doc/source/tutorial/stats/discrete_poisson.rst
.. _discrete-poisson:
The Poisson random variable counts the number of successes in :math:n independent Bernoulli trials in the limit as :math:n\rightarrow\infty and :math:p\rightarrow0 where the probability of success in each trial is :math:p and :math:np=\lambda\geq0 is a constant. It can be used to approximate the Binomial random
variable or in its own right to count the number of events that occur
in the interval :math:\left[0,t\right] for a process satisfying certain "sparsity" constraints. The functions are:
.. math:: :nowrap:
\begin{eqnarray*} p\left(k;\lambda\right) & = & e^{-\lambda}\frac{\lambda^{k}}{k!}\quad k\geq0,\\ F\left(x;\lambda\right) & = & \sum_{n=0}^{\left\lfloor x\right\rfloor }e^{-\lambda}\frac{\lambda^{n}}{n!}=\frac{1}{\Gamma\left(\left\lfloor x\right\rfloor +1\right)}\int_{\lambda}^{\infty}t^{\left\lfloor x\right\rfloor }e^{-t}dt,\\ \mu & = & \lambda\\ \mu_{2} & = & \lambda\\ \gamma_{1} & = & \frac{1}{\sqrt{\lambda}}\\ \gamma_{2} & = & \frac{1}{\lambda}.\end{eqnarray*}
.. math::
M\left(t\right)=\exp\left[\lambda\left(e^{t}-1\right)\right].
Implementation: scipy.stats.poisson