Page:Biometrika - Volume 6, Issue 1.djvu/11

Hence to find the area of the curve between any limits we must find

${\frac {n-2}{n-3}}\cdot {\frac {n-4}{n-5}}{\textrm {...}}\;{\textrm {etc.}}\times \int {\textrm {cos}}^{n-2}\theta d\theta$
$={\frac {n-2}{n-3}}\cdot {\frac {n-4}{n-5}}{\textrm {...}}\;{\textrm {etc.}}\left\{{\frac {n-3}{n-2}}\int {\textrm {cos}}^{n-4}\theta d\theta +\left[{\frac {{\textrm {cos}}^{n-3}\theta {\textrm {sin}}\theta }{n-2}}\right]\right\}$⁠
$={\frac {n-4}{n-5}}\cdot {\frac {n-6}{n-7}}{\textrm {...}}\;{\textrm {etc.}}\int {\textrm {cos}}^{n-4}\theta d\theta +{\frac {1}{n-3}}\cdot {\frac {n-4}{n-5}}{\textrm {...}}\;{\textrm {etc.}}[{\textrm {cos}}^{n-3}\theta {\textrm {sin}}\theta ]$,

and by continuing the process the integral may be evaluated.

For example, if we wish to find the area between $0$ and $\theta$ for $n=8$ we have

area	$={\frac {6}{5}}\cdot {\frac {4}{3}}\cdot {\frac {2}{1}}\cdot {\frac {1}{\pi }}\int _{0}^{\theta }{\textrm {cos}}^{6}\theta d\theta$
	$={\frac {4}{3}}\cdot {\frac {2}{\pi }}\cdot \int _{0}^{\theta }{\textrm {cos}}^{4}\theta d\theta +{\frac {1}{5}}\cdot {\frac {4}{3}}\cdot {\frac {2}{\pi }}{\textrm {cos}}^{5}\theta \,{\textrm {sin}}\theta$
	$={\frac {\theta }{\pi }}+{\frac {1}{\pi }}{\textrm {cos}}\theta \,{\textrm {sin}}\theta +{\frac {1}{3}}\cdot {\frac {2}{\pi }}{\textrm {cos}}^{3}\theta \,{\textrm {sin}}\theta +{\frac {1}{5}}\cdot {\frac {4}{3}}\cdot {\frac {2}{\pi }}{\textrm {cos}}^{5}\theta \,{\textrm {sin}}\theta$,

and it will be noticed that for $n=10$ we shall merely have to add to this same expression the term ${\frac {1}{7}}\cdot {\frac {6}{5}}\cdot {\frac {4}{3}}\cdot {\frac {2}{\pi }}{\textrm {cos}}^{7}\theta \,{\textrm {sin}}\theta$.

The tables at the end of the paper give the area between $-\infty$ and $z$

(or $\theta =-{\frac {\pi }{2}}$ and $\theta ={\textrm {tan}}^{-1}z$).

This is the same as $.5+$the area between $\theta =0$, and $\theta ={\textrm {tan}}^{-1}z$, and as the whole area of the curve is equal to $1$, the tables give the probability that the mean of the sample does not differ by more than $z$ times the standard deviation of the sample from the mean of the population.

The whole area of the curve is equal to

${\frac {n-2}{n-3}}\cdot {\frac {n-4}{n-5}}{\textrm {...}}\;{\textrm {etc.}}\times \int _{-{\frac {\pi }{2}}}^{+{\frac {\pi }{2}}}{\textrm {cos}}^{n-2}\theta d\theta$,

and since all the parts between the limits vanish at both limits this reduces to $1$.

Similarly the second moment coefficient is equal to

${\frac {n-2}{n-3}}\cdot {\frac {n-4}{n-5}}{\textrm {...}}\;{\textrm {etc.}}\times \int _{-{\frac {\pi }{2}}}^{+{\frac {\pi }{2}}}{\textrm {cos}}^{n-2}\theta {\textrm {tan}}^{2}\theta d\theta$
$={\frac {n-2}{n-3}}\cdot {\frac {n-4}{n-5}}{\textrm {...}}\;{\textrm {etc.}}\times \int _{-{\frac {\pi }{2}}}^{+{\frac {\pi }{2}}}({\textrm {cos}}^{n-4}\theta -{\textrm {cos}}^{n-2}\theta )d\theta$
$={\frac {n-2}{n-3}}-1={\frac {1}{n-3}}$.⁠