I recently sketched a quick proof of Stirling’s asymptotic formula for the factorial. But what does this formula really mean?
Above is my favorite explanation. You can’t see the number in the words here, but it’s hiding in the formula for a Gaussian probability distribution. But what about the factorial, and the rest?
My description in words was informal. I’m really talking about a
Poisson distribution. If raindrops land at an average rate this says that after time the probability of having landed is
This is where the factorial comes from, and also the number
Since the average rate of rainfall is at time the expected number of drops that have landed will be Since I said “wait until the expected number of drops that have landed is ”, we want Then the probability of having landed is
Next, what’s the formula for a Gaussian with mean and standard deviation ? Written as a function of it’s
If this matches the Poisson distribution above in the limit of large the two functions must match at the point at least asymptotically, so
And this becomes Stirling’s formula after a tiny bit of algebra!
I learned about this on Twitter: Ilya Razenshtyn showed how to prove Stirling’s formula starting from probability theory this way. But it’s much easier to use his ideas to check that my paragraph in words implies Stirling’s formula, as I’ve just done.