# Why use Poisson distribution

I learned from my textbook that Poisson Distribution was invented to approximate binomial distribution, so two questions arose:

1. How to get Poisson distribution from binomial distribution ? It doesn't make sense for me to do so although I know how to get binomial distribution from Poisson distribution using some approximation technique.
2. Since we still can't calculate $$k!$$ precisely in a reasonable amount of time, what's the point of choosing Poisson distribution over binomial distribution ?
• What textbook said that the purpose of Poisson was to approximate the binomial? While it is true that you can sometimes get a useful approximation in that sense, that's a bizarre description of Poisson.
– lulu
Oct 11, 2020 at 13:01
• Wikipedia has a much more accurate description. You can read about the connection with the binomial there as well. Oct 11, 2020 at 13:05
• Here's how to calculate $k!$ precisely: $k!=\prod_{j=1}^kj$ Oct 11, 2020 at 13:05
• If $X_n \sim \text{Bin}\left(n,\dfrac\lambda n\right)$ so with expectation $\lambda$ then it is true that $X_n$ converges in distribution to $\text{Poisson}(\lambda)$ as $n$ increases Oct 11, 2020 at 13:06

The Poisson distribution can be seen as a limit case of the binomial for $$n$$ large ($$n \to \infty$$) and $$p$$ a small fraction of $$n$$, so e.g. if we have $$n=1000$$ and $$p=\frac{1}{1000}$$ we can approximate $$X$$ with Poisson with rate $$1$$. Wikipedia has more explanation. I won't derive the limit of the binomial here, but it's related to $$(1+\frac{c}{n})^n \to e^c$$ as $$n \to \infty$$...