This section contains the parameterizations for the probability density function (pdf) and cumulative distribution function (cdf) for each distribution. For more information about the Weibull distribution, see Life Distribution in the Reliability and Survival Methods book.
The lognormal family is parameterized by a location parameter, μ, and a shape parameter, σ. The lognormal pdf and cdf are given as follows, where the logarithm is to the base e:
,
where α is a scale parameter, and β is a shape parameter. The Weibull distribution reduces to an exponential distribution when β = 1.
In JMP, the Weibull parameterization defines σ as the scale parameter and μ as the location parameter. These are related to the α and β Weibull parameterization as follows:
With these parameters, the pdf and the cdf of the Weibull distribution are expressed as a log-transformed smallest extreme value distribution (SEV) using a location-scale parameterization with μ = log(α) and σ = 1/β:
are the pdf and cdf, respectively, for the standardized smallest extreme value (μ = 0, σ = 1) distribution.