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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 ?
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    $\begingroup$ 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. $\endgroup$
    – lulu
    Oct 11, 2020 at 13:01
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    $\begingroup$ Wikipedia has a much more accurate description. You can read about the connection with the binomial there as well. $\endgroup$ Oct 11, 2020 at 13:05
  • $\begingroup$ Here's how to calculate $k!$ precisely: $k!=\prod_{j=1}^kj$ $\endgroup$ Oct 11, 2020 at 13:05
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    $\begingroup$ 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 $\endgroup$
    – Henry
    Oct 11, 2020 at 13:06

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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 $n \to \infty$...

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