PROBABIU I Y CURVES 101 



is the domain in which the ordinary formulas are not convenient for 

 calculation. This is formula (1) below which involved transforming 

 the normal probability integral to fit the skew probability summation 

 of Poisson's exponential binomial limit. The reason for thinking 

 that this transformation would prove useful is made clear by noting 

 that in Figs. 1 and 2 the curves become more and more uniformly 

 spaced with increasing values of the average a and thus the prob- 

 ability approximates more and more closely to the normal probability 

 integral, since this is the scale employed for the ordinates. The results 

 of the mathematical work are summed up in the following formula: 



For Poisson's exponential binomial limit the average a is expressed 

 as a function of the probability P of at least c occurrences by the infinite 

 series 



X 



a = c^Q n c-*, (1) 



n = o 



where the coefficients Q n are functions of the argument t corresponding to 

 the probability P expressed in the form of the normal probability integral, 



P -J5i£r**; (2) 



twelve of these coefficients are given in the following table: 



Table I Coefficients in Formula (1) for the Average 



n Qn 



1 



1 / 



2 (/ 2 -l)/3 



3 (/ 3 -7/)/2 2 3 2 



4 (-3t 4 -7t 2 +l6)/2 l 3 4 r> 



5 (9/ 5 + 256/ 3 -433/)/2 5 3 5 5 



6 (12* 6 - 243/ 4 - 923/ 2 + 1 ,472)/2 3 3 6 5 1 7 



7 (-3,753/ 7 -4,353/ 5 + 289,517^ + 289,7170/2 7 3 8 o 2 7 



8 (270/ 8 +4,614/ 6 -9,513/ 4 - 104,989/ 2 + 35,968)/2 4 3 9 5 2 7 



9 (-5,139/ 9 -547,848/ 7 -2,742,210/ s + 7,()16,224^ + 37,501,3250/ 



211310527 



