In the ALT design platform, you can select either a lognormal or Weibull failure distribution. The parameterizations for the probability density function (pdf) and cumulative distribution function (cdf) for each distribution are given in this section. For additional detail on 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.
An alternative parameterization is commonly used in the literature and in JMP. In the JMP parameterization, σ is the scale parameter and μ is the location parameter. These are related to the α and β 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.