638 BELL SYSTEM TECHNICAL JOURNAL 



For every assumption of a and c in the a priori distribution of h 

 there is, of course, a corresponding a priori distribution, 0(o-), of the 

 standard deviation. Here 



and 



^(1/2) c+l 



K' = 



2a/2ur(ic + 1) 



The distributions of a corresponding to each of the four frequency 

 curves of h in Fig. 3 are shown in Fig. 4 with similar designations. 



In many cases, too, it is obvious that very Httle is known concerning 

 the shape and the parameters of the mean's a priori distribution 

 beyond that it is generally unimodal and quite likely to be fairly 

 symmetrical about its most probable value; a mathematical expression 

 of this has been set up in equation (9). In this circumstance we 

 may not introduce serious restrictions if we make two further assump- 

 tions which greatly simplify (9). 



The first is that we set M = x which says that, a priori, the most 

 probable value of the unknown mean was the same as that which was 

 later calculated as the mean in the sample.^ It is admitted that the 

 chance of exactly fixing on M = x from a priori information is very 

 small, yet if our knowledge is so slight that we must introduce some 

 guesswork here, the selection of the value of x at least has the advan- 

 tage of being a possible one which M might assume and, except in 

 rare cases, it will not be greatly distant from the true m in that particu- 

 lar lot. The logical difficulty here also may be minimized by selecting 

 a form of Wi(m) of such flatness that over a considerable range of 

 values in the neighborhood of x the existence probability does not 

 take on widely difl"ering magnitudes.'^ 



The second assumption can more readily be allowed, and consists 

 in empirically defining 



a -f- ns^ 



This removes a degree of freedom from the Wi(m) function but, as 

 far as its form is concerned, except in special cases, the one variable, 

 N, may serve quite well in characterizing the pre-existing information. 

 As is clearly shown in Fig. 1, the increase of N indicates a greater 



* While it does not matter in this particular problem, the authors wish to carefully 

 distinguish, at least in thought, between an "observed" parameter and a parameter 

 calculated from individual observations. 



^ The setting oi M = x, it should be noted, has no effect if all values of the mean 

 are made a priori equally likely by setting N = (that is, Wi{,m) = Ai). 



