Stirling's Formula/Proof 3

Theorem

The factorial function can be approximated by the formula:

$n! \sim \sqrt {2 \pi n} \paren {\dfrac n e}^n$

where $\sim$ denotes asymptotically equal.

Proof

From Poisson Distribution Approximated by Normal Distribution, we have:

Let $X$ be a discrete random variable which has the Poisson distribution $\Poisson \lambda$.

Then for large $\lambda$:

$\Poisson \lambda \approx \Gaussian \lambda \lambda$

where $\Gaussian \lambda \lambda$ denotes the normal distribution.

Since the Poisson Distribution with parameter $\lambda$ converges to the Normal Distribution with mean $\lambda$ and variance $\lambda$, their density functions will be asymptotically equal.

$\dfrac 1 {k!} \lambda^k e^{-\lambda} \approx \dfrac 1 {\sqrt {2 \pi \lambda} } \map \exp {-\dfrac {\paren {k - \lambda}^2} {2 \lambda} }$

Evaluating this expression at the mean simplifies to the expression:

$\dfrac 1 {\lambda!} \lambda^\lambda e^{-\lambda} \approx \dfrac 1 {\sqrt {2 \pi \lambda} }$

Rearranging, we obtain:

$\lambda! \approx \sqrt {2 \pi \lambda} \lambda^\lambda e^{-\lambda}$

$\blacksquare$