PROBABILITY CURVES 99 



Curves similar to Fig. 4 showing the exact number of trials pro- 

 ducing a given relative or absolute shift in the average would be 

 useful. Still another variation is shown by Fig. 5 where the curves 

 give the first coefficient in the expansion, in descending powers of n, 

 of the ratio of the increments in probability, due to a decrease in 

 n and to unit increase in c. These curves therefore show the initial 

 rate at which any c curve approaches the c+1 curve above it, if the 

 scale of ordinates were made linear; below the curve A =0 the initial 

 shift is downward as indicated by the negative sign for the A's. If 

 sets of curves corresponding to Figs. 1 and 2 were drawn for the 

 number of trials «=400 and 2000, respectively, no curve would be 

 shifted by as much as the original distance between the curves shown, 

 since the maximum values on Fig. 5 up to a = 15 and 200 are 400 and 

 10,000, respectively; Fig. 2 shows only every fifth curve; the second 

 term of the series indicates that the initial maximum rate of shift 

 is not maintained as n decreases at these points. 



The second question arising in connection with the use of the curves 

 is their accuracy. Fig. 1 was drawn with the greatest care on a 

 scale somewhat larger than that of the reproduction, and errors are 

 believed to be only of the order of uncertainty of reading such curves 

 with the unaided eye. Fig. 2 was drawn with less skill and shows 

 larger deviations but it has proved accurate enough for ordinary 

 applications. 2 



The third question which may arise is that of going beyond the 

 curves either in range or in accuracy. 3 The exact calculated values 

 employed in plotting the curves up to c= 101 are contained in Table II, 

 every entry having been independently checked by two persons. 

 The greater part of the table was calculated by means of a new formula 

 which so expresses the average in terms of P and c as to readily give 

 accurate results for the central range of P with large values of c, which 



1 Cf. Soper, H.E., The Numerical Evaluation of the Incomplete B-Function, 1921, 

 p. 41, and Fisher, A., Mathematical Theory of Probabilities, 2nd Edition, 1922, p. 276. 



2 These claims for the accuracy of the curves of Figs. 1 and 2 have been confirmed 

 by comparison with Pearson's Tables of the Incomplete T-Function, 1922, which 

 has been received during the proof-reading of this paper. His tabulated function 

 / (m, p) is, in the notation of the present paper, the probability P corresponding 

 to the average a — u^Jp + l and the number of occurrences c=p-\-l. 



3 When c is not greater than 51, Pearson's tables may be employed. If the prob- 

 ability is assigned, as in many practical engineering problems, finding the corre- 

 sponding average from the tables requires interpolation. Formula (1) of the present 

 paper gives the average directly, that is, it gives the inverse incomplete gamma 

 function. The following formula gives c in terms of a: 



-i 



l-to-' + >(' ! +2)a-' + i(/'+2/)a-H . . 



