Blurb::
Aleatory uncertain discrete variable - Poisson

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 :math:`\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, :math:`\lambda`  would be 2.  However, if we were interested in the distribution of events occuring over 5 years, :math:`\lambda`  would be 20.

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

.. math:: f(x) = \frac{\lambda^{x} e^{-\lambda}}{x!},

where

- :math:`\lambda`  is the expected number of events occuring in a single time interval
- :math:`x` is the number of events that occur in this time period
- f(x) is the probability that :math:`x` events occur in this time period

Topics::
discrete_variables, aleatory_uncertain_variables

Examples::

Theory::
When used with some methods such as design of experiments and
multidimensional parameter studies, distribution bounds are inferred
to be [0, :math:`\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.

Faq::

See_Also::
