608 



BELL SYSTEM TECHNICAL JOURNAL 



parameter, the average a; Its standard deviation is \/a, its skewness 

 is 1 /-y/a, and its kurtosis is 3+1 /a.' 



One consequence of this sinipHcity is that there Is no difificulty In 

 deciding on a definition of the corresponding Poisson distribution with 

 which any other distribution should be compared. It Is naturally 

 the Poisson distribution having the same average as the given dis- 

 tribution. , - 



If) I.U 



0.6 



O.Z 



—^ .^1 ^\ 



11 <i 



n 



<| 



<| 



\ M I I I M i i „ 



O.G 



<D 0.4 



io.z 



05 

 X) 

 o 



• \ 1 — I — I \ 



II 



II 

 il 



!! 



i ; 



il 



il 



I o 



, — 1 — I — 1 — 1 — ; , — s — A — II — I 



Fig. 3- 



2 4 6 6 10 12 14 



(a) 



2 4 6 8 10 12 (4 

 c 



.(b).. 



-Poisson distribution with the average a = 5 shown (a) in the individual term 

 form and (b) in the summation form 



Poisson Probability Curves 



Another advantage Is that It Is possible to represent the whole family 



of Poisson distributions graphically by a chart such as Fig. 1 or Fig. 2, 



In which the value of the average a Is read on the horizontal scale, the 



value of the probability P on the vertical scale, and the number of 



occurrences c on the individual curves of the set. Any two of these 



three variables may then be taken as the Independent variables and 



the values assigned to them will determine the value of the third 



variable, which can be read off at once. The only ambiguity occurs 



* The standard deviation (o-), skewness {k), and kurtosis (/S-i) of any distribution 

 are defined as 



4 



S(.v,-a)2 2(.ri-a)3 



N 



and 



S(:cr 



respectively, N being the number of samples in the series, and .r, the actual number 

 of occurrences in the fth sample. For any point binomial 



ff = yjnpq, k = 



■>inpq 



h = i + 



^-6pq 

 npq 



