Stirling’s Approximation
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
where we have used the property of logarithms that log(abc) =
log(a) + log(b) + log(c). The sum is shown in figure below.
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.
To solve the integral use integration by parts
Here we let u = ln x and dv = dx. Then v = x and du = dx/x.
Notice that x/x = 1 in the last integral and x ln x is 0 when
evaluated at zero, so we have
ln N!= lnm
Σ
m
=1
N
ln N!= lnm
Σ
m=1
N
≈ln xd
x
1
N
udv=uv –vdu
ln xdx
0
N=xln x0
N–xdx
x
0
N
