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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 it’s 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