PROBABILITY CURVES 107 



cernible on Fig. 1, but Table II gives for c= 1, 2, 3 . . . 100, the differ- 

 ences (c-a) =0.3069, 0.3217, 0.3259, ... 0.3331, which differ but 

 little from 0.3333 . . . , which is approached more and more closely 

 for large values of c. 



At P = 0.5 and large values of c the derivative along the c curve is 

 dP/da = \/\/2Trc, as found by differentiating (3), substituting 

 a = c— 1/3 and Stirling's expression for the gamma function. Thus, 

 for large values of c the slope of the curve at P = 0.5 decreases nu- 

 merically as the square root of c increases. For large values of c the 

 curves are approximately straight over the wide range of proba- 

 bility shown in the figures. This, in connection with the additional 

 fact that the standard deviation \/npq is always equal to\/a for the 

 Poisson exponential, is an alternative way of arriving at the ex- 

 pression for the derivative given above. 



I am indebted to Miss Edith Clarke for extending the series of 

 formula (1) to seven terms, for making all of the original computa- 

 tions and for drawing Fig. 1, and to Miss Sallie E. Pero for extending 

 the formula to its present eleven terms, and for checking all of the 

 preceding work; the single error which she found occurred in the 

 seventh term of the expansion where it was without effect on the final 

 numerical results. Finally, the work was entirely rechecked, with- 

 out discovering additional errors, by Mr. Ronald M. Foster, who also 

 put the mathematical work into its present form, pointed out the 

 asymptotic nature of the expansion and compared the overlapping 

 numerical results with those obtained by direct summation by Miss 

 Lucy Whitaker 5 and more recently by Mr. E. C. Molina, as well as 

 with his earlier table. 6 



5 Tables for Statisticians and Biometricians, 1914, Table LI I. 



6 Computation Formula for the Probability of an Event Happening at Least C 

 Times in N Trials, American Mathematical Monthly, XX, June, 1913, p. 193. 



