UNKNOWN MEAN OF SAMPLED UNIVERSE 637 



consequence of the three assumptions made regarding the form of 

 the a priori existence probabiHty function W{m, h) ; the three assump- 

 tions being embodied in equations (5), (7) and (9). 



III. Practical Selection of A Priori Frequency 

 Distributions 



In equation (10) we have first to assign a numerical value to each 

 of the five constants a, c, N, B, M, before the probability P(m) can 

 be evaluated for any desired range of m. Obviously, in actual practise, 

 the selection of their values is extremely important and too much 

 care cannot be exercised in an attempt to satisfy the engineering 

 judgment that all of the a priori information at hand has been nicely 

 comprehended. 



In an endeavor to reduce the number of constants to which we 

 must assign values we shall consider first the a priori function 



Setting 



h = c/2a (11) 



makes W2(h) a maximum. On the other hand, the value of h which 

 would make the observed set of values of x most probable is given 

 by the equation 



1 ^ 2Z(xi - my 



h n 



or, if m be set equal to x, we obtain the simpler equation 



h = l/2s\ (12) 



Upon eliminating h from (11) and (12),^ 



a = cs\ (13) 



In Fig. 3 are shown four frequency curves of W2{h). Curve / is 

 plotted for c = 3 according to equation (13), and to illustrate the 

 wide possibility of forms, curves // and /// have been constructed, 

 keeping c = 3, after changing equation (13) to 



cs^ , cs- 



and 



" ~ 1 - .W ^ ^ 1 + .W 



respectively. Curve IV again satisfies equation (13) but has c in- 

 creased from 3 to 6. 



* It should be carefully noted that there is no necessary relation between the 

 a priori most probable value of h and the value of h which would make the observed event 

 most probable. The elimination of h between (11) and (12) is justified solely by the 

 practical consideration that a tentative relation between a and c will reduce by 

 one the number of arbitrary constants to which numerical values niust be assigned. 



