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