May 22, 2022

# Stirling formula in words | Azimuth

Related This entry was posted on Friday, October 8th, 2021 at 6:40 pm and is filed under Mathematics, You can follow any responses to this entry through the RSS You can leave a response, or trackback from…

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 $e$ i $2 pi$ 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 $r,$ this says that after time $t$ the probability of $k$ to have landed is

$displaystyle { frac {(rt) ^ ke ^ {- rt}} {k!}}$

Hence the factorial.

At the time $t$ the expected number of falls they have landed is clear $rt.$ Since I said “wait until the expected number of drops they have landed is $n$“, we want $rt = n.$ Then the probability of $k$ to have landed is

$displaystyle { frac {n ^ ke ^ {- n}} {k!}}$

Here is the formula for an average Gaussian $n$ and standard deviation $sqrt {n}$? Written based on $k,$ is

$displaystyle { frac {e ^ {- (kn) ^ 2 / 2n}} { sqrt {2 pi n}}}$

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

$displaystyle { frac {n ^ ne ^ {- n}} {n!} sim frac {1} { sqrt {2 pi n}}}$

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.

This entry was posted on Friday, October 8th, 2021 at 6:40 pm and is filed under Mathematics, Probability. You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback from your own site.

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