• Yekti Widyaningsih Department of Mathematics, Faculty of Mathematics and Natural Science, University of Indonesia, Indonesia https://orcid.org/0000-0002-1309-6916
  • Rugun Ivana Department of Mathematics, Faculty of Mathematics and Natural Science, University of Indonesia, Indonesia
Keywords: Exponential-Poisson Distribution, Poisson Distribution, Weibull Distribution, Maximum Likelihood Method, Mathematics topic


The Weibull-Poisson distribution represents a continuous distribution type applicable to various forms of hazard, including monotone up, monotone down, and upside-down bathtub shapes that ascend. The distribution characterizes lifetimes and can effectively model failures within a series of systems, which evolves from the Exponential-Poisson distribution. This distribution emerges through the compounding of the Weibull Distribution and Zero Truncated Poisson Distribution. The compounding itself integrates several mathematical properties, such as statistical order and Taylor’s number expansion, to reach its final form. Alongside the formulation of the Weibull-Poisson distribution, this paper includes the probability density function, distribution function, rth moment, rth central moment, mean, and variance. For illustration, the Weibull-Poisson distribution is applied to guinea pig survival data after being infected with Turblece virus Bacilli.


Download data is not yet available.


S. L. K. Adamidis, "A lifetime distribution with decreasing failure rate," Statistics & Probability Letters, pp. 35-42, 1998.

F. C.-N. Wagner Barreto-Souza, "A generalization of the exponential-Poisson distribution," Statistics & Probability Letters, pp. 2493-2500, 2009.

S. R. Rasool Tahmasbi, "A two-parameter lifetime distribution with decreasing failure rate," Computational Statistics & Data Analysis, pp. 3889-3901, 2008.

W. Barreto-Souza, A. L. d. Morais and G. M. Cordeiro, "The Weibull-geometric distribution," Journal of Statistical Computation and Simulation, pp. 645-657, 2011.

V. P. Roxana Ciumara, "The Weibull-Logarithmic Distribution in Lifetime Analysis and It's Properties," ASMDA. Proceedings of the International Conference Applied Stochastic Models and Data Analysis, p. 395, 2009.

D. S. Wanbo Lu, "A new compounding life distribution: the Weibull–Poisson distribution," Journal of Applied Statistics, pp. 21-38, 2012.

T. Bjerkedal, "Acquisition of Resistance in Guinea Pies infected with Different Doses of Virulent Tubercle Bacilli," American Journal of Hygiene, pp. 130-148, 1960.

V. P. K. A. Sotirios Loukas, "A Generalization of the Exponential-Logarithmic Distribution," Journal of Statistical Theory and Practice, p. 395, 2009.

J. W. M. A. T. C. Robert V. Hogg, Introduction to Mathematical Statistics, 2019.

R. L. Burden, J. D. Faires, and A. M. Burden, Numerical Analysis, 10th ed. Singapore: Cengage Learning Asia Pte Ltd, 2015.

W. J. Conover, Practical Nonparametric Statistics. New York City, New York: Wiley, 1999.

D C M Dickson, M. Hardy, and H. R. Waters, Actuarial mathematics for life contingent risks. New York: Cambridge University Press, 2013.

R. E. Glaser, "Bathtub and Related Failure Rate Characterizations," Journal of the American Statistical Association, pp. 667-672, 1980.

M. K. David G. Kleinbaum, Survival Analysis: A Self-Learning Text. United State: Springer, 2005.

H. H. P. G. E. W. Stuart A. Klugman, Loss models: from data to decisions. United State: John Wiley & Sons, 2012.

How to Cite
Y. Widyaningsih and R. Ivana, “WEIBULL-POISSON DISTRIBUTION AND THEIR APPLICATION TO SYSTEMATIC PARALLEL RISK”, BAREKENG: J. Math. & App., vol. 18, no. 1, pp. 0029-0042, Mar. 2024.