# 10.10: The Binomial Distribution and Stirling's Appromixation

Stirling's approximation is named after the Scottish mathematician James Stirling (1692-1770). In confronting statistical problems we often encounter factorials of very large numbers. The factorial N! is a product N(N-1)(N-2)..(2)(1). Therefore, $$\ln \,N!$$ is a sum

$\left.\ln N!\right. = \ln 1 + \ln 2 + \ln 3 + ... + \ln N = \sum_{k=1}^N \ln k. \label{1}$

where we have used the property of logarithms that $$\log(abc) =\ log(a) + \log(b) +\log(c)$$. The sum is shown in figure below. Using Euler-MacLaurin formula one has

$\sum_{k=1}^N \ln k=\int_1^N \ln x\,dx+\sum_{k=1}^p\frac{B_{2k}}{2k(2k-1)}\left(\frac{1}{n^{2k-1}}-1\right)+R , \label{2}$

where B1 = −1/2, B2 = 1/6, B3 = 0, B4 = −1/30, B5 = 0, B6 = 1/42, B7 = 0, B8 = −1/30, ... are the Bernoulli numbers, and R is an error term which is normally small for suitable values of p.

Then, for large N,

$\ln N! \sim \int_1^N \ln x\,dx \approx N \ln N -N . \label{3}$

after some further manipulation one arrives at (apparently Stirling's contribution was the prefactor of $$\sqrt{2\pi})$$

$N! = \sqrt{2 \pi N} \; N^{N} e^{-N} e^{\lambda_N} \label{4}$

where

$\dfrac{1}{12N+1} < \lambda_N < \frac{1}{12N}. \label{5}$

The sum of the area under the blue rectangles shown below up to N is ln N!. As you can see the rectangles begin to closely approximate the red curve as m gets larger. The area under the curve is given the integral of ln x.

$\ln N! = \sum_{m=1}^N \ln m \approx \int_1^N \ln x\, dx \label{6}$

To solve the integral use integration by parts

$\int u\,dv=uv-\int v\,dy \label{7A}$

Here we let $$u = \ln x$$ and $$dv = dx$$. Then $$v = x$$ and $$du = \frac{dx}{x}$$.

$\int_0^N \ln x \, dx = x \ln x|_0^N - \int_0^N x \dfrac{dx}{x} \label{7B}$

Notice that $$x/x = 1$$ in the last integral and $$x \ln x$$ is 0 when evaluated at zero, so we have

$\int_0^N \ln x \, dx = N \ln N - \int_0^N dx \label{8}$

Which gives us Stirling’s approximation: $$\ln N! = N \ln N – N$$. As is clear from the figure above Stirling’s approximation gets better as the number N gets larger (Table $$\PageIndex{1}$$).

Table $$\PageIndex{1}$$: Evaluation of Approximation with absolute values
N N! ln N! N ln N – N Error
10 3.63 x 106 15.1 13.02 13.8%
50 3.04 x 1064 148.4 145.6 1.88%
100 9.33 x 10157 363.7 360.5 0.88%
150 5.71 x 10262 605.0 601.6 0.56%

Calculators often overheat at 200!, which is all right since clearly result are converging. In thermodynamics, we are often dealing very large N (i.e., of the order of Avagadro’s number) and for these values Stirling’s approximation is excellent.

## References

1. J. Stirling "Methodus differentialis, sive tractatus de summation et interpolation serierum infinitarium", London (1730). English translation by J. Holliday "The Differential Method: A Treatise of the Summation and Interpolation of Infinite Series" (1749)