A1. Reminder of basic formulae

Let us start reminding the well known binomial and Poisson distributions, taken verbatim from Ref. [13], just to introduce the notation used in this note.

Binomial distribution

$X\sim$   Binom$(n,p)$ (hereafter “$\sim$” stands for “follows”); Binom$(n,p)$ stands for binomial with parameters $n$ and $p$:

$$f(x\,\vert\,n,p) = \frac{n!}{(n-x)!\,x!}\cdot p^x\cdot (1-p)^{n-x... ...\le p \le 1 \\ x = 0, 1, \ldots, n \end{array}\right.\,. %\label{eq:binomial}$$

Expected value, standard deviation and variation coefficient $[$  $v\equiv\sigma(X)/$E$(X)$ $]$:

$$(X) = n\cdot p$$

$$\sigma(X) = \sqrt{n\cdot p\cdot (1-p)}$$

$$v = \frac{\sqrt{n\cdot p\cdot (1-p)}}{n\cdot p} \propto \frac{1}{\sqrt{n}}\, .$$

Poisson distribution

$X\sim$   Poisson$(\lambda)$:

$$f(x\,\vert\,\lambda)=\frac{\lambda^x}{x!}\cdot e^{-\lambda} \hspa... ... x = 0, 1, \ldots, \infty\\ \end{array} \right.\,. % \label{eq:poisson_distr}$$

Expected value, standard deviation and variation coefficient

$$(X) = \lambda$$

$$\sigma(X) = \sqrt{\lambda}$$

$$v = {1}/{\sqrt{\lambda}}.$$

Binomial $\rightarrow$ Poisson

Binom$$(n,p) \xrightarrow [\begin{array}{l}n\rightarrow \infty \\ p\rightarrow 0 \\ (n\cdot p = \lambda)\end{array}]{} {\mbox{Poisson}}(\lambda) \,.$$