# Poisson Distribution¶

The Poisson random variable counts the number of successes in $$n$$ independent Bernoulli trials in the limit as $$n\rightarrow\infty$$ and $$p\rightarrow0$$ where the probability of success in each trial is $$p$$ and $$np=\lambda\geq0$$ 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 $$\left[0,t\right]$$ for a process satisfying certain “sparsity” constraints. The functions are:

\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*}
$M\left(t\right)=\exp\left[\lambda\left(e^{t}-1\right)\right].$

Implementation: scipy.stats.poisson

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Geometric Distribution