.. _variables-weibull_uncertain:

"""""""""""""""""
weibull_uncertain
"""""""""""""""""


Aleatory uncertain variable - Weibull



**Topics**


continuous_variables, aleatory_uncertain_variables


.. toctree::
   :hidden:
   :maxdepth: 1

   variables-weibull_uncertain-alphas
   variables-weibull_uncertain-betas
   variables-weibull_uncertain-initial_point
   variables-weibull_uncertain-descriptors


**Specification**

- *Alias:* None

- *Arguments:* INTEGER

- *Default:* no weibull uncertain variables


**Child Keywords:**

+-------------------------+--------------------+--------------------+---------------------------------------------+
| Required/Optional       | Description of     | Dakota Keyword     | Dakota Keyword Description                  |
|                         | Group              |                    |                                             |
+=========================+====================+====================+=============================================+
| 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                    |
+----------------------------------------------+--------------------+---------------------------------------------+

.. __: variables-weibull_uncertain-alphas.html
__ variables-weibull_uncertain-betas.html
__ variables-weibull_uncertain-initial_point.html
__ variables-weibull_uncertain-descriptors.html



**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:

.. math::  f(x) = \frac{\alpha}{\beta}
         \left(\frac{x}{\beta}\right)^{\alpha-1}
    \exp \left( -\left(\frac{x}{\beta}\right)^{\alpha} \right),

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