Deriving Stirling's approximation using the Poisson distribution

elseif has not written about this yet · Abstract Nonsense describes it this way

I discovered a neat way to derive Stirling’s approximation for the factorial in a succinct manner. I came across this neat derivation in Information Theory, Inference and Learning Algorithms by David J.C. MacKay (though I’ve also just noticed that it’s listed on the Wikipedia page too). This is an addendum of sorts to my Factorial Overflow post. Consider an indexed family of Poisson random variables $N_{\lambda }\sim \operatorname{Pn}(\lambda), \lambda \in \mathbb N$ with probability mass functions