# Discrete Weibull Distribution

The discrete Weibull distribution is the discrete variant of the Weibull distribution. It is used to model lifetime data expressed by discrete random variables. For example, it can be used to model equipment that operates in cycles, devices with on/off switches, or equipment with to-and-from motion.

## Types of Discrete Weibull Distribution

Three types of discrete Weibull distribution are well studied, but none are exact analogs of the continuous Weibull distribution:

• The type-I discrete Weibull distribution, introduced by Nakagawa and Osaki in 1975 [1], retains the cumulative distribution function (CDF) of the continuous Weibull distribution.
• The type-II discrete Weibull distribution, introduced by Stein and Dattero in 1984 [2], retains the continuous distribution’s hazard rate. They specified the hazard rate function and probability mass function as:

It is impossible to construct a discrete Weibull distribution that retains both the hazard rate and PMF of the continuous Weibull distribution [3].

• Type III, introduced by Padgett and Spurrier [4], has a flexible hazard rate function that can take on a variety of shapes. However, the PMF is more complex than Type I and Type II:

Other discrete types have appeared in the literature, including Nooghabi et al.’s discrete modified version [5] and exponentiated discrete Weibull [6].

## References

[1] Nakagawa and Osaki (1975). The discrete Weibull distribution; IEEE Transactions on Reliability. Volume R-25, Issue 5.

[2] Stein and Dattero (1984). A new discrete Weibull. IEEE Transactions on Reliability. Volume R-33 Issue 2.

[3] Horst Rinne. 20 Nov 2008, Related distributions from: The Weibull Distribution, A Handbook CRC Press. Accessed on: 08 May 2022
https://www.routledgehandbooks.com/doi/10.1201/9781420087444.ch3

[4] Padgett and Spurrier (1985). Discrete Failure Models. IEEE Transactions on Reliability. R-34, Issue 3.

[5] Nooghabi, M.S., Roknabadi, A.H.R. and Borzadaran, G.M. (2011). Discrete modified Weibull distribution. Metron, LXIX, 207–222.

[6] Nekoukhou & Bidram. (2015). The exponentiated discrete Weibull distribution. SORT 39 (1) January-June, 127-146 .