98 BELL SYSTEM TECHNICAL JOURNAL 



n is an integer which cannot be less than the average a or the number 

 of occurrences c. 



Fig. 3, making use of Fig. 2 as a background, shows for c = l, 10, 

 50, 100, 150 and 200, the curves of the point binomial with n= oo and 

 n = c, as heavy full and dashed lines, respectively. Each pair of 

 curves, with the exception of the first, crosses in the neighborhood of 

 P = %, and, except near this crossing, all of the intermediate curves of 

 each family of c curves lie between these extreme curves. The rela- 

 tive change in the probability P or 1— P, when these probabilities 

 are small, due to reducing n to this lower limit c, for the c curve, is 

 great, but the relative increase in the average a is only moderate over 

 the greater part of the range covered by Fig. 3. The extreme rela- 

 tive change in the average a is shown by dots placed on each of the 

 Poisson exponential curves, each dot being located at the point where 

 the extreme relative increase in the average is ±.25, ±.50, or ±.75. 

 The relative increment in the average ranges, for Fig. 3, from a de- 

 crease of 93 per cent at P = .999999 on c = 1 to an increase of 97 per 

 cent at P=. 000001 on c = 9 and 10, but the greater part of the field 

 is included between the beaded curves for =•= 50 per cent. Having 

 thus obtained, by examining Fig. 3, a general idea of the relative and 

 absolute numerical magnitudes of the extreme changes to which the 

 probability curves are subject, we are in a better position to make 

 practical use of the curves of Figs. 4 and 5 for the small initial shift 

 in the curves occurring when the number of trials is finite but still 

 large compared with c. 



The rate at which the probability curves start to shift, when the 

 number of trials is decreased from infinity, is shown by Fig. 4, which 

 gives the value of the first coefficient A in the expansion, in descending 

 powers of n, for the relative increment in the average. In the upper 

 part of the curves the shift is to the right and in the lower part of the 

 curves it is to the left. The point at which the curve remains in- 

 itially at rest is shown by the intersection of the c curve with the 

 curve for .4=0. Since A =35 is the largest arithmetical value occur- 

 ring on Fig. 4 and n = 700 will make the first term of the series equal 

 1/20, and the next term is then still smaller, it follows that Fig. 2 

 redrawn for 700 trials would not show a difference of more than about* 

 5 per cent in any value of the average. For Fig. 1 the corresponding 

 number of trials is 220; it may be shown by direct computation that 

 n may even be reduced to the lower limit 1 with only a small 

 percentage change in the abscissas of the upper portion of the 

 curve c = 1 . 



