I recently outlined a quick test of Stirling’s asymptotic formula for factorial. But what does this formula really mean?

Above is my favorite explanation. Unable to see numbers i in the words here, but they are hidden in the formula of a Gaussian probability distribution.

My description in words was informal. I’m really talking about one

Poisson distribution. If the raindrops land at a medium rate this says that after time the probability of to have landed is

Hence the factorial.

At the time the expected number of falls they have landed is clear Since I said “wait until the expected number of drops they have landed is “, we want Then the probability of to have landed is

Here is the formula for an average Gaussian and standard deviation ? Written based on is

If it matches the Poisson distribution above the grain boundary the two functions must match when at least asymptotically, therefore

And that becomes Stirling’s formula after a bit of algebra!

I found out about this on Twitter: Ilya Razenshtyn showed how to do it *demonstrate* Stirling’s formula from this theory of probability. But it’s much easier to use your ideas to check that my paragraph in words involves Stirling’s formula, as I just did.

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