212 Explanation q/ Deviations from J*oissons Law 
As to these three coDclitions (1) is seldom or never true. I propose to show 
that this is generally unimportant ; unless the chances of some individuals falling 
in a particular division are relatively high the Poisson law holds; the tendency 
however is towards a positive binomial. 
Next (2) is comparatively seldom true except in the case of artificial divisions. 
The result of this, as Pearson has shown, is that a negative binomial fits the 
results better than the exponential. 
Lastly (3) is often untrue. It will be shown that if the presence of an individual 
makes another less likely to fall into a division the positive binomial, but if more 
likely, the negative binomial will fit the figures best. 
We may start from the fact that if the chance of an event happening be q and 
of its not happening p, then the chances of its happening 0, 1, 2, etc. times in 
n trials are given by the terms of the expansion of {'p + q)''\ viz. 
pji, . iipn-i g . _J^^) pn-2 ^2 . g^g_ 
As the moment coefficients of this series about the zero end of the range are 
Vj = vq, 
2'o = vpq + n^q^ whence fi., = npq, 
the binomial is completely determined if we know 7'i and /z., for 
« = --o'=l — «=1 — — and n = - = , 
and in particular the binomial is positive (i.e. n and q are positive) if — < 1 and 
negative if — " > 1. In the particular case when — = 1 the binomial becomes the 
Poisson exponential. 
It is therefore unnecessary to deal with higher moments than the second for 
the purpose in hand. 
Let us first consider the result of each individual having a different chance of 
falling in a given division : — 
Let the chances of n individuals falling in a given division be qi, q.,, q-i-- - qn- 
The chances of their not doing so are therefore (1 — q^), (1 — q.^, (1 - q-.) ... (1 — q.,i), 
and the chances that 0, 1, 2 ... n of them will fall in that division are given by the 
various terms of the expansion of 
{(1 - ?0 + 'h] 1(1 - q.) + 7.1 {(1 - + qM ) Ri - g,,) + qn}, 
i.e. by 
( 1 - 50 (1 - g.) • ■ • (1 - qn) + ^' ( 1 -?.)(•••)( 1 - qn)] 
+ S {q^q, (1 - q,) ... (1 - q,,)] + ...+S [q.q.q, ... r/^ (1 - g^+O ... (1 - q^)] + ... 
+ qiqoq.,... qn, 
the term *S {(^i^a^s — 5'r (1 ~ ^r+i) •■• (1 ~ giving the chance that exactly r 
individuals will fall in the division. 
