Does the Poisson distribution arise from something deeper?

Question

Is the Poisson distribution a mere approximation to the binomial distribution or a result of something more fundamental?

Answer 1

Im not sure what you mean but "something more fundamental", but the poisson distribution is related with two important concepts: exponencial distribución and poisson process. I really like this reference of the latter:

https://youtu.be/3z-M6sbGIZ0

Answer 2

Describing Radioactive Decay

Suppose we have an extremely large number of atoms, $n$, each atom having a very small chance of radioactive decay in a second. Suppose that the large number and small individual probability end up giving a measurable expected number of decay events, $v$, in a second.

For all intents and purposes, the radioactive decay of one atom is independent of the decay of the other atoms. The Linearity of Expectation then says that the individual probability of decay in a second is $\frac vn$. The Binomial Theorem says the probability that $k$ decay events are observed in a second is $$ \binom{n}{k}\left(\frac vn\right)^k\left(1-\frac vn\right)^{n-k} $$ Since $$ \lim_{n\to\infty}\binom{n}{k}\left(\frac vn\right)^k\left(1-\frac vn\right)^{n-k}=\frac{v^k}{k!}e^{-v} $$ for large $n$, we get the Poisson Distribution.

This is not an artificial argument, cooked up to give the Poisson Distribution; this is the description of a physical process, which, due to the large numbers involved, follows the Poisson Distribution.

It is, however, the limit of Binomial Distributions.