binomial_uncertain

Aleatory uncertain discrete variable - binomial

Topics

discrete_variables, aleatory_uncertain_variables

Specification

• Alias: None

• Arguments: INTEGER

• Default: no binomial uncertain variables

Child Keywords:

Required/Optional	Description of Group	Dakota Keyword	Dakota Keyword Description
Required		probability_per_trial	A distribution parameter for the binomial distribution
Required		num_trials	A distribution parameter
Optional		initial_point	Initial values for variables
Optional		descriptors	Labels for the variables

Description

The binomial distribution describes probabilities associated with a series of independent Bernoulli trials. A Bernoulli trial is an event with two mutually exclusive outcomes, such as 0 or 1, yes or no, success or fail. The probability of success remains the same (the trials are independent).

The density function for the binomial distribution is given by:

\[\begin{split}f(x) = \left(\begin{array}{c}n\\x\end{array}\right){p^x}{(1-p)^{(n-x)}},\end{split}\]

where \(p\) is the probability of failure per trial, \(n\) is the number of trials and \(x\) is the number of successes.

Theory

The binomial distribution is typically used to predict the number of failures or defective items in a total of \(n\) independent tests or trials, where each trial has the probability \(p\) of failing or being defective.