ELEMENTARY SAMPLING THEORY FOR ENGINEERING 47 



w{X) without being able or willing to assign any single value to p. 

 In such cases we may, however, proceed to make assumptions con- 

 cerning the probability that p has a given value. Let 



s{X,p) =f(P)(^^)pHl -pr-""; 



then 



w{x) = C s{x,p)dp = (^^^rf(p)pHi - pr-'^dp, 



where 



Cf{p)dp = 1 and £ w{X) = 1. 



Suppose we assume J{p) constant for all values between and 1; 

 we have 



wiX) = (^^^ J p^{l - p)^-^dp 



N -\- \' 



which we note to be a constant which assigns to all of the iV + 1 

 possible a priori hypotheses concerning X an equal weight. This 

 pair of assumptions in Case II amounts, therefore, to the same thing 

 analytically as the assumption of Case I. 



Any number of possible hypotheses concerning /(/>) might be 

 made. Some of these would complicate the analysis considerably, 

 others might be carried through fairly simply. One of these hypothe- 

 ses might fit one class of physical problems, another some other class. 

 To consider these all in detail in this paper would be outside of the 

 scope of a general treatment. The methods outlined here would, 

 however, hold for such extensions. Such difficulties as might be 

 encountered would be of an analytical rather than a logical nature. 



In closing, the author wishes to express his appreciation to his 

 numerous friends and associates in the Bell System, whose suggestions 

 and cooperation have been of material assistance in the preparation of 

 this work, and particularly the work of Miss Nelliemae Z. Pearson of 

 the Department of Development and Research, under whose direction 

 most of the computations were carried out and who has checked 

 through the various proofs. 



