the logit — the log odds of a probability — of a beta random variable. The distribution was introduced by Nadarajah and Kotz [1]. With two extra shape parameters, the beta exponential distribution can fit a wider range of data than the exponential distribution; therefore, it has been widely used in life testing [2]. 

Other names for the beta exponential distribution include Gompertz-Verhulst, generalized Gompertz-Verhulst type III, log-beta, and exponential generalized beta type I distribution [3, 4, 5].

Beta Exponential distribution properties

CDF of the beta exponential distribution
CDF of the beta exponential distribution at parameters α = 3, β = 4, λ = 1.8.

The probability density function (PDF) of the beta exponential distribution is given by [1]

pdf of the beta exponential distribution

Where α, β = shape parameters, although note that its shape depends only on α. Also, some authors use various other notation for the parameterization such as θ or m1 for λ, e.g., [6, 7].

The cumulative distribution function (CDF) is

beta exponential distribution cdf

The exponentiated exponential distribution is a particular case when β = 1; the exponential distribution (with parameter βλ) is the particular case when α = 1. Other special cases include the Nadarajah-Kotz distribution and hyperbolic sine distribution. In addition, the beta exponential distribution is a limit of the generalized beta distribution [8].

The hazard rate is

beta exponential hazard rate


is also the incomplete beta function.


[1] Saralees Nadarajah, Samuel Kotz, The beta exponential distribution, Reliability Engineering & System Safety, Volume 91, Issue 6, 2006, Pages 689-697, ISSN 0951-8320,

[2] Guan, R., Cheng, W., Rong, Y. et al. Parameter Estimation of Beta-Exponential Distribution Using Linear Combination of Order Statistics. Commun. Math. Stat. (2023).

[3] J. C. Ahuja and Stanley W. Nash. The generalized Gompertz-Verhulst family of distributions. Sankhy¯a, 29:141–156 (1967).
stable/25049460. (pages 102, 102, 198, 198, 198, and 199).

[4] Saralees Nadarajah and Samuel Kotz. The beta exponential distribution. Reliability Eng. Sys. Safety, 91:689–697 (2006). doi:10.1016/j.ress.2005.05.008.
(pages 102, 104, 106, 106, 106, 106, 106, 106, and 106).

[5] Srividya Iyer-Biswas, Gavin E. Crooks, Norbert F. Scherer, and Aaron R.
Dinner. Universality in stochastic exponential growth. Phys. Rev. Lett.,
113:028101 (2014). doi:10.1103/PhysRevLett.113.028101. (page 102).

[6] Singh, B. & Goel, E. The Beta Inverted Exponential Distribution: Properties and Applications. International Journal of Applied Science and Mathematics. Volume 2, Issue 5, ISSN (Online): 2394-2894

[7] Mahmoud, M. & Amer, N. New Mixture of Two Beta Exponential Distributions and Income Distribution. Journal of Statistics Applications & Probability. 11, No. 2, 395-402 (2022)

[8] Crooks, G. (2019). Field Guide to Continuous Probability Distributions. Berkeley Institute for Theoretical Sciences (BITS).

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