THEORY OF STATISTICS. 
within the bracket give us the proportional frequencies of 0, 1, 2, 
etc. failures. In other words, (8) is the limit of the binomial 
(p+ ¢)* when ¢ is very small but ng finite. 
The investigation contained in the preceding paragraphs was 
published in 1837 by Poisson, so that (8) may be termed Poisson’s 
limit to the binomial ; but the result has been reached indepen- 
dently by several writers since Poisson’s time, and we shall give 
one of the methods of proof adopted by modern statisticians, which 
the student may perhaps find easier to follow than that of Poisson 
(see ref. 19, p. 273). 
x 2 
(par =(-g+oy=Q-gp(1+ 2). © 
The first bracket on the right is equal to e=* when ¢ is inde- 
finitely small. Expanding the second bracket, we have 
A/A 
Ag oe E 1) 7 \ 
143.00 00 JSF. 
HET ar (Z x 
The ratio of the (r+ 1)™ to the 7 term is 
2 Rl 
gt + (9a) 
1-9 2 
which reduces to 2 when ¢ is very small. The convergence of 
gq 
the series is seen from the fact that » cannot exceed > and the 
substitution of this value in (9a) reduces it to 
g? 
(1-g\ 
which vanishes with gq. 
Hence the second bracket on the right of (9) may be written 
X23 y 
(1 + A+ Tit Tide 
and (9) is 
; AZ As 
e (T+d4g ++ o (was 
identical with (8). 
268