.. _variables-triangular_uncertain:

""""""""""""""""""""
triangular_uncertain
""""""""""""""""""""


Aleatory uncertain variable - triangular




**Topics**


continuous_variables, aleatory_uncertain_variables



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

   variables-triangular_uncertain-modes
   variables-triangular_uncertain-lower_bounds
   variables-triangular_uncertain-upper_bounds
   variables-triangular_uncertain-initial_point
   variables-triangular_uncertain-descriptors


**Specification**

- *Alias:* None

- *Arguments:* INTEGER

- *Default:* no triangular uncertain variables


**Child Keywords:**

+-------------------------+--------------------+--------------------+---------------------------------------------+
| Required/Optional       | Description of     | Dakota Keyword     | Dakota Keyword Description                  |
|                         | Group              |                    |                                             |
+=========================+====================+====================+=============================================+
| Required                                     | `modes`__          | Distribution parameter                      |
+----------------------------------------------+--------------------+---------------------------------------------+
| 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-triangular_uncertain-modes.html
__ variables-triangular_uncertain-lower_bounds.html
__ variables-triangular_uncertain-upper_bounds.html
__ variables-triangular_uncertain-initial_point.html
__ variables-triangular_uncertain-descriptors.html



**Description**


The triangular distribution is often used when one does not have much
data or information, but does have an estimate of the most likely
value and the lower and upper bounds. The number of triangular uncertain
variables, the modes, and the distribution lower and upper bounds are
required specifications, whilevariable descriptors is an optional
specification.

The density function for the triangular distribution is:

.. math:: f(x) = \frac{2(x-L)}{(U-L)(M-L)}

if :math:`L\leq x \leq M` , and

.. math:: f(x) = \frac{2(U-x)}{(U-L)(U-M)}

if :math:`M\leq x \leq U` , and 0 elsewhere. In these equations, :math:`L`  is the lower bound, :math:`U`  is the upper bound,
and :math:`M`  is the mode of the triangular distribution.