POISSON'S PRO B ABILITY SUMMATION 609 



when a and P are the independent variables. The point determined 

 by their values will, in p:eneral, fall between two of the c curves and 

 the interpretation of P must be known to determine which of the tw^o 

 values of c should be taken. The desired value of c is read from the 

 lower curv^e if P means a probability of P or more, from tlie ui)per 

 curve if P means a probability of not more than P. 



These charts may then be used conveniently in place of unwieldy 

 double-entry tallies to obtain theoretical values needed either for 

 comparison with experimental data or to take the place of experi- 

 mental data. Examples of such uses of the Poisson exponential are 

 discussed in detail by Karl Pearson,^ W. A. Shewhart/ and E. C. 

 Molina.^ The use of these curves in the study of telephone trunking, 

 letting a represent the average number of simultaneous calls from a 

 large group of subscribers, c—\ the number of trunks provided for 

 them, and P the probability that all the trunks will be in use when a 

 subscriber attempts to make a call, is suggested by Mr. Molina's 

 paper. Other possible applications might be found in connection 

 with the control of errors in service, defects in a manufactured article, 

 the stock on hand of staple articles such as ink, shoe-polish, or spark 

 plugs, or the number of copies of reference books in a library serving 

 a large number of people. Still others may be suggested by Table I, 

 which is a summary of the actual data now brought together for the 

 first time for comparison with the theory. 



The comparison of any actual distribution with the corresponding 

 Poisson distribution may easily be made graphically, using these 

 curves as a background. In fact the charts will often be found 

 useful as coordinate paper on which to plot any frequency-distribu- 

 tion, theoretical or observed, provided the values of the variate are 

 inherently limited to the positive integers and zero. 



When the curves are used in this way the corresponding Poisson 

 distribution is represented by the points in which the vertical line for 

 the observed value of a cuts the c curves, or for convenience simply 

 by the vertical line itself. The other distribution may then be plotted 

 with c and P as the independent variables, and the horizontal devia- 

 tions of these points from the vertical line serve as a measure of the 

 discrepancy between the two distributions.^ If the comparison is 

 to be made with an observed frequency-distribution the values used 



" Introduction to "Tables of the Incomplete Gamma Function," London, 1922. 



"* "Some Applications of Statistical Methods to the Analysis of Physical and Engi- 

 neering Data," Bell System Technical Journal, Vol. 3, No. 1, pp. 43-87, January, 1924. 



^ "The Theory of Probabilities Applied to Telephone Trunking Problems," Bell 

 System Technical Journal, Vol. 1, No. 2, pp. 69-81, November, 1922. 



* The distributions might be plotted in other ways, e.g., letting P or c be the de- 

 pendent variable, but the method used here is the simplest. 



