634 BELL SYSTEM TECHNICAL JOURNAL 



that the true mean of a sampled normal universe lies 

 within a given range. 

 II. Certain mild restrictions are placed on the general equation of 

 (I) to facilitate its use in practice. 



III. The selection of a priori frequency functions in practice is dis- 



cussed. 



IV. A typical example is selected and solved for various a priori 



existence probability distributions with a discussion of the 

 ranges of errors. 

 V. Conclusions. 



I. The General A Posteriori Equation 



It is common, unless information is known to the contrary, to 

 assume that the universe from which the sample is to be made is 

 composed of an infinite number of items all having a particular 

 characteristic whose numerical value from item to item follows the 

 normal frequency law. In the remainder of this paper we shall 

 limit ourselves to a discussion involving only this type of universe. 

 The problem may now be precisely stated: 



A set of n observations has been made on a variable quantity 

 drawn from a universe wherein the normal law of errors 





is satisfied but the values of the mean and the precision constant, 

 or standard deviation, are unknown; before the observations were 

 made the probability in favor of the simultaneous existence of the 

 inequalities 



m < mean < m + dm (1) 



h < precision constant < h -\- dh (2) 



was some function of m and h, say Wim, h)dmdh; what is the proba- 

 bility that after the observations were made the unknown mean 

 satisfies the inequality (1)? 



Let Xi, X2 ■ • • Xn be the values for x given by the n observations. 

 Set 



n n 



1 1 



Now if m and h were known the probability that a set of n observa- 

 tions, noi yet made, would give values Xi, X2, • • • x„ would be 



( ^ ) e-^-^'^-'^^'dxidx, ■ ■ ' dxn. (3) 



