weibull_uncertain

Aleatory uncertain variable - Weibull

Topics

continuous_variables, aleatory_uncertain_variables

Specification

  • Alias: None

  • Arguments: INTEGER

  • Default: no weibull uncertain variables

Child Keywords:

Required/Optional

Description of Group

Dakota Keyword

Dakota Keyword Description

Required

alphas

First parameter of the Weibull distribution

Required

betas

Second parameter of the Weibull distribution

Optional

initial_point

Initial values for variables

Optional

descriptors

Labels for the variables

Description

The Weibull distribution is also referred to as the Type III Smallest Extreme Value distribution. The Weibull distribution is commonly used in reliability studies to predict the lifetime of a device. It is also used to model capacity variables such as material strength.

The density function for the Weibull distribution is given by:

\[f(x) = \frac{\alpha}{\beta} \left(\frac{x}{\beta}\right)^{\alpha-1} \exp \left( -\left(\frac{x}{\beta}\right)^{\alpha} \right),\]

where \(\mu = \beta \Gamma\left( 1+\frac{1}{\alpha} \right),\) and \(\sigma = \mu \sqrt{\frac{\Gamma(1+\frac{2}{\alpha})}{\Gamma^2(1+\frac{1}{\alpha})} - 1}\)