P-Distribution

Weibull Distribution

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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 Weibull distribution PDF Can take on many shapes.

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 [1].

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.

PDF

Weibull distribution three parameter PDF
  • 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 When μ = 0 and α = 1, the formula reduces to:

The two parameter Weibull omits μ:

two parameter Weibull distribution PDF
Two parameter Weibull is often used in failure analysis, where no failure can happen before time = 0

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 [2].

Weibull-Rician

The Weibull-Rician distribution can is a mixture distribution that may be a better model for fast fading components [3]. Its density functions are derived by a conditional probability [4].

Fréchet distribution

The Fréchet distribution, also called the inverse Weibull distribution or extreme value distribution(EVD) Type II, can be used to model maximum values in a data set for a wide range of phenomena such as human lifespans, maximum rainfalls and river discharges.

The PDF is:

PDF for the Fréchet distribution

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

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

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

[3] 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

[4] 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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