SUPPLEMENTS—THE LAW OF SMALL CHANCES. 729 
The frequent rediscovery of this theorem is due to the fact that 
its value is felt in the study of problems involving small, inde- 
pendent probabilities. For instance, if we desired to find the 
distribution of = things in IV pigeon-holes (all the pigeon-holes 
being of equal size and equally accessible), I being large, the dis- 
tribution given by the binomial 
1, ¥-1} 
7+) 
FTF 
would be effectively represented by (8), tables of which for 
different values of A have been published by v. Bortkewitsch and 
others. 
The theorem has also been applied to cases in which, although 
the actual value of ¢ (or p) is unknown, it may safely be assumed 
to be very small. It should be noticed that, if (8) is the real law 
of distribution, certain relations must obtain between the con- 
stants of the statistics (see par. 12, Chapter XIIL). Using the 
method of par. 6, Chapter XV., we have for the mean 
23 
eM A +A+ 2 pF ed 
2! 
2 : 
=re A(1+A+ 2+ eta ud 
21 
=A 
and for o? 
3 
eA(A+200 +30 ‘on ) - AZ 
. AZ A A 222 3A g 
=e SYRER EL. LY pe (A +2020 ceee)=A 
2 
=A+e-M(1 Xe 2 piri x 
2! 
=A+A2-A2 
= A. 
Hence any statistics produced by causes conforming to Poisson’s 
limit should, within the limits of sampling, have the mean equal 
to the square of the standard deviation. For instance, in the 
statistics used in par. 12 of Chapter XIII, the mean is ‘61, 
a="78, ¢2="6079, 
2% 
24