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## Weibull distribution |

The Weibull distribution has previously been mentioned in connection with the Exponential distribution, as it is essentially a generalization of the Exponential. As we noted there, if the variable Y=(X-θ)c has an Exponential distribution then X has a Weibull distribution. There are various modifications of this transform, with the most common being θ=0 and the use of an additional parameter, α, as a divisor. If α=1 and θ=0 the standard Weibull pdf is then:

The more general form is:

The distribution was originally popularized by Swedish physicist Waloddi Weibull as a model for the breaking strength of materials, but has come into more widespread use as a general model for reliability and quality control applications. In particular, where an Exponential model does not fit the data satisfactorily, a Weibull, with its power transform and extra parameters, may be used as an alternative model. The chart below shows the varied forms of the Weibull for α=1 (sometimes called the scale parameter or characteristic life parameter) and c (the shape parameter) varying.

Weibull distribution, α=1 and c varying

Kizilersü, Kreer and Thomas (2018) note that as the Weibull is an extreme value distribution, it is useful in predicting events such as floods, earthquakes and high winds, and has uses in many other fields, for example in predicting disease survival rates. They use a slightly different, but equivalent, formulation to that provided above. Kizilersü, Kreer and Thomas interpret the shape parameter, c in our formulation, in terms of the hazard function (the conditional density given that the event of interest has not yet occurred) as follows: when c is <1 events are likely to fail at the start; for c=1 the failure rate is fairly constant, and for c>1 the failure rate increases as time goes by.

Key measures for the distribution are provided in the table below:

References

[JOH1] Johnson N L, Kotz S (1970) Continuous Univariate Distributions, I. Houghton Mifflin/J Wiley & Sons, New York

Mathworld: Weisstein E W: Weibull distribution: https://mathworld.wolfram.com/WeibullDistribution.html

NIST/SEMATECH Engineering Statistics Handbook: Weibull: https://www.itl.nist.gov/div898/handbook/apr/section1/apr162.htm

Wikipedia: Weibull distribution: https://en.wikipedia.org/wiki/Weibull_distribution