214 Explanation of Deviations from Poissons Law 
If now the distribution of chances is to be represented by the binomial 
{P + Qy"^. Then 
Q = 1 _ ^ = 1 _ -q-(^qlq) 
■ =q + "f (5). 
Since the original qs are the chances of events happening they are always 
positive so that the above expression must be positive and the binomial positive. 
If now we introduce the Poisson condition that q though positive is negligibly 
small (5) becomes in general zero, for o-,^ is usually of the same order as q, and in 
that case Poisson's law holds in spite of the inequality of the original q's. If 
however is appreciably greater than zero ^as in the extreme case 
ig2 = 23=... = gn=0 when ^ = ^^^ = -iJ, 
1 
_ 1 
" s 
the distribution of chances is to be represented by a positive binomial. 
Next we have to consider the effect of disregarding condition (2), namely that 
the chance of an individual falling into it must be the same for each division. 
Let us suppose then that the q's are all different for each division so that nq is 
also different. 
Then writing m for nq and m, 7ii-, nq- for the means of in, m" and 7iq- taken 
over all the divisions. 
We get from (1) v^ = Wi (6), 
from (2) ■ = w + — nq- 
= m + 711 + G,^ -nq- (7), 
.-. /x. = m + o-,„- - nq' * (8). 
As before if (P + Q)"^ is the best fitting binomial, 
0=1 -^' = ^^g' ~ . 
I'l 711 
Hence if o-,„- > ncf, which if there is any appreciable variation in m is probable, 
since as explained above 7iq- is generally negligible, a negative binomial will be 
found to fit better than the exponential. 
Clearly condition (2) is usually not fulfilled in the vital and demographic 
statistics; divisions either of space or time are generally governed by different 
* If we suppose that (/ does not vary with the individual but that nq ( = to) varies with the division, 
the moment-coetHcients of the m distribution being written ,„^<, then the moment-coefficients of the 
resulting distribution of divisions are as follows : 
/X4 = TO + 3m2 + (7 + 6nt) ,,^2 + <^> »iM3 + »,M4 ■ 
