.. _variables-negative_binomial_uncertain:

"""""""""""""""""""""""""""
negative_binomial_uncertain
"""""""""""""""""""""""""""


Aleatory uncertain discrete variable - negative binomial




**Topics**


discrete_variables, aleatory_uncertain_variables



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

   variables-negative_binomial_uncertain-probability_per_trial
   variables-negative_binomial_uncertain-num_trials
   variables-negative_binomial_uncertain-initial_point
   variables-negative_binomial_uncertain-descriptors


**Specification**

- *Alias:* None

- *Arguments:* INTEGER

- *Default:* no negative binomial uncertain variables


**Child Keywords:**

+-------------------------+--------------------+---------------------------+-----------------------------------------------+
| Required/Optional       | Description of     | Dakota Keyword            | Dakota Keyword Description                    |
|                         | Group              |                           |                                               |
+=========================+====================+===========================+===============================================+
| Required                                     | `probability_per_trial`__ | A negative binomial distribution parameter    |
+----------------------------------------------+---------------------------+-----------------------------------------------+
| Required                                     | `num_trials`__            | A negative binomial distribution parameter    |
+----------------------------------------------+---------------------------+-----------------------------------------------+
| Optional                                     | `initial_point`__         | Initial values for variables                  |
+----------------------------------------------+---------------------------+-----------------------------------------------+
| Optional                                     | `descriptors`__           | Labels for the variables                      |
+----------------------------------------------+---------------------------+-----------------------------------------------+

.. __: variables-negative_binomial_uncertain-probability_per_trial.html
__ variables-negative_binomial_uncertain-num_trials.html
__ variables-negative_binomial_uncertain-initial_point.html
__ variables-negative_binomial_uncertain-descriptors.html



**Description**


The density function for the negative binomial distribution is given by:

.. math:: f(x) = \left(\begin{array}{c}{n+x-1}\\{x}\end{array}\right){p^n}{(1-p)^{x}},

where

- :math:`p`  is the probability of success per trial
- :math:`n`  is the number of successful trials
- :math:`x`  is the number of failures




**Theory**


The negative binomial distribution is typically used to predict
the number of failures observed when repeating a test until a
total of :math:`n` successes have occurred, where each test has a
probability :math:`p` of success.



