I've been researching on the application and origin of irrational numbers in probability theory and statistical distributions, so far having derived a unique proof of Stirling's approximation, and establishing a graphical link between binomial and normal distributions. I'd like to explore some more distributions with e and pi in their functions, so as to mathematically explain their origin and hopefully understand their importance in the calculation of probability. Any help would be much appreciated!
2026-03-25 16:03:48.1774454628
What are some statistical distributions with the irrational numbers e and pi in their functions? (apart from the most common - Normal, Poisson)
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Some that come to mind:
$e:$ Exponential, Gamma, Beta, hyperbolic, log-normal, weibull, Laplace
$\sqrt{\pi}:$ Chi-square with 1df, t distribution, log-normal
You may find this list useful:
https://en.wikipedia.org/wiki/List_of_probability_distributions
For distributions defined using the Gamma function, you may be interested to know $\Gamma(n+0.5) \propto \sqrt{\pi}$ for any natural number $n$. So such distributions can depend on $\pi$ for certain parameters.