Editing Stirling's approximation (section)

=== Using the Central Limit Theorem and the Poisson distribution ===
An alternative version uses the fact that the [[Poisson distribution]] converges to a [[normal distribution]] by the [[Central limit theorem|Central Limit Theorem]].<ref>{{Cite book |last=MacKay |first=David J. C. |title=Information theory, inference, and learning algorithms |date=2019 |publisher=Cambridge University Press |isbn=978-0-521-64298-9 |edition=22nd printing |location=Cambridge}}</ref>

Since the Poisson distribution with parameter <math>\lambda</math> converges to a normal distribution with mean <math>\lambda</math> and variance <math>\lambda</math>, their [[Probability density function|density functions]] will be approximately the same:

<math>\frac{\exp(-\mu)\mu^x}{x!}\approx \frac{1}{\sqrt{2\pi\mu}}\exp(-\frac{1}{2}(\frac{x-\mu}{\sqrt{\mu}}))</math>

Evaluating this expression at the mean, at which the approximation is particularly accurate, simplifies this expression to:

<math>\frac{\exp(-\mu)\mu^\mu}{\mu!}\approx \frac{1}{\sqrt{2\pi\mu}}</math>

Taking logs then results in:

<math>-\mu+\mu\ln\mu-\ln\mu!\approx -\frac{1}{2}\ln 2\pi\mu</math>

which can easily be rearranged to give:

<math>\ln\mu!\approx  \mu\ln\mu - \mu + \frac{1}{2}\ln 2\pi\mu</math>

Evaluating at <math>\mu=n</math> gives the usual, more precise form of Stirling's approximation.