# Poisson's approximation to the binomial

I'm assuming that Poisson came up with his approximation to the binomial to ease the burden of computing the binomial coefficients for large n. I'm also assuming that as a student of Laplace, who revived and generalized de Moivre's approximation, he must have been aware of it. So, why not take a perfectly good approximation off the shelf?

• Please document your preliminary research to avoid duplication of effort. Commented Aug 15, 2023 at 4:45
• Stephen M. Stigler, "Poisson on the Poisson Distribution." Statistics & Probability Letters, Vol. 1, No. 1, July 1982, pp. 33-35: " This equation can be seen to agree with $\frac{1}{2}=(1+\omega+\frac{1}{2}\omega^2+\frac{1}{6}\omega^3)e^{-\omega}$, and some [...] have felt the distribution should be attributed to De Moivre. It must be admitted that Poisson added little to De Moivre's mathematical approximation, with which he was quite familiar, although one would have to stretch the point to claim the discrete distribution $e^{-\omega}\omega^{n}/n!$ is found in De Moivre." Commented Aug 15, 2023 at 6:56
• @njuffa: Thank you, this is great. Once again I find I'm not a good at asking questions as I think. I was thinking about de Moirvre's normal approximation. I'd forgotten that he also near enough found a discrete approximation. With that clarification, any ideas about why he didn't use the normal approximation, one his mentor had generalized? Commented Aug 15, 2023 at 15:08
• A noble effort and thank you again. Commented Aug 15, 2023 at 22:27