# Weibull Distribution

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## 1. Weibull Distribution

The Weibull distribution is a continuous probability distribution for assessing product reliability, analyzing life data and modeling failure times.

It is an example of an extreme value distribution (EVD) and is sometimes called EVD Type III. Extreme values are found in a distribution’s tails; EVDs are the limiting distributions for these values. The other two EVDs are the Gumbel distribution (EVD Type I) and the Fréchet distribution (EVD Type II).

The distribution was originally designed by the Swedish mathematician Waloddi Weibull to model material breaking strength; he recognized the distribution’s potential in his 1951 paper A Statistical Distribution Function of Wide Applicability. The Weibull distribution is applied to a wide range of data from disciplines such as biology, economics, engineering sciences, and hydrology .

The Weibull isn’t an appropriate model for every situation. For example, chemical reactions and corrosion failures are usually modeled with the log-normal distribution.

## Probability density function (PDF)

The Probability Density Function (PDF) PDF for the three parameter Weibull distribution is

• γ = shape parameter (also called the Weibull slope or the threshold parameter).
• α = scale parameter (also called the characteristic life parameter).
• μ =location parameter (also called the waiting time parameter or shift parameter).

Note: Different notation exists for the Weibull distribution PDF. For example, you might see β, m, or k for the shape parameter; c, ν, η , or γ as the scale parameter.

When μ = 0 and α = 1, the formula reduces to:

The two parameter Weibull omits μ: Two parameter Weibull is often used in failure analysis, where no failure can happen before time = 0

## 2. 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 — variables that have individual and distinct data points (as opposed to continuous). 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.

Three types of discrete Weibull distribution are well studied — Type I, Type II and Type III, but none are exact analogs of the continuous Weibull distribution.

## Type I Discrete Weibull distribution PMF of the Type I discrete Weibull distribution with p = 0.5 and β = 1. Graphed with Desmos.com.

The type-I discrete Weibull distribution, introduced by Nakagawa and Osaki in 1975 , retains the cumulative distribution function (CDF) of the continuous Weibull distribution.

The probability mass function (PMF) depends on whether support for zero (0) is included or not :

Similarly, two versions of the CDF exist depending on whether or not zero is included in the support:

Note that many alternate parameterizations have been proposed for the discrete Weibull distribution. For example, Nakagawa and Osaki specified the PMF as q = e, which means that the CDF is 1 – q(x + 1)β. In infectious disease modeling, Endo et al.  proposed the following PMF 

Endo et al’s PMF contains an additional parameter κ = β/αβ.

## Types II and III

The type-II discrete Weibull distribution, introduced by Stein and Dattero in 1984 , retains the continuous distribution’s hazard rate. They specified the hazard rate function and PMF as:

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

Type III, introduced by Padgett and Spurrier , 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  and exponentiated discrete Weibull .

## References

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

 Barbiero, A. (2022). Package ‘DiscreteWeibull. Retrieved April 23, 2023 from: https://cran.r-project.org/web/packages/DiscreteWeibull/DiscreteWeibull.pdf

 Endo A, Murayama H, Abbott S, et al. (2022). “Heavy-tailed sexual contact networks and monkeypox epidemiology in the global outbreak, 2022”Science378 (6615): 90–94. doi:10.1126/science.add4507PMID 36137054.

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

 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

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

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

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

## 3. Weibull-Gnedenko

Another name for the Weibull distribution. Soviet mathematician Boris Vladimirovich Gnedenko wrote about the same distribution at about the same time as Weibull, proving the existence of several classes of limiting distributions for extreme ordered statistics. Therefore, both names are associated with the same distribution .

## 4. Weibull-Rician

The Weibull-Rician distribution can is a mixture distribution that may be a better model for fast fading components . Its density functions are derived by a conditional probability .

## References

PDF of Weibull graph: No machine-readable author provided. Anarkman~commonswiki assumed (based on copyright claims)., CC BY-SA 3.0 http://creativecommons.org/licenses/by-sa/3.0/, via Wikimedia Commons

 Rinne, H., (2008). The Weibull Distribution: A Handbook. CRC Press.

 Pecht, M. (1995). Product Reliability, Maintainability, and Supportability Handbook. CRC Press.

 Meng, Y. (2009). VHF and UHF wireless channel measurement and modeling for foliage environment. Doctoral thesis, Nanyang Technological University, Singapore. Retrieved August 6, 2022 from: https://dr.ntu.edu.sg/bitstream/10356/20613/3/MengYusong2009.pdf

 W. A. Skiliman, “Comments on “On the derivation and numerical
evaluation of the Weibull-Rician distribution”,” IEEE Trans. Aerosp.
Electron. Syst., vol. AES-21, no. 3, pp. 427–429, May 1985

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