Intuition for distribution of terms of exponential power series

Question

One way to express $e^x$ is via a power series:

$$e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!}$$

If we plot the first terms of this series for increasing $x$ we observe a distribution of the terms that resembles a normal distribution.

The factorial grows faster than a power, eventually. But is there an intuition or interpretation of the terms' distribution?

Code for animation: https://gist.github.com/miku/d83be6ec61d05f1fa4ed5b70cf5b59b9

Answer 1

What you are seeing can be recognized as a case of the central limit theorem. The distribution with $P(X=n)=\frac{\frac{x^n}{n!}}{\sum_{n=0}^N \frac{x^n}{n!}}$ converges as $N \to \infty$ to a Poisson(x) distribution. Because of infinite divisibility of the Poisson distribution, the Poisson(x) distribution behaves according to the central limit theorem as $x \to \infty$. By estimating the error you can make the comparison to the normal distribution for finite $N$ and $x$.

You can also make a comparison to a Binomial($N,x/N$), which will also converge to a normal distribution.