Applications of Poisson's Probability Summation 



By FRANCES THORNDIKE 



Synopsis: The applicability of Poisson's exponential summation to a 

 variety of actual data is illustrated by thirty-two examples of actual fre- 

 quency-distributions to which the Poisson distribution is a fairly good 

 approximation. The comparison of actual and theoretical distributions is 

 made graphically, using as a background new probability curves showing 

 Poisson's exponential summation with a logarithmic scale for the avera^.e. 

 To suggest possible explanations of the observed deviations from the theo- 

 retical Poisson distribution consideration is given to the effect on the 

 theoretical distribution of certain modifications in the underlying assump- 

 tions, corresponding to conditions under which much actual data must be 

 obtained. 



IN an earlier number of The Bell System Technical Journal 

 there were published two sets of curves showing Poisson's expo- 

 nential summation. 1 These charts, which are shown on a reduced 

 scale in Figs. 1 and 2, give the relation between a, the average number 

 of occurrences of an e\'ent in a large group of trials, the number of 

 trials being very great compared with the average a, and the proba- 

 bility P that the actual number of occurrences in any such group of 

 trials will equal or exceed any given number c. The purpose of this 

 paper is to facilitate the use of these curves by making clear the char- 

 acteristics of the Poisson summation, especially the assumptions on 

 which it is based, and the precautions which must be observed in 

 applying it, these points being illustrated by a number of actual 

 frequency-distributions for which the Poisson distribution furnishes 

 a fairly good working approximation. 



Poisson's Exponential Summation 



Three assumptions underlie the mathematical treatment of Poisson's 

 exponential summation 



and its application to practical problems. The first is that the quan- 

 tity measured is the number of occurrences of a particular event 

 which always definitely happens or fails to happen, so that the actual 

 number of occurrences c is either zero or a positive integer. The 

 second assumption is that we may imagine the group of trials con- 



' F"igs. 1 and 2 of "Probability Curves Showing Poisson's Exponential Summa- 

 tion," by G. A. Campbell, Bell System Technical Journal, Vol. 2, No. 1, pp. 95-113, 

 January, 1923. 



604 



