Page:Biometrika - Volume 6, Issue 1.djvu/9

The second moment about the end of the range is

${\frac {I_{n}}{I_{n-2}}}={\frac {(n-1)\sigma ^{2}}{n}}$.

The third moment about the end of the range is equal to

${\frac {I_{n+1}}{I_{n-2}}}$	$={\frac {I_{n+1}}{I_{n-1}}}\cdot {\frac {I_{n-1}}{I_{n-2}}}$
	$=\sigma ^{2}\times$ the mean.

The fourth moment about the end of the range is equal to

${\frac {I_{n+2}}{I_{n-2}}}={\frac {(n-1)(n+1)}{n^{2}}}\sigma ^{4}$.

If we write the distance of the mean from the end of the range ${\frac {D\sigma }{\sqrt {n}}}$ and the moments about the end of the range $v_{1}$, $v_{2}$, etc.

then

$v_{1}={\frac {D\sigma }{\sqrt {n}}}$, $v_{2}={\frac {n-1}{n}}\sigma ^{2}$, $v_{3}={\frac {D\sigma ^{3}}{\sqrt {n}}}$, $v_{4}={\frac {n^{2}-1}{n^{2}}}\sigma ^{4}$.

From this we get the moments about the mean

$\mu _{2}$	$={\frac {\sigma ^{2}}{n}}(n-1-D^{2})$,
$\mu _{3}$	$={\frac {\sigma ^{3}}{n{\sqrt {n}}}}\{nD-3(n-1)D+2D^{3}\}={\frac {\sigma ^{3}D}{n{\sqrt {n}}}}\{2D^{2}-2n+3\}$,
$\mu _{4}$	$={\frac {\sigma ^{2}}{n^{2}}}\{n^{2}-1-4D^{2}n+6(n-1)D^{2}-3D^{4}\}={\frac {\sigma ^{4}}{n^{2}}}\{n^{2}-1-D^{2}(3D^{2}-2n+6)\}$.

It is of interest to find out what these become when $n$ is large.

In order to do this we must find out what is the value of $D$.

Now Wallis’s expression for $\pi$ derived from the infinite product value of sin $x$ is

${\frac {\pi }{2}}(2n+1)={\frac {2^{2}\cdot 4^{2}\cdot 6^{2}...(2n)^{2}}{1^{2}\cdot 3^{2}\cdot 5^{2}...(2n-1)^{2}}}$.

If we assume a quantity $\theta \left(=a_{0}+{\frac {a_{1}}{n}}+{\textrm {etc.}}\right)$ which we may add to the $2n+1$ in order to make the expression approximate more rapidly to the truth, it is easy to show that $\theta =-{\frac {1}{2}}+{\frac {1}{16n}}-$etc. and we get

${\frac {\pi }{2}}\left(2n+{\frac {1}{2}}+{\frac {1}{16n}}\right)={\frac {2^{2}\cdot 4^{2}\cdot 6^{2}...(2n)^{2}}{1^{2}\cdot 3^{2}\cdot 5^{2}...(2n-1)^{2}}}$.^[1]

From this we find that whether $n$ be even or odd $D^{2}$ approximates to $n-{\frac {3}{2}}+{\frac {1}{8n}}$ when $n$ is large.

1. This expression will be found to give a much closer approximation to $\pi$ than Wallis’s.