.. _variables-beta_uncertain:

""""""""""""""
beta_uncertain
""""""""""""""


Aleatory uncertain variable - beta




**Topics**


continuous_variables, aleatory_uncertain_variables



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

   variables-beta_uncertain-alphas
   variables-beta_uncertain-betas
   variables-beta_uncertain-lower_bounds
   variables-beta_uncertain-upper_bounds
   variables-beta_uncertain-initial_point
   variables-beta_uncertain-descriptors


**Specification**

- *Alias:* None

- *Arguments:* INTEGER

- *Default:* no beta uncertain variables


**Child Keywords:**

+-------------------------+--------------------+--------------------+-----------------------------------------------+
| Required/Optional       | Description of     | Dakota Keyword     | Dakota Keyword Description                    |
|                         | Group              |                    |                                               |
+=========================+====================+====================+===============================================+
| Required                                     | `alphas`__         | First parameter of the beta distribution      |
+----------------------------------------------+--------------------+-----------------------------------------------+
| Required                                     | `betas`__          | Second parameter of the beta distribution     |
+----------------------------------------------+--------------------+-----------------------------------------------+
| Required                                     | `lower_bounds`__   | Specify minimum values                        |
+----------------------------------------------+--------------------+-----------------------------------------------+
| Required                                     | `upper_bounds`__   | Specify maximium values                       |
+----------------------------------------------+--------------------+-----------------------------------------------+
| Optional                                     | `initial_point`__  | Initial values for variables                  |
+----------------------------------------------+--------------------+-----------------------------------------------+
| Optional                                     | `descriptors`__    | Labels for the variables                      |
+----------------------------------------------+--------------------+-----------------------------------------------+

.. __: variables-beta_uncertain-alphas.html
__ variables-beta_uncertain-betas.html
__ variables-beta_uncertain-lower_bounds.html
__ variables-beta_uncertain-upper_bounds.html
__ variables-beta_uncertain-initial_point.html
__ variables-beta_uncertain-descriptors.html



**Description**


The number of beta uncertain variables, the alpha and beta parameters, and the distribution upper and lower bounds are required specifications, while the variable descriptors is an optional specification.
The beta distribution can be helpful when the actual distribution of an uncertain variable is unknown, but the user has a good idea of the bounds, the mean, and the standard deviation of the uncertain variable.
The density function for the beta distribution is

.. math:: f(x)= \frac{\Gamma(\alpha+\beta)}{\Gamma(\alpha)\Gamma(\beta)}\frac{(x-L)^{\alpha-1}(U-x)^{\beta-1}}{(U-L)^{\alpha+\beta-1}},

where :math:`\Gamma(\alpha)` is the gamma function and

.. math:: `B(\alpha, \beta) = \frac{\Gamma(\alpha)\Gamma(\beta)}{\Gamma(\alpha+\beta)}` 

is the beta function. To calculate the mean and standard deviation from the alpha, beta, upper bound, and lower bound parameters of the beta distribution, the following expressions may be used.

.. math:: \mu = L+\frac{\alpha}{\alpha+\beta}(U-L)

.. math:: \sigma^2 =\frac{\alpha\beta}{(\alpha+\beta)^2(\alpha+\beta+1)}(U-L)^2

Solving these for :math:`\alpha`  and :math:`\beta`  gives:

.. math:: \alpha = (\mu-L)\frac{(\mu-L)(U-\mu)-\sigma^2}{\sigma^2(U-L)}

.. math:: \beta = (U-\mu)\frac{(\mu-L)(U-\mu)-\sigma^2}{\sigma^2(U-L)}

Note that the uniform distribution is a special case of this distribution for parameters :math:`\alpha = \beta = 1` .




**Theory**


For some methods, including vector and centered parameter studies, an
initial point is needed for the uncertain variables. When not given
explicitly, these variables are initialized to their means.