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 introduced 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

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: http://mathworld.wolfram.com/WeibullDistribution.html

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

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