Poisson limit theorem

In probability theory, the law of rare events or Poisson limit theorem states that the Poisson distribution may be used as an approximation to the binomial distribution, under certain conditions. The theorem was named after Siméon Denis Poisson (1781–1840).

As ${\displaystyle n\rightarrow \infty }$ and ${\displaystyle p\rightarrow 0}$ such that the mean value ${\displaystyle np=\lambda }$ remains constant, we can approximate

Using Stirling's approximation, we can write:

...
Wikipedia