# poisson_uncertain

Aleatory uncertain discrete variable - Poisson

**Topics**

discrete_variables, aleatory_uncertain_variables

**Specification**

*Alias:*None*Arguments:*INTEGER*Default:*no poisson uncertain variables

**Child Keywords:**

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

Required |
The parameter for the Poisson distribution, the expected number of events in the time interval of interest |
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Optional |
Initial values for variables |
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Optional |
Labels for the variables |

**Description**

The Poisson distribution is used to predict the number of discrete events that happen in a single time interval. The random events occur uniformly and independently. The expected number of occurences in a single time interval is \(\lambda\) , which must be a positive real number. For example, if events occur on average 4 times per year and we are interested in the distribution of events over six months, \(\lambda\) would be 2. However, if we were interested in the distribution of events occuring over 5 years, \(\lambda\) would be 20.

The probability mass function for the poisson distribution is given by:

where

\(\lambda\) is the expected number of events occuring in a single time interval

\(x\) is the number of events that occur in this time period

f(x) is the probability that \(x\) events occur in this time period

**Theory**

When used with some methods such as design of experiments and multidimensional parameter studies, distribution bounds are inferred to be [0, \(\mu + 3 \sigma\) ].

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.